Michio Kaku`s Hyperspace

Transcription

HYPERSPACE
A Scientific Odyssey
Through
Parallel Universes,
Time Warps, and
the Tenth Dimension
Michio Kaku
Illustrations by Robert O'Keefe
ANCHOR
BOOKS
D O U B L E D A Y
New York
London
Toronto
Sydney Auckland
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A N C H O R B O O K S , D O U B L E D A Y , a n d t h e portrayal o f a n a n c h o r are
trademarks of Doubleday, a division of Bantam Doubleday Dell
Publishing G r o u p , Inc.
Hyperspace was originally published in hardcover by O x f o r d University Press in 1994.
T h e A n c h o r Books edition is published by arrangement with O x f o r d University Press.
"Cosmic Gall." From Telephone Poles and Other Poems by John Updike. Copyright © 1 9 6 0
by John Updike. Reprinted by permission of Alfred A. Knopf, Inc. Originally
appeared in The New Yorker.
Excerpt from "Fire and Ice." From The Poetry of Robert Frost, edited by Edward
C o n n e r y Lathem. Copyright 1951 by Robert Frost. Copyright 1923, © 1969 by
Henry Holt a n d C o m p a n y , Inc. Reprinted by permission of Henry Holt a n d
C o m p a n y , Inc.
Library of Congress Cataloging-in-Publication Data
Kaku, Michio.
Hyperspace: a scientific odyssey t h r o u g h parallel universes, time
warps, and the tenth d i m e n s i o n / Michio Kaku; illustrations by
Robert O'Keefe.
p. cm.
Includes bibliographical references and index.
1. Physics. 2. Astrophysics. 3. Mathematical physics.
I. Title.
QC21.2.K3
1994
530.1'42—dc20
94-36657
CIP
ISBN 0-385-47705-8
Copyright © 1994 by O x f o r d University Press
All Rights Reserved
Printed in the United States of America
First A n c h o r B o o k s Edition: March 1995
1 0
9
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7
6
5
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3
2
1
This book is dedicated
to my parents
Preface
Scientific revolutions, almost by definition, defy c o m m o n sense.
If all o u r common-sense notions about the universe were correct,
t h e n science would have solved the secrets of the universe thousands of
years ago. T h e purpose of science is to peel back the layer of the appearance of objects to reveal their underlying nature. In fact, if appearance
and essence were the same thing, there would be no n e e d for science.
Perhaps the most deeply e n t r e n c h e d common-sense notion about
o u r world is that it is three dimensional. It goes without saying that
length, width, and b r e a d t h suffice to describe all objects in o u r visible
universe. Experiments with babies and animals have shown that we are
b o r n with an innate sense that our world is three dimensional. If we
include time as a n o t h e r dimension, then four dimensions are sufficient
to record all events in the universe. No matter where o u r instruments
have probed, from d e e p within the atom to the farthest reaches of the
galactic cluster, we have only found evidence of these four dimensions.
To claim otherwise publicly, that o t h e r dimensions might exist or that
our universe may coexist with others, is to invite certain scorn. Yet this
deeply ingrained prejudice a b o u t our world, first speculated on by
ancient Greek philosophers 2 millennia ago, is a b o u t to succumb to the
progress of science.
This book is about a scientific revolution created by the theory of hyperspace, which states that dimensions exist beyond the commonly accepted
four of space and time. T h e r e is a growing acknowledgment a m o n g
physicists worldwide, including several Nobel laureates, that the universe
may actually exist in higher-dimensional space. If this theory is proved
correct, it will create a profound conceptual and philosophical revolution in our u n d e r s t a n d i n g of the universe. Scientifically, the hyperspace
theory goes by the n a m e s of Kaluza-Klein theory a n d supergravity. But
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Preface
its most advanced formulation is called superstring theory, which even
predicts the precise n u m b e r of dimensions: ten. T h e usual three dimensions of space (length, width, and breadth) a n d o n e of time are now
e x t e n d e d by six m o r e spatial dimensions.
We caution that the theory of hyperspace has n o t yet b e e n experimentally confirmed and would, in fact, be exceedingly difficult to prove
in the laboratory. However, the theory has already swept across the major
physics research laboratories of the world a n d has irrevocably altered
the scientific landscape of m o d e r n physics, generating a staggering number of research papers in the scientific literature (over 5,000 by o n e
c o u n t ) . However, almost n o t h i n g has b e e n written for the lay audience
to explain the fascinating properties of higher-dimensional space.
Therefore, the general public is only dimly aware, if at all, of this revolution. In fact, the glib references to o t h e r dimensions a n d parallel universes in the popular culture are often misleading. This is regrettable
because the theory's importance lies in its power to unify all known
physical p h e n o m e n a in an astonishingly simple framework. This book
makes available, for the first time, a scientifically authoritative b u t accessible account of t h e current fascinating research on hyperspace.
To explain why the hyperspace theory has generated so m u c h excitem e n t within the world of theoretical physics, I have developed four fundamental themes that r u n t h r o u g h this book like a thread. These four
themes divide the book into four parts.
In Part I, I develop the early history of hyperspace, emphasizing the
t h e m e that the laws of nature b e c o m e simpler a n d m o r e elegant when
expressed in h i g h e r dimensions.
To u n d e r s t a n d how adding higher dimensions can simplify physical
problems, consider the following example: To the ancient Egyptians,
the weather was a complete mystery. What caused the seasons? Why did
it get warmer as they traveled south? Why did the winds generally blow
in o n e direction? T h e weather was impossible to explain from the limited
vantage point of the ancient Egyptians, to whom the earth a p p e a r e d flat,
like a two-dimensional plane. But now imagine sending the Egyptians in
a rocket into outer space, where they can see the earth as simple and
whole in its orbit a r o u n d the sun. Suddenly, the answers to these questions b e c o m e obvious.
F r o m outer space, it is clear that the earth's axis is tilted a b o u t 23
degrees from the vertical (the 'vertical" being the perpendicular to the
plane of the earth's orbit a r o u n d the sun). Because of this tilt, the northe r n h e m i s p h e r e receives m u c h less sunlight during o n e part of its orbit
than d u r i n g a n o t h e r part. H e n c e we have winter and s u m m e r . A n d since
Preface
ix
the e q u a t o r receives m o r e sunlight t h e n the n o r t h e r n or s o u t h e r n polar
regions, it becomes warmer as we a p p r o a c h the equator. Similarly, since
the earth spins counterclockwise to s o m e o n e sitting on the n o r t h pole,
the cold, polar air swerves as it moves south toward the equator. T h e
m o t i o n of h o t a n d cold masses of air, set in motion by the earth's spin,
thus helps to explain why the winds generally blow in o n e direction,
d e p e n d i n g on where you are on the earth.
In summary, the r a t h e r obscure laws of the weather are easy to understand o n c e we view the earth from space. T h u s the solution to the problem is to go up into space, into t h e third dimension. Facts that were impossible to u n d e r s t a n d in a flat world suddenly b e c o m e obvious when
viewing a three-dimensional earth.
Similarly, the laws of gravity a n d light seem totally dissimilar. They
obey different physical assumptions a n d different mathematics.
Attempts to splice these two forces have always failed. However, if we
a d d o n e m o r e dimension, a fifth dimension, to the previous four dimensions of space a n d time, t h e n the equations governing light a n d gravity
a p p e a r to m e r g e together like two pieces of a jigsaw puzzle. Light, in
fact, can be explained as vibrations in the fifth dimension. In this way,
we see that the laws of light a n d gravity b e c o m e simpler in five dimensions.
Consequently, many physicists are now convinced that a conventional
four-dimensional theory is " t o o small" to describe adequately the forces
that describe o u r universe. In a four-dimensional theory, physicists have
to squeeze together the forces of nature in a clumsy, u n n a t u r a l fashion.
F u r t h e r m o r e , this hybrid theory is incorrect. W h e n expressed in dimensions beyond four, however, we have " e n o u g h r o o m " to explain the
fundamental forces in an elegant, self-contained fashion.
In Part II, we further elaborate on this simple idea, emphasizing that
the hyperspace theory may be able to unify all known laws of n a t u r e into
o n e theory. T h u s the hyperspace theory may be the crowning achievem e n t of 2 millennia of scientific investigation: the unification of all
known physical forces. It may give us the Holy Grail of physics, the " t h e ory of everything" that eluded Einstein for so many decades.
For the past half-century, scientists have b e e n puzzled as to why the
basic forces that hold together the cosmos—gravity, electromagnetism,
a n d the strong a n d weak nuclear forces—differ so greatly. Attempts by
the greatest minds of the twentieth century to provide a unifying picture
of all the known forces have failed. However, the hyperspace theory
allows the possibility of explaining the four forces of n a t u r e as well as
the seemingly r a n d o m collection of subatomic particles in a truly elegant
X
Preface
fashion. In the hyperspace theory, " m a t t e r " can be also viewed as t h e
vibrations that ripple t h r o u g h the fabric of space a n d time. T h u s follows
the fascinating possibility that everything we see a r o u n d us, from the
trees a n d m o u n t a i n s to the stars themselves, are n o t h i n g b u t vibrations
in hyperspace. If this is true, t h e n this gives us an elegant, simple, a n d
geometric m e a n s of providing a c o h e r e n t and compelling description
of the entire universe.
In Part III, we explore the possibility that, u n d e r extreme circumstances, space may be stretched until it rips or tears. In o t h e r words,
hyperspace may provide a m e a n s to t u n n e l t h r o u g h space a n d time.
Although we stress that this is still highly speculative, physicists are seriously analyzing the properties of " w o r m h o l e s , " of tunnels that link distant parts of space a n d time. Physicists at the California Institute of Technology, for example, have seriously p r o p o s e d the possibility of building
a time m a c h i n e , consisting of a wormhole that connects the past with
the future. Time machines have now left the realm of speculation a n d
fantasy a n d have b e c o m e legitimate fields of scientific research.
Cosmologists have even p r o p o s e d the startling possibility that o u r
universe is j u s t o n e a m o n g an infinite n u m b e r of parallel universes.
These universes might be c o m p a r e d to a vast collection of soap bubbles
suspended in air. Normally, contact between these bubble universes is
impossible, but, by analyzing Einstein's equations, cosmologists have
shown that there might exist a web of wormholes, or tubes, that c o n n e c t
these parallel universes. On each bubble, we can define o u r own distinctive space a n d time, which have m e a n i n g only on its surface; outside
these bubbles, space a n d time have no m e a n i n g .
Although many consequences of this discussion are purely theoretical, hyperspace travel may eventually provide the most practical application of all: to save intelligent life, including ours, from t h e d e a t h of
the universe. Scientists universally believe that the universe must eventually die, a n d with it all life that has evolved over billions of years. For
example, according to the prevailing theory, called the Big Bang, a cosmic explosion 15 to 20 billion years ago set the universe e x p a n d i n g ,
hurling stars a n d galaxies away from us at great velocities. However, if
the universe o n e day stops e x p a n d i n g a n d begins to contract, it will
eventually collapse into a fiery cataclysm called the Big C r u n c h , in which
all intelligent life will be vaporized by fantastic heat. Nevertheless, some
physicists have speculated that the hyperspace theory may provide the
o n e a n d only h o p e of a refuge for intelligent life. In the last seconds of
the d e a t h of o u r universe, intelligent life may escape the collapse by
fleeing into hyperspace.
Preface
xi
In Part IV, we conclude with a final, practical question: If the theory
is proved correct, then when will we be able to harness the power of the
hyperspace theory? This is not just an academic question, because in the
past, the harnessing of just o n e of the four fundamental forces irrevocably changed the course of h u m a n history, lifting us from the ignorance
and squalor of ancient, preindustrial societies to m o d e r n civilization. In
some sense, even the vast sweep of h u m a n history can be viewed in a
new light, in terms of the progressive mastery of each of the four forces.
T h e history of civilization has u n d e r g o n e a profound change as each of
these forces was discovered a n d mastered.
For example, when Isaac Newton wrote down the classical laws of
gravity, he developed the theory of mechanics, which gave us the laws
governing machines. This, in turn, greatly accelerated the Industrial Revolution, which unleashed political forces that eventually overthrew the
feudal dynasties of Europe. In the mid-1860s, when J a m e s Clerk Maxwell
wrote down the fundamental laws of the electromagnetic force, he ushered in the Electric Age, which gave us the dynamo, radio, television,
radar, household appliances, the telephone, microwaves, consumer electronics, the electronic computer, lasers, and many other electronic marvels. Without the u n d e r s t a n d i n g a n d utilization of the electromagnetic
force, civilization would have stagnated, frozen in a time before the discovery of the light bulb and the electric motor. In the mid-1940s, when
the nuclear force was harnessed, the world was again t u r n e d upside
down with the development of the atomic and hydrogen bombs, the
most destructive weapons on the planet. Because we are n o t on the verge
of a unified u n d e r s t a n d i n g of all the cosmic forces governing the universe, o n e might expect that any civilization that masters the hyperspace
theory will become lord of the universe.
Since the hyperspace theory is a well-defined body of mathematical
equations, we can calculate the precise energy necessary to twist space
and time into a pretzel or to create wormholes linking distant parts of
our universe. Unfortunately, the results are disappointing. T h e energy
required far exceeds anything that our planet can muster. In fact, the
energy is a quadrillion times larger than the energy of our largest atom
smashers. We must wait centuries or even millennia until o u r civilization
develops the technical capability of manipulating space-time, or h o p e
for contact with an advanced civilization that has already mastered
hyperspace. T h e book therefore ends by exploring the intriguing b u t
speculative scientific question of what level of technology is necessary
for us to b e c o m e masters of hyperspace.
Because the hyperspace theory takes us far beyond normal, c o m m o n -
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Preface
sense conceptions of space a n d time, I have scattered t h r o u g h o u t the
text a few purely hypothetical stories. I was inspired to utilize this pedagogical t e c h n i q u e by the story of Nobel Prize winner Isidore I. Rabi
addressing an a u d i e n c e of physicists. He l a m e n t e d the abysmal state of
science education in the U n i t e d States a n d scolded the physics community for neglecting its duty in popularizing the adventure of science
for the general public a n d especially for the young. In fact, he a d m o n ished, science-fiction writers h a d d o n e m o r e to c o m m u n i c a t e t h e
r o m a n c e of science than all physicists c o m b i n e d .
In a previous book, Beyond Einstein: The Cosmic Quest for the Theory of
the Universe (coauthored with Jennifer T r a i n e r ) , I investigated superstring theory, described the n a t u r e of subatomic particles, a n d discussed
at length the visible universe a n d how all the complexities of m a t t e r might
be explained by tiny, vibrating strings. In this book, I have e x p a n d e d on
a different t h e m e a n d explored the invisible universe—that is, the world
of geometry and space-time. T h e focus of this b o o k is n o t the n a t u r e of
subatomic particles, b u t the higher-dimensional world in which they
probably live. In the process, readers will see that higher-dimensional
space, instead of being an empty, passive b a c k d r o p against which quarks
play o u t their eternal roles, actually becomes the central actor in the
d r a m a of n a t u r e .
In discussing the fascinating history of the hyperspace theory, we will
see that the search for the ultimate n a t u r e of matter, b e g u n by the
Greeks 2 millennia ago, has b e e n a long a n d tortuous o n e . W h e n the
final c h a p t e r in this long saga is written by future historians of science,
they may well r e c o r d that the crucial b r e a k t h r o u g h was the defeat of
common-sense theories of three or four dimensions and the victory of
the theory of hyperspace.
New York
May 1993
M.K.
Acknowledgments
In writing this book, I have b e e n fortunate to have Jeffrey Robbins as
my editor. He was the editor who skillfully guided the progress of three
of my previous textbooks in theoretical physics written for the scientific
community, c o n c e r n i n g the unified field theory, superstring theory, and
q u a n t u m field theory. This book, however, marks the first popular science book aimed at a general audience that I have written for him. It
has always b e e n a rare privilege to work closely with him.
I would also like to thank Jennifer Trainer, who has b e e n my coauthor on two previous popular books. O n c e again, she has applied her
considerable skills to make the presentation as smooth a n d c o h e r e n t as
possible.
I am also grateful to n u m e r o u s other individuals who have helped
to strengthen and criticize earlier drafts of this book: Burt Solomon,
Leslie Meredith, Eugene Mallove, and my agent, Stuart Krichevsky.
Finally, I would like to thank the Institute for Advanced Study at
Princeton, where m u c h of this book was written, for its hospitality. T h e
Institute, where Einstein spent the last decades of his life, was an appropriate place to write about the revolutionary developments that have
e x t e n d e d a n d embellished m u c h of his pioneering work.
Contents
Part I Entering the Fifth Dimension
1. Worlds Beyond Space and Time, 3
2. Mathematicians and Mystics, 30
3. The Man Who "Saw" the Fourth Dimension, 55
The Secret of Light: Vibrations in the Fifth Dimension,
Part II Unification in Ten Dimensions
5. Quantum Heresy, 111
6. Einstein's Revenge, 136
7. Superstrings, 151
8. Signals from the Tenth Dimension, 178
9. Before Creation, 191
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Contents
PART III WORMHOLES: GATEWAYS TO ANOTHER UNIVERSE?
10. Black Holes and Parallel Universes, 217
11. To Build a Time Machine, 232
12. Colliding Universes, 252
PART IV MASTERS OF HYPERSPACE
13. Beyond the Future, 273
14. The Fate of the Universe, 301
15. Conclusion, 313
Notes, 335
References and Suggested Reading, 353
Index, 355
But the creative principle resides in mathematics. In a certain
sense, therefore, I hold it true that pure thought can grasp
reality, as the ancients dreamed.
Albert Einstein
PART I
Entering
the Fifth Dimension
1
Worlds Beyond Space
and Time
I w a n t t o k n o w h o w G o d c r e a t e d this w o r l d . I a m n o t i n t e r e s t e d
i n this o r that p h e n o m e n o n . I w a n t t o k n o w H i s t h o u g h t s , t h e
rest a r e d e t a i l s .
Albert Einstein
The Education of a Physicist
T
WO incidents from my childhood greatly enriched my understanding of the world a n d sent me on course to b e c o m e a theoretical
physicist.
I r e m e m b e r that my parents would sometimes take me to visit the
famous J a p a n e s e T e a Garden in San Francisco. O n e of my happiest
childhood memories is of crouching next to the p o n d , mesmerized by
the brilliantly colored carp swimming slowly b e n e a t h the water lilies.
In these quiet m o m e n t s , I felt free to let my imagination wander; I
would ask myself silly questions that a only child might ask, such as how
the carp in that p o n d would view the world a r o u n d them. I thought,
W h a t a strange world theirs must be!
Living their entire lives in the shallow p o n d , the carp would believe
that their " u n i v e r s e " consisted of the murky water a n d the lilies. Spending most of their time foraging on the b o t t o m of the p o n d , they would
be only dimly aware that an alien world could exist above the surface.
3
4
ENTERING THE FIFTH D I M E N S I O N
T h e n a t u r e of my world was beyond their c o m p r e h e n s i o n . I was
intrigued that I could sit only a few inches from the carp, yet be separated
from t h e m by an immense chasm. T h e carp a n d I spent o u r lives in two
distinct universes, never entering each other's world, yet were separated
by only the thinnest barrier, the water's surface.
I once imagined that there may be carp "scientists" living a m o n g
the fish. They would, I thought, scoff at any fish who proposed that a
parallel world could exist j u s t above the lilies. To a carp "scientist," the
only things that were real were what the fish could see or touch. T h e
p o n d was everything. An unseen world beyond the p o n d m a d e no scientific sense.
O n c e I was caught in a rainstorm. I noticed that the p o n d ' s surface
was b o m b a r d e d by thousands of tiny raindrops. T h e p o n d ' s surface
b e c a m e turbulent, a n d the water lilies were being p u s h e d in all directions by water waves. Taking shelter from the wind a n d the rain, I wond e r e d how all this a p p e a r e d to the carp. To t h e m , the water lilies would
a p p e a r to be moving a r o u n d by themselves, without anything pushing
t h e m . Since the water they lived in would a p p e a r invisible, m u c h like
the air and space a r o u n d us, they would be baffled that the water lilies
could move a r o u n d by themselves.
T h e i r "scientists," I imagined, would concoct a clever invention
called a " f o r c e " in o r d e r to h i d e their ignorance. Unable to compreh e n d that there could be waves on the unseen surface, they would conclude that lilies could move without being t o u c h e d because a mysterious,
invisible entity called a force acted between them. They might give this
illusion impressive, lofty n a m e s (such as action-at-a-distance, or the ability of the lilies to move without anything touching t h e m ) .
O n c e I imagined what would h a p p e n if I r e a c h e d down a n d lifted
o n e of the carp "scientists" out of the p o n d . Before I threw h i m back
into the water, he might wiggle furiously as I e x a m i n e d him. I w o n d e r e d
how this would a p p e a r to the rest of the carp. To t h e m , it would be a
truly unsettling event. They would first notice that o n e of their "scientists" h a d disappeared from their universe. Simply vanished, without
leaving a trace. Wherever they would look, there would be no evidence
of the missing carp in their universe. T h e n , seconds later, when I threw
him back into the p o n d , the "scientist" would abruptly r e a p p e a r out of
nowhere. To the o t h e r carp, it would appear that a miracle h a d h a p
pened.
After collecting his wits, the "scientist" would tell a truly amazing
story. "Without warning," he would say, "I was somehow lifted out of
the universe (the p o n d ) a n d hurled into a mysterious n e t h e r w o r l d , with
Worlds Beyond Space and Time
5
blinding lights a n d strangely shaped objects that I h a d never seen before.
T h e strangest of all was the creature who held me prisoner, who did n o t
resemble a fish in the slightest. I was shocked to see that it h a d no fins
whatsoever, b u t nevertheless could move without t h e m . It struck me that
the familiar laws of n a t u r e no longer applied in this n e t h e r world. T h e n ,
j u s t as suddenly, I found myself thrown back into o u r universe." (This
story, of course, of a j o u r n e y beyond the universe would be so fantastic
that most of the carp would dismiss it as utter poppycock.)
I often think that we are like the carp swimming contentedly in that
p o n d . We live out o u r lives in o u r own " p o n d , " confident that o u r universe consists of only those things we can see or touch. Like the carp,
o u r universe consists of only the familiar a n d the visible. We smugly
refuse to admit that parallel universes or dimensions can exist next to
ours, just beyond o u r grasp. If o u r scientists invent concepts like forces,
it is only because they cannot visualize the invisible vibrations that fill
the empty space a r o u n d us. Some scientists sneer at the m e n t i o n of
higher dimensions because they cannot be conveniently measured in
the laboratory.
Ever since that time, I have b e e n fascinated by the possibility of o t h e r
dimensions. Like most children, I devoured adventure stories in which
time travelers e n t e r e d o t h e r dimensions a n d explored u n s e e n parallel
universes, where the usual laws of physics could be conveniently susp e n d e d . I grew up wondering if ships that wandered into the B e r m u d a
Triangle mysteriously vanished into a hole in space; I marveled at Isaac
Asimov's Foundation Series, in which the discovery of hyperspace travel
led to the rise of a Galactic Empire.
A second incident from my childhood also m a d e a d e e p , lasting
impression on m e . W h e n I was 8 years old, I h e a r d a story that would
stay with me for the rest of my life. I r e m e m b e r my schoolteachers telling
the class a b o u t a great scientist who h a d just died. They talked about
him with great reverence, calling him o n e of the greatest scientists in all
history. They said that very few people could u n d e r s t a n d his ideas, b u t
that his discoveries c h a n g e d the entire world a n d everything a r o u n d us.
I d i d n ' t u n d e r s t a n d m u c h of what they were trying to tell us, b u t what
most intrigued me a b o u t this m a n was that he died before he could
complete his greatest discovery. They said he spent years on this theory,
b u t he died with his unfinished papers still sitting on his desk.
I was fascinated by the story. To a child, this was a great mystery.
What was his unfinished work? What was in those papers on his desk?
What p r o b l e m could possibly be so difficult a n d so i m p o r t a n t that such
a great scientist would dedicate years of his life to its pursuit? Curious, I
6
decided to learn all I could a b o u t Albert Einstein a n d his unfinished
theory. I still have warm memories of spending many quiet h o u r s reading
every b o o k I could find a b o u t this great m a n a n d his theories. W h e n I
exhausted the books in o u r local library, I began to scour libraries a n d
bookstores across the city, eagerly searching for m o r e clues. I soon
learned that this story was far m o r e exciting than any m u r d e r mystery
a n d m o r e important t h a n anything I could ever imagine. I decided that
I would try to get to the r o o t of this mystery, even if I h a d to b e c o m e a
theoretical physicist to do it.
I soon learned that the unfinished papers on Einstein's desk were an
a t t e m p t to construct what he called the unified field theory, a theory
that could explain all the laws of n a t u r e , from the tiniest atom to the
largest galaxy. However, being a child, I d i d n ' t u n d e r s t a n d that p e r h a p s
there was a link between the carp swimming in the Tea Garden a n d the
unfinished papers lying on Einstein's desk. I d i d n ' t u n d e r s t a n d that
higher dimensions might be the key to solving the unified field theory.
Later, in high school, I exhausted most of the local libraries a n d often
visited the Stanford University physics library. T h e r e , I came across the
fact that Einstein's work m a d e possible a new substance called antimatter, which would act like ordinary matter b u t would annihilate u p o n
contact with matter in a burst of energy. I also read that scientists h a d
built large machines, or " a t o m smashers," that could p r o d u c e microscopic quantities of this exotic substance in the laboratory.
O n e advantage of youth is that it is u n d a u n t e d by worldly constraints
that would ordinarily seem insurmountable to most adults. Not appreciating the obstacles involved, I set out to build my own atom smasher.
I studied the scientific literature until I was convinced that I could build
what was called a betatron, which could boost electrons to millions of
electron volts. (A million electron volts is the energy attained by electrons accelerated by a field of a million volts.)
First, I purchased a small quantity of sodium-22, which is radioactive
a n d naturally emits positrons (the antimatter c o u n t e r p a r t of electrons).
T h e n I built what is called a cloud chamber, which makes visible the
tracks left by subatomic particles. I was able to take h u n d r e d s of beautiful
p h o t o g r a p h s of the tracks left b e h i n d by antimatter. Next, I scavenged
a r o u n d large electronic warehouses in the area, assembled the necessary
hardware, including h u n d r e d s of p o u n d s of scrap transformer steel, a n d
built a 2.3-million-electron-volt betatron in my garage that would be powerful e n o u g h to p r o d u c e a b e a m of antielectrons. To construct the monstrous magnets necessary for the betatron, I convinced my parents to
help me wind 22 miles of cooper wire on the high-school football field.
7
We spent Christmas vacation on the 50-yard line, winding a n d assembling the massive coils that would b e n d the paths of the high-energy
electrons.
W h e n finally constructed, t h e 300-pound, 6-kilowatt betatron cons u m e d every o u n c e of energy my house p r o d u c e d . W h e n I t u r n e d it on,
I would usually blow every fuse, a n d the house would suddenly b e c a m e
dark. With the house p l u n g e d periodically into darkness, my m o t h e r
would often shake h e r head. (I imagined that she probably w o n d e r e d
why she c o u l d n ' t have a child who played baseball or basketball, instead
of building these h u g e electrical machines in the garage.) I was gratified
that the m a c h i n e successfully p r o d u c e d a magnetic field 20,000 times
m o r e powerful than the earth's magnetic field, which is necessary to
accelerate a b e a m of electrons.
Confronting the Fifth Dimension
Because my family was poor, my parents were c o n c e r n e d that I w o u l d n ' t
be able to continue my experiments a n d my education. Fortunately, the
awards that I won for my various science projects caught the attention
of the atomic scientist Edward Teller. His wife generously arranged for
me to receive a 4-year scholarship to Harvard, allowing me to fulfill my
dream.
Ironically, although at Harvard I began my formal training in theoretical physics, it was also where my interest in h i g h e r dimensions gradually died out. Like o t h e r physicists, I began a rigorous a n d t h o r o u g h
p r o g r a m of studying the higher mathematics of each of the forces of
n a t u r e separately, in complete isolation from o n e another. I still rememb e r solving a p r o b l e m in electrodynamics for my instructor, a n d t h e n
asking him what the solution might look like if space were curved in a
higher dimension. He looked at me in a strange way, as if I were a bit
cracked. Like others before m e , I soon learned to p u t aside my earlier,
childish notions a b o u t higher-dimensional space. Hyperspace, I was told,
was n o t a suitable subject of serious study.
I was never satisfied with this disjointed a p p r o a c h to physics, a n d my
thoughts would often drift back to the the carp living in the Tea Garden.
Although the equations we used for electricity a n d magnetism, discovered by Maxwell in the n i n e t e e n t h century, worked surprisingly well, the
equations seemed rather arbitrary. I felt that physicists (like the carp)
invented these "forces" to hide o u r ignorance of how objects can move
each o t h e r without touching.
8
In my studies, I learned that o n e of the great debates of the ninet e e n t h century h a d b e e n a b o u t how light travels t h r o u g h a vacuum.
(Light from the stars, in fact, can effortlessly travel trillions u p o n trillions
of miles t h r o u g h the vacuum of o u t e r space.) Experiments also showed
beyond question that light is a wave. But if light were a wave, t h e n it
would require s o m e t h i n g to be "waving." S o u n d waves require air, water
waves require water, b u t since there is n o t h i n g to wave in a vacuum, we
have a paradox. How can light be a wave if t h e r e is n o t h i n g to wave? So
physicists conjured up a substance called the aether, which filled the
vacuum a n d acted as the m e d i u m for light. However, experiments conclusively showed that the " a e t h e r " does n o t exist.*
Finally, w h e n I b e c a m e a graduate student in physics at the University
of California at Berkeley, I learned quite by accident that there was an
alternative, albeit controversial, explanation of how light can travel
t h r o u g h a vacuum. This alternative theory was so outlandish that I
received quite a j o l t when I stumbled across it. T h a t shock was similar
to the o n e experienced by many Americans when they first h e a r d that
President J o h n Kennedy h a d b e e n shot. They can invariably r e m e m b e r
the precise m o m e n t when they h e a r d the shocking news, what they were
doing, a n d to w h o m they were talking at that instant. We physicists, too,
receive quite a shock when we first stumble across Kaluza-Klein theory
for t h e first time. Since the theory was considered to be a wild speculation, it was never taught in graduate school; so young physicists are left
to discover it quite by accident in their casual readings.
This alternative theory gave the simplest explanation of light: that it
was really a vibration of the fifth dimension, or what used to called the
fourth dimension by the mystics. If light could travel t h r o u g h a vacuum,
it was because t h e vacuum itself was vibrating, because t h e " v a c u u m "
really existed in four dimensions of space a n d o n e of time. By adding
the fifth dimension, the force of gravity a n d light could be unified in a
startlingly simple way. Looking back at my childhood experiences at the
T e a G a r d e n , I suddenly realized that this was the mathematical theory
for which I h a d been looking.
T h e old Kaluza-Klein theory, however, h a d many difficult, technical
problems that r e n d e r e d it useless for over half a century. All this, however, has c h a n g e d in the past d e c a d e . More advanced versions of the
theory, like supergravity theory a n d especially superstring theory, have
*Surprisingly, even today physicists still do not have a real answer to this puzzle, but
over the decades we have simply gotten used to the idea that light can travel through a
vacuum even if there is nothing to wave.
9
finally eliminated the inconsistencies of the theory. Rather abruptly, the
theory of h i g h e r dimensions is now b e i n g c h a m p i o n e d in research laboratories a r o u n d the globe. Many of the world's leading physicists now
believe that dimensions beyond the usual four of space a n d time m i g h t
exist. This idea, in fact, has b e c o m e the focal point of intense scientific
investigation. I n d e e d , many theoretical physicists now believe that
h i g h e r dimensions may be the decisive step in creating a comprehensive
theory that unites the laws of n a t u r e — a theory of hyperspace.
If it proves to be correct, t h e n future historians of science may well
record that o n e of the great conceptual revolutions in twentieth-century
science was the realization that hyperspace may be the key to unlock the
deepest secrets of n a t u r e a n d Creation itself.
This seminal c o n c e p t has sparked an avalanche of scientific research:
Several thousand papers written by theoretical physicists in the major
research laboratories a r o u n d the world have been devoted to exploring
the properties of hyperspace. T h e pages of Nuclear Physics a n d Physics
Letters, two leading scientific j o u r n a l s , have b e e n flooded with articles
analyzing the theory. More t h a n 200 international physics conferences
have b e e n sponsored to explore the consequences of higher dimensions.
Unfortunately, we are still far from experimentally verifying that o u r
universe exists in higher dimensions. (Precisely what it would take to
prove the correctness of the theory a n d possibly harness the power of
hyperspace will be discussed later in this book.) However, this theory
has now b e c o m e firmly established as a legitimate b r a n c h of m o d e r n
theoretical physics. T h e Institute for Advanced Study at Princeton, for
example, where Einstein spent the last decades of his life (and where
this book was written), is now o n e of the active centers of research on
higher-dimensional space-time.
Steven Weinberg, who won the Nobel Prize in physics in 1979, summarized this conceptual revolution when he c o m m e n t e d recently that
theoretical physics seems to be b e c o m i n g m o r e a n d m o r e like science
fiction.
Why Can't We See Higher Dimensions?
T h e s e revolutionary ideas seem strange at first because we take for
granted that o u r everyday world has three dimensions. As the late physicist Heinz Pagels noted, " O n e feature of o u r physical world is so obvious
that most p e o p l e are not even puzzled by it—the fact that space is threed i m e n s i o n a l . " Almost by instinct alone, we know that any object can be
1
10
described by giving its height, width, a n d d e p t h . By giving three numbers, we can locate any position in space. If we want to m e e t s o m e o n e
for lunch in New York, we say, " M e e t me on the twenty-fourth floor of
the building at the c o r n e r of Forty-second Street a n d First A v e n u e . " Two
n u m b e r s provide us the street corner; a n d the third, the height off the
ground.
Airplane pilots, too, know exactly where they are with three n u m bers—their altitude a n d two coordinates that locate their position on a
grid or m a p . In fact, specifying these three n u m b e r s can p i n p o i n t any
location in o u r world, from the tip of o u r nose to the e n d s of the visible
universe. Even babies u n d e r s t a n d this: Tests with infants have shown that
they will crawl to the edge of a cliff, p e e r over the edge, a n d crawl back.
In addition to u n d e r s t a n d i n g "left" a n d " r i g h t " a n d "forward" a n d
" b a c k w a r d " instinctively, babies instinctively u n d e r s t a n d " u p " a n d
" d o w n . " T h u s the intuitive c o n c e p t of three dimensions is firmly embedd e d in o u r brains from an early age.
Einstein e x t e n d e d this c o n c e p t to include time as the fourth dimension. For example, to m e e t that s o m e o n e for lunch, we must specify that
we should m e e t at, say, 12:30 P.M. in Manhattan; that is, to specify an
event, we also n e e d to describe its fourth dimension, the time at which
the event takes place.
Scientists today are interested in going beyond Einstein's conception
of the fourth dimension. C u r r e n t scientific interest centers on the fifth
dimension (the spatial dimension beyond time and the three dimensions of space) a n d beyond. (To avoid confusion, t h r o u g h o u t this book
I have bowed to custom a n d called the fourth dimension the spatial
dimension beyond length, b r e a d t h , a n d width. Physicists actually refer
to this as the fifth dimension, b u t I will follow historical p r e c e d e n t . We
will call time the fourth temporal dimension.)
How do we see the fourth spatial dimension?
T h e p r o b l e m is, we can't. Higher-dimensional spaces are impossible
to visualize; so it is futile even to try. T h e p r o m i n e n t G e r m a n physicist
H e r m a n n von Helmholtz c o m p a r e d the inability to " s e e " the fourth
dimension with t h e inability of a blind m a n to conceive of the c o n c e p t
of color. No matter how eloquently we describe " r e d " to a blind person,
words fail to impart the m e a n i n g of anything as rich in m e a n i n g as color.
Even experienced mathematicians a n d theoretical physicists who have
worked with higher-dimensional spaces for years admit that they c a n n o t
visualize t h e m . Instead, they retreat into the world of mathematical equations. But while mathematicians, physicists, a n d c o m p u t e r s have no
p r o b l e m solving equations in multidimensional space, h u m a n s find it
impossible to visualize universes beyond their own.
II
At best, we can use a variety of mathematical tricks, devised by mathematician a n d mystic Charles H i n t o n at the turn of the century, to visualize shadows of higher-dimensional objects. O t h e r mathematicians, like
T h o m a s Banchoff, chairman of the mathematics d e p a r t m e n t at Brown
University, have written c o m p u t e r programs that allow us to manipulate
higher-dimensional objects by projecting their shadows o n t o flat, twodimensional c o m p u t e r screens. Like the Greek p h i l o s o p h e r Plato, w h o
said that we are like cave dwellers c o n d e m n e d to see only the dim, gray
shadows of the rich life outside o u r caves, Banchoff's c o m p u t e r s allow
only a glimpse of the shadows of higher-dimensional objects. (Actually,
we c a n n o t visualize h i g h e r dimensions because of an accident of evolution. O u r brains have evolved to handle myriad emergencies in three
dimensions. Instantly, without stopping to think, we can recognize a n d
react to a leaping lion or a charging elephant. In fact, those h u m a n s
w h o could better visualize how objects move, turn, a n d twist in three
dimensions h a d a distinct survival advantage over those who could not.
Unfortunately, there was no selection pressure placed on h u m a n s to
master motion in four spatial dimensions. Being able to see the fourth
spatial dimension certainly did n o t h e l p s o m e o n e fend off a charging
saber-toothed tiger. Lions a n d tigers do n o t lunge at us t h r o u g h the
fourth dimension.)
The Laws of Nature Are Simpler in Higher Dimensions
O n e physicist who delights in teasing audiences a b o u t the properties of
higher-dimensional universes is Peter Freund, a professor of theoretical
physics at the University of Chicago's r e n o w n e d Enrico Fermi Institute.
F r e u n d was o n e of the early pioneers working on hyperspace theories
when it was considered too outlandish for mainstream physics. For years,
F r e u n d a n d a small g r o u p of scientists dabbled in the science of higher
dimensions in isolation; now, however, it has finally b e c o m e fashionable
a n d a legitimate b r a n c h of scientific research. To his delight, he is finding that his early interest is at last paying off.
F r e u n d does n o t fit t h e traditional image of a narrow, crusty, disheveled scientist. Instead, he is u r b a n e , articulate, a n d cultured, a n d has a
sly, impish grin that captivates nonscientists with fascinating stories of
fast-breaking scientific discoveries. He is equally at ease scribbling on a
blackboard littered with dense equations or exchanging light b a n t e r at
a cocktail party. Speaking with a thick, distinguished Romanian accent,
F r e u n d has a rare knack for explaining the most arcane, convoluted
concepts of physics in a lively, engaging style.
12
Traditionally, F r e u n d r e m i n d s us, scientists have viewed higher
dimensions with skepticism because they could n o t be m e a s u r e d a n d did
n o t have any particular use. However, the growing realization a m o n g
scientists today is that any three-dimensional theory is " t o o small" to
describe the forces that govern o u r universe.
As F r e u n d emphasizes, o n e fundamental t h e m e r u n n i n g t h r o u g h the
past decade of physics has b e e n that the laws of nature become simpler and
elegant when expressed in higher dimensions, which is their natural h o m e .
T h e laws of light a n d gravity find a natural expression when expressed
in higher-dimensional s p a c e - t i m e . T h e key step in unifying the laws of
n a t u r e is to increase the n u m b e r of dimensions of space-time until m o r e
a n d m o r e forces can be a c c o m m o d a t e d . In higher dimensions, we have
e n o u g h " r o o m " to unify all known physical forces.
Freund, in explaining why higher dimensions are exciting the imagination of the scientific world, uses the following analogy: " T h i n k , for a
m o m e n t , of a cheetah, a sleek, beautiful animal, o n e of the fastest on
earth, which roams freely on the savannas of Africa. In its natural habitat,
it is a magnificent animal, almost a work of art, unsurpassed in speed or
grace by any o t h e r animal. Now," he continues,
think of a c h e e t a h that has b e e n captured a n d thrown i n t o a miserable
c a g e i n a z o o . I t has lost its o r i g i n a l g r a c e a n d b e a u t y , a n d i s p u t o n d i s p l a y
f o r o u r a m u s e m e n t . W e s e e o n l y t h e b r o k e n spirit o f t h e c h e e t a h i n t h e
c a g e , n o t its o r i g i n a l p o w e r a n d e l e g a n c e . T h e c h e e t a h c a n b e c o m p a r e d
t o t h e laws o f p h y s i c s , w h i c h a r e b e a u t i f u l i n t h e i r n a t u r a l s e t t i n g . T h e
n a t u r a l h a b i t a t o f t h e laws o f p h y s i c s i s h i g h e r - d i m e n s i o n a l s p a c e - t i m e .
H o w e v e r , w e c a n o n l y m e a s u r e t h e laws o f p h y s i c s w h e n t h e y h a v e b e e n
b r o k e n a n d p l a c e d on display in a c a g e , w h i c h is o u r t h r e e - d i m e n s i o n a l
l a b o r a t o r y . W e o n l y s e e t h e c h e e t a h w h e n its g r a c e a n d b e a u t y h a v e b e e n
stripped away.
2
For decades, physicists have w o n d e r e d why t h e four forces of n a t u r e
a p p e a r to be so fragmented—why the " c h e e t a h " looks so pitiful a n d
b r o k e n in his cage. T h e fundamental reason why these four forces seem
so dissimilar, notes Freund, is that we have b e e n observing the " c a g e d
c h e e t a h . " O u r three-dimensional laboratories are sterile zoo cages for
the laws of physics. But when we formulate the laws in higher-dimensional space-time, their natural habitat, we see their true brilliance a n d
power; the laws b e c o m e simple a n d powerful. T h e revolution now sweeping over physics is the realization that the natural h o m e for the cheetah
may be hyperspace.
13
To illustrate how a d d i n g a higher dimension can make things simpler, imagine how major wars were fought by ancient Rome. T h e great
R o m a n wars, often involving many smaller battlefields, were invariably
fought with great confusion, with rumors a n d misinformation p o u r i n g
in on both sides from many different directions. With battles raging on
several fronts, R o m a n generals were often operating blind. Rome won
its battles m o r e from b r u t e strength than from the elegance of its strategies. T h a t is why o n e of the first principles of warfare is to seize the
high g r o u n d — t h a t is, to go up into the third dimension, above the twodimensional battlefield. From the vantage point of a large hill with a
p a n o r a m i c view of the battlefield, the chaos of war suddenly becomes
vastly reduced. In o t h e r words, viewed from the third dimension (that
is, from the top of the hill), the confusion of the smaller battlefields
becomes integrated into a c o h e r e n t single picture.
A n o t h e r application of this principle—that n a t u r e becomes simpler
when expressed in higher dimensions—is the central idea b e h i n d Einstein's special theory of relativity. Einstein revealed time to be the fourth
dimension, a n d he showed that space a n d time could conveniently be
unified in a four-dimensional theory. This, in turn, inevitably led to the
unification of all physical quantities measured by space a n d time, such
as matter a n d energy. He then found the precise mathematical expression for this unity between matter a n d energy: E = mc , perhaps the most
celebrated of all scientific equations.*
3
To appreciate the e n o r m o u s power of this unification, let us now
describe the four fundamental forces, emphasizing how different they
are, a n d how higher dimensions may give us a unifying formalism. Over
the past 2,000 years, scientists have discovered that all p h e n o m e n a in
o u r universe can be r e d u c e d to four forces, which at first bear no resemblance to o n e another.
The Electromagnetic Force
T h e electromagnetic force takes a variety of forms, including electricity,
magnetism, a n d light itself. T h e electromagnetic force lights o u r cities,
fills the air with music from radios a n d stereos, entertains us with television, reduces housework with electrical appliances, heats o u r food with
*The theory of higher dimensions is certainly not merely an academic one, because
the simplest consequence of Einstein's theory is the atomic bomb, which has changed the
destiny of humanity. In this sense, the introduction of higher dimensions has been one of
the pivotal scientific discoveries in all human history.
14
microwaves, tracks o u r planes a n d space probes with radar, a n d electrifies o u r power plants. More recently, the power of the electromagnetic
force has b e e n used in electronic c o m p u t e r s (which have revolutionized
the office, h o m e , school, a n d military) a n d in lasers (which have introd u c e d new vistas in communications, surgery, c o m p a c t disks, advanced
Pentagon weaponry, a n d even the check-out stands in groceries). More
than half the gross national p r o d u c t of the earth, representing the accumulated wealth of our planet, d e p e n d s in some way on the electromagnetic force.
The Strong Nuclear Force
T h e strong nuclear force provides the energy that fuels the stars; it makes
the stars shine a n d creates the brilliant, life-giving rays of the sun. If the
strong force suddenly vanished, the sun would darken, e n d i n g all life
on earth. In fact, some scientists believe that the dinosaurs were driven
to extinction 65 million years ago when debris from a c o m e t impact was
blown high into the a t m o s p h e r e , d a r k e n i n g the earth a n d causing the
t e m p e r a t u r e a r o u n d the planet to p l u m m e t . Ironically, it is also the
strong nuclear force that may o n e day take back the gift of life.
Unleashed in the hydrogen b o m b , the strong nuclear force could o n e
day e n d all life on earth.
The Weak Nuclear Force
T h e weak nuclear force governs certain forms of radioactive decay.
Because radioactive materials emit h e a t when they decay or break apart,
the weak nuclear force contributes to heating the radioactive rock d e e p
within the earth's interior. This heat, in turn, contributes to the heat
that drives the volcanoes, the rare b u t powerful eruptions of molten rock
that reach the earth's surface. T h e weak a n d electromagnetic forces are
also exploited to treat serious diseases: Radioactive iodine is used to kill
tumors of the thyroid gland a n d fight certain forms of cancer. T h e force
of radioactive decay can also be deadly: It wreaked havoc at T h r e e Mile
Island a n d Chernobyl; it also creates radioactive waste, the inevitable byp r o d u c t of nuclear weapons p r o d u c t i o n a n d commercial nuclear power
plants, which may remain harmful for millions of years.
The Gravitational Force
T h e gravitational force keeps the earth a n d the planets in their orbits
a n d binds the galaxy. Without the gravitational force of the earth, we
15
would be flung into space like rag dolls by the spin of the earth. T h e air
we b r e a t h e would be quickly diffused into space, causing us to asphyxiate
a n d making life on earth impossible. Without the gravitational force of
the sun, all the planets, including the earth, would be flung from the
solar system into the cold reaches of d e e p space, w h e r e sunlight is too
dim to support life. In fact, without the gravitational force, the sun itself
would explode. T h e sun is the result of a delicate balancing act between
the force of gravity, which tends to crush the star, a n d the nuclear force,
which tends to blast the sun apart. W i t h o u t gravity, the sun would deto n a t e like trillions u p o n trillions of hydrogen b o m b s .
T h e central challenge of theoretical physics today is to unify these four
forces into a single force. Beginning with Einstein, t h e giants of twentieth-century physics have tried a n d failed to find such a unifying scheme.
However, the answer that eluded Einstein for the last 30 years of his life
may lie in hyperspace.
The Quest for Unification
Einstein o n c e said, " N a t u r e shows us only the tail of the lion. But I do
n o t d o u b t that the lion belongs to it even t h o u g h he c a n n o t at once
reveal himself because of his e n o r m o u s size." If Einstein is correct, t h e n
p e r h a p s these four forces are t h e "tail of t h e l i o n , " a n d t h e " l i o n " itself
is higher-dimensional space-time. This idea has fueled the h o p e that
the physical laws of the universe, whose consequences fill entire library
walls with books densely packed with tables a n d graphs, may o n e day be
explained by a single equation.
3
Central to this revolutionary perspective on the universe is the realization that higher-dimensional geometry may be the ultimate source of
unity in the universe. Simply put, the matter in the universe a n d the
forces that hold it together, which a p p e a r in a bewildering, infinite variety of complex forms, may be n o t h i n g b u t different vibrations of hyperspace. This concept, however, goes against the traditional thinking
a m o n g scientists, who have viewed space a n d time as a passive stage on
which the stars a n d the atoms play the leading role. To scientists, the
visible universe of matter seemed infinitely richer a n d m o r e diverse than
the empty, u n m o v i n g arena of the invisible universe of space-time.
Almost all the intense scientific effort a n d massive g o v e r n m e n t funding
in particle physics has historically gone to cataloging the properties of
subatomic particles, such as " q u a r k s " a n d " g l u o n s , " rather than fath-
16
o m i n g the n a t u r e of geometry. Now, scientists are realizing that the "useless" concepts of space and time may be the ultimate source of beauty
and simplicity in n a t u r e .
T h e first theory of h i g h e r dimensions was called Kaluza-Klein theory,
after two scientists who p r o p o s e d a new theory of gravity in which light
could be explained as vibrations in the fifth dimension. W h e n e x t e n d e d
to N-dimensional space (where N can stand for any whole n u m b e r ) , the
clumsy-looking theories of subatomic particles dramatically take on a
startling symmetry. T h e old Kaluza-Klein theory, however, could n o t
d e t e r m i n e the correct value of N, a n d there were technical problems in
describing all the subatomic particles. A m o r e advanced version of this
theory, called supergravity theory, also had problems. T h e recent interest
in the theory was sparked in 1984 by physicists Michael Green a n d J o h n
Schwarz, who proved the consistency of the most advanced version of
Kaluza-Klein theory, called superstring theory, which postulates that all
matter consists of tiny vibrating strings. Surprisingly, the superstring theory predicts a precise n u m b e r of dimensions for space and time: ten.*
T h e advantage of ten-dimensional space is that we have " e n o u g h
r o o m " in which to a c c o m m o d a t e all four fundamental forces. Furtherm o r e , we have a simple physical picture in which to explain the confusi n g j u m b l e of subatomic particles p r o d u c e d by o u r powerful atom smashers. Over the past 30 years, h u n d r e d s of subatomic particles have been
carefully cataloged a n d studied by physicists a m o n g the debris created
by smashing together protons a n d electrons with atoms. Like b u g collectors patiently giving names to a vast collection of insects, physicists
have at times b e e n overwhelmed by the diversity and complexity of these
subatomic particles. Today, this bewildering collection of subatomic particles can be explained as m e r e vibrations of the hyperspace theory.
Traveling Through Space and Time
T h e hyperspace theory has also r e o p e n e d the question of w h e t h e r hyperspace can be used to travel t h r o u g h space a n d time. To u n d e r s t a n d this
* F r e u n d c h u c k l e s w h e n a s k e d w h e n w e w i l l b e a b l e t o see these h i g h e r d i m e n s i o n s . W e
c a n n o t see these h i g h e r d i m e n s i o n s b e c a u s e t h e y h a v e " c u r l e d u p " i n t o a t i n y b a l l s o s m a l l
t h a t t h e y c a n n o l o n g e r b e d e t e c t e d . A c c o r d i n g t o K a l u z a - K l e i n t h e o r y , t h e size o f these
c u r l e d u p d i m e n s i o n s i s c a l l e d t h e Planck length, w h i c h i s 100 b i l l i o n b i l l i o n t i m e s s m a l l e r
4
t h a n t h e p r o t o n , t o o small t o b e p r o b e d b y even b y o u r largest a t o m smasher. H i g h - e n e r g y
physicists h a d h o p e d t h a t t h e $ 1 1 b i l l i o n s u p e r c o n d u c t i n g s u p e r c o l l i d e r (SSC) ( w h i c h was
c a n c e l e d b y C o n g r e s s i n O c t o b e r 1 9 9 3 ) m i g h t h a v e b e e n able t o reveal s o m e i n d i r e c t
g l i m m e r s o f hyperspace.
17
concept, imagine a race of tiny flatworms living on the surface of a large
apple. It's obvious to these worms that their world, which they call Appleworld, is flat a n d two dimensional, like themselves. O n e worm, however,
n a m e d Columbus, is obsessed by the notion that Appleworld is somehow
finite a n d curved in something he calls the third dimension. He even
invents two new words, up a n d down, to describe motion in this invisible
third dimension. His friends, however, call h i m a fool for believing that
Appleworld could be b e n t in some unseen dimension that no o n e can
see or feel. O n e day, Columbus sets out on a long a n d a r d u o u s j o u r n e y
a n d disappears over the horizon. Eventually he returns to his starting
point, proving that the world is actually curved in the u n s e e n third
dimension. His j o u r n e y proves that Appleworld is curved in a higher
unseen dimension, the third dimension. Although weary from his travels, Columbus discovers that there is yet a n o t h e r way to travel between
distant points on the apple: By burrowing into the apple, he can carve
a tunnel, creating a convenient shortcut to distant lands. These tunnels,
which considerably r e d u c e the time and discomfort of a long journey,
he calls wormholes. They demonstrate that the shortest path between two
points is n o t necessarily a straight line, as h e ' s b e e n taught, b u t a wormhole.
O n e strange effect discovered by Columbus is that when he enters
o n e of these tunnels a n d exits at the o t h e r end, he finds himself back
in the past. Apparently, these wormholes c o n n e c t parts of the
apple where time beats at different rates. Some of the worms even
claim that these wormholes can be m o l d e d into a workable time
machine.
Later, Columbus makes an even m o r e m o m e n t o u s discovery—his
Appleworld is actually n o t the only o n e in the universe. It is but o n e
apple in a large apple orchard. His apple, he finds out, coexists with
h u n d r e d s of others, some with worms like themselves, a n d some
without worms. U n d e r certain rare circumstances, he conjectures, it
may even be possible to j o u r n e y between the different apples in the
orchard.
We h u m a n beings are like the flatworms. C o m m o n sense tells us that
o u r world, like their apple, is flat a n d three dimensional. No matter
where we go with o u r rocket ships, the universe seems flat. However, the
fact that o u r universe, like Appleworld, is curved in an unseen dimension
beyond o u r spatial c o m p r e h e n s i o n has b e e n experimentally verified by
a n u m b e r of rigorous experiments. These experiments, performed on
the path of light beams, show that starlight is b e n t as it moves across the
universe.
18
Multiply Connected Universes
W h e n we wake up in the m o r n i n g a n d o p e n the window to let in some
fresh air, we expect to see the front yard. We do not expect to face the
towering pyramids of Egypt. Similarly, when we o p e n the front door, we
expect to see the cars on the street, n o t the craters a n d dead volcanoes
of a bleak, l u n a r landscape. Without even thinking a b o u t it, we assume
that we can safely o p e n windows or doors without being scared out of
o u r wits. O u r world, fortunately, is n o t a Steven Spielberg movie. We act
on a deeply ingrained prejudice (which is invariably correct) that o u r
world is simply connected, that o u r windows a n d doorways are n o t
entrances to wormholes connecting o u r h o m e to a far-away universe. (In
ordinary space, a lasso of r o p e can always be shrunk to a point. If this is
possible, t h e n the space is called simply connected. However, if the lasso
is placed a r o u n d the e n t r a n c e of the wormhole, then it c a n n o t be s h r u n k
to a point. T h e lasso, in fact, enters the wormhole. Such spaces, where
lassos are n o t contractible, are called multiply connected. Although the
b e n d i n g of o u r universe in an u n s e e n dimension has b e e n experimentally measured, the existence of wormholes a n d whether o u r universe is
multiply c o n n e c t e d or not is still a topic of scientific controversy.)
Mathematicians dating back to Georg B e r n h a r d R i e m a n n have studied the properties of multiply c o n n e c t e d spaces in which different
regions of space a n d time are spliced together. And physicists, w h o once
t h o u g h t this was merely an intellectual exercise, are now seriously studying multiply c o n n e c t e d worlds as a practical model of o u r universe.
These models are the scientific analogue of Alice's looking glass. W h e n
Lewis Carroll's White Rabbit falls down the rabbit hole to e n t e r Wonderland, he actually falls down a wormhole.
Wormholes can be visualized with a sheet of p a p e r a n d a pair of
scissors: Take a piece of paper, cut two holes in it, a n d then r e c o n n e c t
the two holes with a long tube (Figure 1.1). As long as you avoid walking
into the wormhole, o u r world seems perfectly n o r m a l . T h e usual laws of
geometry taught in school are obeyed. However, if you fall into the
wormhole, you are instantly transported to a different region of space
a n d time. Only by retracing your steps a n d falling back into t h e wormhole can you return to your familiar world.
Time Travel and Baby Universes
Although wormholes provide a fascinating area of research, p e r h a p s the
most intriguing concept to emerge from this discussion of hyperspace
19
Figure 1.1. Parallel universes may be graphically represented by two parallel
planes. Normally, they never interact with each other. However, at times wormholes or tubes may open up between them, perhaps making communication and
travel possible between them. This is now the subject of intense interest among
theoretical physicists.
is the question of time travel. In the film Back to the Future, Michael J.
Fox j o u r n e y s back in time a n d meets his parents as teenagers before they
were married. Unfortunately, his m o t h e r falls in love with him a n d spurns
his father, raising the ticklish question of how he will be b o r n if his
parents never marry a n d have children.
Traditionally, scientists have held a dim opinion of anyone who
raised the question of time travel. Causality (the notion that every effect
is p r e c e d e d , n o t followed, by a cause) is firmly enshrined in the foun-
20
dations of m o d e r n science. However, in the physics of wormholes, "acausal" effects show up repeatedly. In fact, we have to make strong assumptions in o r d e r to prevent time travel from taking place. T h e main
p r o b l e m is that wormholes may c o n n e c t n o t only two distant points in
space, b u t also t h e future with the past.
In 1988, physicist Kip T h o r n e of t h e California Institute of Technology a n d his collaborators m a d e t h e astonishing (and risky) claim that
time travel is i n d e e d n o t only possible, but probable u n d e r certain conditions. They published their claim n o t in an obscure " f r i n g e " j o u r n a l ,
b u t in the prestigious Physical Review Letters. This m a r k e d the first time
that reputable physicists, a n d n o t crackpots, were scientifically advancing
a claim about changing the course of time itself. T h e i r a n n o u n c e m e n t
was based on the simple observation that a w o r m h o l e connects two
regions that exist in different time periods. T h u s t h e w o r m h o l e may
c o n n e c t the present to the past. Since travel t h r o u g h the w o r m h o l e is
nearly instantaneous, o n e could use the wormhole to go backward in
time. Unlike the m a c h i n e portrayed in H. G. Wells's The Time Machine,
however, which could hurl the protagonist h u n d r e d s of thousands of
years into England's distant future with the simple twist of a dial, a wormhole may require vast a m o u n t s of energy for its creation, beyond what
will be technically possible for centuries to c o m e .
A n o t h e r bizarre c o n s e q u e n c e of wormhole physics is the creation of
"baby universes" in t h e laboratory. We are, of course, u n a b l e to re-create
the Big Bang a n d witness the birth of o u r universe. However, Alan Guth
of the Massachusetts Institute of Technology, who has m a d e many
i m p o r t a n t contributions in cosmology, shocked many physicists a few
years ago w h e n he claimed that the physics of wormholes may m a k e it
possible to create a baby universe of o u r own in the laboratory. By concentrating intense heat a n d energy in a chamber, a wormhole may eventually o p e n u p , serving as an umbilical cord connecting o u r universe to
a n o t h e r , m u c h smaller universe. If possible, it would give a scientist an
u n p r e c e d e n t e d view of a universe as it is created in the laboratory.
Mystics and Hyperspace
Some of these concepts are not new. For the past several centuries, mystics a n d philosophers have speculated a b o u t the existence of o t h e r universes a n d tunnels between t h e m . They have long b e e n fascinated by
the possible existence of o t h e r worlds, undetectable by sight or sound,
yet coexisting with o u r universe. They have been intrigued by the pos-
21
sibility that these u n e x p l o r e d , n e t h e r worlds may even be tantalizingly
close, in fact s u r r o u n d i n g us a n d p e r m e a t i n g us everywhere we move,
yet j u s t beyond o u r physical grasp a n d eluding o u r senses. Such idle talk,
however, was ultimately useless because there was no practical way in
which to mathematically express a n d eventually test these ideas.
Gateways between o u r universe and o t h e r dimensions are also a
favorite literary device. Science-fiction writers find higher dimensions to
be an indispensable tool, using t h e m as a m e d i u m for interstellar travel.
Because of the astronomical distances separating the stars in the heavens, science-fiction writers use higher dimensions as a clever shortcut
between the stars. Instead of taking the long, direct route to o t h e r galaxies, rockets merely zip along in hyperspace by warping the space
a r o u n d them. For instance, in the film Star Wars, hyperspace is a refuge
where Luke Skywalker can safely evade the Imperial Starships of the
Empire. In the television series "Star Trek: D e e p Space N i n e , " a wormhole opens up near a r e m o t e space station, making it possible to span
e n o r m o u s distances across the galaxy within seconds. T h e space station
suddenly becomes the center of intense intergalactic rivalry over who
should control such a vital link to o t h e r parts of the galaxy.
Ever since Flight 19, a g r o u p of U.S. military t o r p e d o bombers, vanished in the Caribbean 30 years ago, mystery writers too have used higher
dimensions as a convenient solution to the puzzle of the B e r m u d a Triangle, or Devil's Triangle. Some have conjectured that airplanes a n d
ships disappearing in the B e r m u d a Triangle actually e n t e r e d some sort
of passageway to a n o t h e r world.
T h e existence of these elusive parallel worlds has also p r o d u c e d endless religious speculation over t h e centuries. Spiritualists have w o n d e r e d
w h e t h e r the souls of d e p a r t e d loved ones drifted into a n o t h e r dimension. T h e seventeenth-century British philosopher Henry More argued
that ghosts a n d spirits did i n d e e d exist a n d claimed that they inhabited
the fourth dimension. In Enchiridion Metaphysicum (1671), he argued for
the existence of a n e t h e r realm beyond o u r tangible senses that served
as a h o m e for ghosts a n d spirits.
Nineteenth-century theologians, at a loss to locate heaven and hell,
p o n d e r e d w h e t h e r they might be found in a h i g h e r dimension. Some
wrote about a universe consisting of three parallel planes: the earth,
heaven, a n d hell. God himself, according to the theologian Arthur Willink, found his h o m e in a world far removed from these three planes;
he lived in infinite-dimensional space.
Interest in higher dimensions r e a c h e d its peak between 1870 a n d
1920, when the "fourth d i m e n s i o n " (a spatial dimension, different from
22
what we know as the fourth dimension of time) seized the public imagination a n d gradually cross-fertilized every b r a n c h of the arts a n d sciences, b e c o m i n g a m e t a p h o r for the strange a n d mysterious. T h e fourth
dimension a p p e a r e d in the literary works of Oscar Wilde, Fyodor Dostoyevsky, Marcel Proust, H. G. Wells, and J o s e p h Conrad; it inspired
some of the musical works of Alexander Scriabin, Edgard Varese, a n d
George Antheil. It fascinated such diverse personalities as psychologist
William James, literary figure G e r t r u d e Stein, a n d revolutionary socialist
Vladimir Lenin.
T h e fourth dimension also inspired the works of Pablo Picasso a n d
Marcel D u c h a m p a n d heavily influenced the development of Cubism
a n d Expressionism, two of the most influential art movements in this
century. Art historian Linda Dalrymple H e n d e r s o n writes, "Like a Black
Hole, 'the fourth dimension' possessed mysterious qualities that could
n o t be completely u n d e r s t o o d , even by the scientists themselves. Yet, the
impact of 'the fourth d i m e n s i o n ' was far m o r e comprehensive t h a n that
of Black Holes or any o t h e r m o r e recent scientific hypothesis except
Relativity Theory after 1919."
Similarly, mathematicians have long b e e n intrigued by alternative
forms of logic a n d bizarre geometries that defy every convention of comm o n sense. For example, the mathematician Charles L. Dodgson, who
taught at Oxford University, delighted generations of schoolchildren by
writing books—as Lewis Carroll—that incorporate these strange mathematical ideas. W h e n Alice falls down a rabbit hole or steps t h r o u g h the
looking glass, she enters W o n d e r l a n d , a strange place where Cheshire
cats disappear (leaving only their smile), magic m u s h r o o m s t u r n child r e n into giants, a n d Mad Hatters celebrate " u n b i r t h d a y s . " T h e looking
glass somehow connects Alice's world with a strange land where everyo n e speaks in riddles a n d c o m m o n sense isn't so c o m m o n .
5
S o m e of the inspiration for Lewis Carroll's ideas most likely came
from the great nineteenth-century G e r m a n mathematician Georg Bernh a r d Riemann, who was the first to lay the mathematical foundation of
geometries in higher-dimensional space. Riemann c h a n g e d the course
of mathematics for the next century by demonstrating that these universes, as strange as they may a p p e a r to t h e layperson, are completely
self-consistent a n d obey their own i n n e r logic. To illustrate some of these
ideas, think of stacking many sheets of paper, o n e on top of another.
Now imagine that each sheet represents an entire world a n d that each
world obeys its own physical laws, different from those of all the other
worlds. O u r universe, then, would n o t be alone, b u t would be o n e of
23
many possible parallel worlds. Intelligent beings might inhabit some of
these planes, completely unaware of the existence of the others. On o n e
sheet of paper, we might have Alice's bucolic English countryside. On
a n o t h e r sheet might be a strange world populated by mythical creatures
in the world of W o n d e r l a n d .
Normally, life proceeds on each of these parallel planes i n d e p e n d e n t
of the others. On rare occasions, however, the planes may intersect and,
for a brief m o m e n t , tear the fabric of space itself, which o p e n s up a
h o l e — o r gateway—between these two universes. Like the w o r m h o l e
a p p e a r i n g in "Star Trek: D e e p Space N i n e , " these gateways make travel
possible between these worlds, like a cosmic bridge linking two different
universes or two points in the same universe (Figure 1.2). Not surprisingly, Carroll found children m u c h m o r e o p e n to these possibilities t h a n
adults, whose prejudices a b o u t space a n d logic b e c o m e m o r e rigid over
time. In fact, R i e m a n n ' s theory of higher dimensions, as interpreted by
Lewis Carroll, has b e c o m e a p e r m a n e n t p a r t of children's literature a n d
folklore, giving birth to o t h e r children's classics over the decades, such
as Dorothy's Land of Oz a n d Peter Pan's Never Never Land.
Without any experimental confirmation or compelling physical motivation, however, these theories of parallel worlds languished as a b r a n c h
of science. Over 2 millennia, scientists have occasionally picked up the
notion of higher dimensions, only to discard it as an untestable a n d
therefore silly idea. Although R i e m a n n ' s theory of higher geometries
was mathematically intriguing, it was dismissed as clever b u t useless. Scientists willing to risk their reputations on higher dimensions soon found
themselves ridiculed by the scientific community. Higher-dimensional
space b e c a m e the last refuge for mystics, cranks, a n d charlatans.
In this book, we will study the work of these p i o n e e r i n g mystics,
mainly because they devised ingenious ways in which a nonspecialist
could "visualize" what higher-dimensional objects might look like.
These tricks will prove useful to u n d e r s t a n d how these higher-dimensional theories may be grasped by the general public.
By studying the work of these early mystics, we also see m o r e clearly
what was missing from their research. We see that their speculations
lacked two i m p o r t a n t concepts: a physical a n d a mathematical principle.
From the perspective of m o d e r n physics, we now realize that the missing
physical principle is that hyperspace simplifies the laws of n a t u r e , providing the possibility of unifying all the forces of n a t u r e by purely geometric
arguments. T h e missing mathematical principle is called field theory, which
is the universal mathematical language of theoretical physics.
Figure 1.2. Wormholes may connect a universe with itself, perhaps providing a
means of interstellar travel. Since wormholes may connect two different time eras,
they may also provide a means for time travel. Wormholes may also connect an
infinite series of parallel universes. The hope is that the hyperspace theory will be
able to determine whether wormholes are physically possible or merely a mathematical curiosity.
24
25
Field Theory: The Language of Physics
Fields were first introduced by the great nineteenth-century British scientist Michael Faraday. T h e son of a p o o r blacksmith, Faraday was a selftaught genius who c o n d u c t e d elaborate experiments on electricity a n d
magnetism. He visualized "lines of force" that, like long vines spreading
from a plant, e m a n a t e d from magnets a n d electric charges in all directions a n d filled up all of space. With his instruments, Faraday could
measure the strength of these lines of force from a magnetic or an electric charge at any point in his laboratory. T h u s he could assign a series
of n u m b e r s (the strength a n d direction of the force) to that point (and
any point in space). He christened the totality of these n u m b e r s at any
point in space, treated as a single entity, a field. ( T h e r e is a famous story
c o n c e r n i n g Michael Faraday. Because his fame h a d spread far a n d wide,
he was often visited by curious bystanders. W h e n o n e asked what his
work was good for, he answered, " W h a t is the use of a child? It grows to
be a m a n . " O n e day, William Gladstone, t h e n Chancellor of the Exchequer, visited Faraday in his laboratory. Knowing n o t h i n g a b o u t science,
Gladstone sarcastically asked Faraday what use the h u g e electrical contraptions in his laboratory could possibly have for England. Faraday
replied, "Sir, I know n o t what these machines will be used for, b u t I am
sure that o n e day you will tax t h e m . " Today, a large portion of the total
wealth of England is invested in the fruit of Faraday's labors.)
Simply put, a field is a collection of n u m b e r s defined at every p o i n t
in space that completely describes a force at that point. For example,
t h r e e n u m b e r s at each point in space can describe the intensity a n d
direction of the magnetic lines of force. A n o t h e r three n u m b e r s everywhere in space can describe the electric field. Faraday got this concept
when he t h o u g h t of a "field" plowed by a farmer. A farmer's field occupies a two-dimensional region of space. At each p o i n t in the farmer's
field, o n e can assign a series of n u m b e r s (which describe, for example,
how many seeds there are at that p o i n t ) . Faraday's field, however, occupies a three-dimensional region of space. At each point, there is a series
of six n u m b e r s that describes b o t h the magnetic a n d electric lines of
force.
What makes Faraday's field concept so powerful is that all forces of
n a t u r e can be expressed as a field. However, we n e e d o n e m o r e ingredient before we can u n d e r s t a n d the n a t u r e of any force: We must be
able to write down the equations that these fields obey. T h e progress of
the past h u n d r e d years in theoretical physics can be succinctly summarized as the search for the field equations of the forces of n a t u r e .
26
For example, in the 1860s, Scottish physicist J a m e s Clerk Maxwell
wrote down the field equations for electricity a n d magnetism. In 1915,
Einstein discovered the field equations for gravity. After i n n u m e r a b l e
false starts, the field equations for the subatomic forces were finally written down in the 1970s, utilizing the earlier work of C. N. Yang a n d his
student R. L. Mills. These fields, which govern the interaction of all
subatomic particles, are now called Yang-Mills fields. However, the puzzle
that has s t u m p e d physicists within this century is why t h e subatomic field
equations look so vastly different from the field equations of Einstein—
that is, why the nuclear force seems so different from gravity. Some of
the greatest minds in physics have tackled this problem, only to fail.
Perhaps the reason for their failure is that they were trapped by comm o n sense. Confined to three or four dimensions, the field equations
of the subatomic world a n d gravitation are difficult to unify. T h e advantage of the hyperspace theory is that the Yang-Mills field, Maxwell's field,
a n d Einstein's field can all be placed comfortably within the hyperspace
field. We see that these fields fit together precisely within the hyperspace
field like pieces in a jigsaw puzzle. T h e o t h e r advantage of field theory
is that it allows us to calculate the precise energies at which we can expect
space and time to form wormholes. Unlike the ancients, therefore, we
have the mathematical tools to guide us in building the machines that
may o n e day b e n d space a n d time to o u r whims.
The Secret of Creation
Does this m e a n that big-game h u n t e r s can now start organizing safaris
to the Mesozoic era to bag large dinosaurs? No. T h o r n e , Guth, a n d
F r e u n d will all tell you that the energy scale necessary to investigate these
anomalies in space is far beyond anything available on earth. Freund
reminds us that the energy necessary to p r o b e the tenth dimension is a
quadrillion times larger than the energy that can be p r o d u c e d by o u r
largest atom smasher.
Twisting space-time into knots requires energy on a scale that will
n o t be available within the next several centuries or even millennia—if
ever. Even if all the nations of the world were to b a n d together to build
a m a c h i n e that could p r o b e hyperspace, they would ultimately fail. And,
as Guth points out, the temperatures necessary to create a baby universe
in the laboratory is 1,000 trillion trillion degrees, far in excess of anything available to us. In fact, that t e m p e r a t u r e is m u c h greater than
anything found in the interior of a star. So, although it is possible that
27
Einstein's laws and the laws of q u a n t u m theory might allow for time
travel, this is n o t within the capabilities of earthlings like us, who can
barely escape the feeble gravitational field of o u r own planet. While we
can marvel at the implications of wormhole research, realizing its potential is strictly reserved for advanced extraterrestrial civilizations.
T h e r e was only o n e period of time when energy on this e n o r m o u s
scale was readily available, a n d that was at the instant of Creation. In
fact, the hyperspace theory c a n n o t be tested by o u r largest atom smashers because the theory is really a theory of Creation. Only at the instant
of the Big Bang do we see the full power of the hyperspace theory coming into play. This raises the exciting possibility that the hyperspace theory may unlock the secret of t h e origin of the universe.
I n t r o d u c i n g higher dimensions may be essential for prying loose the
secrets of Creation. According to this theory, before the Big Bang, o u r
cosmos was actually a perfect ten-dimensional universe, a world where
interdimensional travel was possible. However, this ten-dimensional
world was unstable, a n d eventually it " c r a c k e d " in two, creating two
separate universes: a four- and a six-dimensional universe. T h e universe
in which we live was b o r n in that cosmic cataclysm. O u r four-dimensional
universe e x p a n d e d explosively, while o u r twin six-dimensional universe
contracted violently, until it shrank to almost infinitesimal size. This
would explain the origin of the Big Bang. If correct, this theory demonstrates that the rapid expansion of the universe was just a r a t h e r m i n o r
aftershock of a m u c h greater cataclysmic event, the cracking of space
a n d time itself. T h e energy that drives the observed expansion of the
universe is t h e n found in the collapse of ten-dimensional space a n d time.
According to the theory, the distant stars a n d galaxies are receding from
us at astronomical speeds because of the original collapse of ten-dimensional space a n d time.
This theory predicts that o u r universe still has a dwarf twin, a comp a n i o n universe that has curled up into a small six-dimensional ball that
is too small to be observed. This six-dimensional universe, far from b e i n g
a useless a p p e n d a g e to o u r world, may ultimately be our salvation.
Evading the Death of the Universe
It is often said that the only constants of h u m a n society are death a n d
taxes. For the cosmologist, the only certainty is that the universe will o n e
day die. Some believe that the ultimate death of the universe will c o m e
in the form of the Big Crunch. Gravitation will reverse the cosmic expan-
28
sion generated by the Big Bang a n d pull the stars a n d galaxies back,
o n c e again, into a primordial mass. As the stars contract, temperatures
will rise dramatically until all matter a n d energy in the universe are concentrated into a colossal fireball that will destroy the universe as we know
it. All life forms will be crushed beyond recognition. T h e r e will be no
escape. Scientists a n d philosophers, like Charles Darwin and Bertrand
Russell, have written mournfully a b o u t the futility of o u r pitiful existence, knowing that o u r civilization will inexorably die when o u r world
ends. T h e laws of physics, apparently, have issued the final, irrevocable
d e a t h warrant for all intelligent life in the universe.
According to the late Columbia University physicist Gerald Feinberg,
there is o n e , a n d perhaps only o n e , h o p e of avoiding the final calamity.
He speculated that intelligent life, eventually mastering the mysteries of
higher-dimensional space over billions of years, will use the o t h e r dimensions as an escape hatch from the Big Crunch. In the final m o m e n t s of
the collapse of o u r universe, o u r sister universe will o p e n up once again,
a n d interdimensional travel will b e c o m e possible. As all matter is
crushed in the final m o m e n t s before doomsday, intelligent life forms
may be able to tunnel into higher-dimensional space or an alternative
universe, avoiding the seemingly inevitable death of o u r universe. T h e n ,
from their sanctuary in higher-dimensional space, these intelligent life
forms may be able to witness the death of the collapsing universe in a
fiery cataclysm. As o u r h o m e universe is crushed beyond recognition,
temperatures will rise violently, creating yet a n o t h e r Big Bang. From
their vantage point in hyperspace, these intelligent life forms will have
front-row seats to the rarest of all scientific p h e n o m e n a , the creation of
a n o t h e r universe a n d of their new h o m e .
Masters of Hyperspace
Although field theory shows that the energy necessary to create these
marvelous distortions of space a n d time is far beyond anything that mode r n civilization can muster, this raises two i m p o r t a n t questions: How l o n g
will it take for o u r civilization, which is growing exponentially in knowledge a n d power, to reach the point of harnessing the hyperspace theory?
A n d what a b o u t o t h e r intelligent life forms in the universe, who may
already have r e a c h e d that point?
What makes this discussion interesting is that serious scientists have
tried to quantify the progress of civilizations far into the future, when
space travel will have b e c o m e c o m m o n p l a c e a n d n e i g h b o r i n g star sys-
29
terns or even galaxies will have b e e n colonized. Although the energy
scale necessary to manipulate hyperspace is astronomically large, these
scientists point o u t that scientific growth will probably c o n t i n u e to rise
exponentially over the n e x t centuries, exceeding the capabilities of
h u m a n minds to grasp it. Since World War II, the sum total of scientific
knowledge has d o u b l e d every 10 to 20 or so years, so the progress of
science a n d technology into the twenty-first century may surpass o u r
wildest expectations. Technologies that can only be d r e a m e d of today
may b e c o m e c o m m o n p l a c e in the next century. Perhaps t h e n o n e can
discuss the question of when we might b e c o m e masters of hyperspace.
T i m e travel. Parallel universes. Dimensional windows.
By themselves, these concepts stand at the edge of o u r u n d e r s t a n d i n g
of the physical universe. However, because the hyperspace theory is a
g e n u i n e field theory, we eventually expect it to p r o d u c e numerical
answers d e t e r m i n i n g w h e t h e r these intriguing concepts are possible. If
the theory produces nonsensical answers that disagree with physical
data, then it must be discarded, no matter how elegant its mathematics.
In the final analysis, we are physicists, n o t philosophers. But if it proves
to be correct a n d explains the symmetries of m o d e r n physics, t h e n it will
usher in a revolution perhaps equal to the Copernican or Newtonian
revolutions.
To have an intuitive u n d e r s t a n d i n g of these concepts, however, it is
i m p o r t a n t to start at the beginning. Before we can feel comfortable with
ten dimensions, we must learn how to manipulate four spatial dimensions. Using historical examples, we will explore the ingenious attempts
m a d e by scientists over the decades to give a tangible, visual representation of higher-dimensional space. T h e first part of the book, therefore,
will stress the history b e h i n d the discovery of higher-dimensional space,
b e g i n n i n g with the mathematician who started it all, Georg B e r n h a r d
Riemann. Anticipating the next century of scientific progress, Riemann
was t h e first to state that n a t u r e finds its natural h o m e in the geometry
of higher-dimensional space.
Mathematicians
and Mystics
M a g i c i s a n y sufficiently a d v a n c e d t e c h n o l o g y .
Arthur C. Clarke
O
N J u n e 10, 1854, a new geometry was b o r n .
T h e theory of higher dimensions was introduced when Georg
B e r n h a r d Riemann gave his celebrated lecture before the faculty of the
University of Gottingen in Germany. In o n e masterful stroke, like opening up a musty, d a r k e n e d r o o m to the brilliance of a warm s u m m e r ' s
sun, R i e m a n n ' s lecture exposed the world to the dazzling properties of
higher-dimensional space.
His profoundly i m p o r t a n t a n d exceptionally elegant essay, " O n the
Hypotheses Which Lie at the Foundation of Geometry," toppled the
pillars of classical Greek geometry, which h a d successfully weathered all
assaults by skeptics for 2 millennia. T h e old geometry of Euclid, in which
all geometric figures are two or three dimensional, came tumbling down
as a new Riemannian geometry e m e r g e d from its ruins. T h e R i e m a n n i a n
revolution would have vast implications for the future of the arts a n d
sciences. Within 3 decades of his talk, the "mysterious fourth dimens i o n " would influence the evolution of art, philosophy, a n d literature
in E u r o p e . Within 6 decades of R i e m a n n ' s lecture, Einstein would use
four-dimensional Riemannian geometry to explain the creation of the
universe a n d its evolution. And 130 years after his lecture, physicists
30
Mathematicians and Mystics
31
would use ten-dimensional geometry to a t t e m p t to unite all the laws of
the physical universe. T h e core of R i e m a n n ' s work was the realization
that physical laws simplify in higher-dimensional space, the very t h e m e
of this book.
Brilliance Amid Poverty
Ironically, R i e m a n n was the least likely person to usher in such a d e e p
a n d thorough-going revolution in mathematical a n d physical thought.
He was excruciatingly, almost pathologically, shy a n d suffered repeated
nervous breakdowns. He also suffered from the twin ailments that have
r u i n e d the lives of so many of the world's great scientists t h r o u g h o u t
history: abject poverty a n d consumption (tuberculosis). His personality
a n d t e m p e r a m e n t showed n o t h i n g of the breath-taking boldness, sweep,
a n d s u p r e m e confidence typical of his work.
R i e m a n n was b o r n in 1826 in Hanover, Germany, the son of a p o o r
L u t h e r a n pastor, the second of six children. His father, who fought in
t h e Napoleonic Wars, struggled as a country pastor to feed a n d clothe
his large family. As biographer E. T. Bell notes, " t h e frail health a n d
early deaths of most of the R i e m a n n children were t h e result of u n d e r n o u r i s h m e n t in their youth a n d were n o t d u e to p o o r stamina. T h e
m o t h e r also died before h e r children were g r o w n . "
1
At a very early age, R i e m a n n exhibited his famous traits: fantastic
calculational ability, c o u p l e d with timidity, a n d a life-long h o r r o r of any
public speaking. Painfully shy, he was the butt of cruel jokes by o t h e r
boys, causing h i m to retreat further into the intensely private world of
mathematics.
He also was fiercely loyal to his family, straining his p o o r health a n d
constitution to buy presents for his parents a n d especially for his beloved
sisters. To please his father, R i e m a n n set o u t to b e c o m e a student of
theology. His goal was to get a paying position as a pastor as quickly as
possible to h e l p with his family's abysmal finances. (It is difficult to imagine a m o r e improbable scenario than that of a tongue-tied, timid young
boy imagining that he could deliver fiery, passionate sermons railing
against sin a n d driving out the devil.)
In high school, he studied the Bible intensely, but his
drifted back to mathematics; he even tried to provide
proof of the correctness of Genesis. He also learned so
kept outstripping the knowledge of his instructors, who
sible to k e e p up with the boy. Finally, the principal of
thoughts always
a mathematical
quickly that he
found it imposhis school gave
32
Riemann a p o n d e r o u s b o o k to keep h i m occupied. T h e b o o k was
Adrien-Marie Legendre's Theory of Numbers, a h u g e 859-page masterpiece, the world's most advanced treatise on the difficult subject of n u m ber theory. Riemann devoured the b o o k in 6 days.
W h e n his principal asked, " H o w far did you r e a d ? " the young Riem a n n replied, " T h a t is certainly a wonderful book. I have mastered it."
Not really believing the bravado of this youngster, the principal several
m o n t h s later asked obscure questions from the book, which Riemann
answered perfectly.
2
Beset by the daily struggle to p u t food on the table, Riemann's father
might have sent the boy to do menial labor. Instead, he scraped together
e n o u g h funds to send his 19-year-old son to the renowned University of
Gottingen, where he first m e t Carl Friedrich Gauss, the acclaimed
" P r i n c e of Mathematicians," o n e of the greatest mathematicians of all
time. Even today, if you ask any mathematician to rank the three most
famous mathematicians in history, the n a m e s of Archimedes, Isaac Newton, a n d Carl Gauss will invariably appear.
Life for Riemann, however, was an endless series of setbacks a n d
hardships, overcome only with the greatest difficulty a n d by straining his
frail health. Each triumph was followed by tragedy a n d defeat. For example, j u s t as his fortunes began to improve a n d he u n d e r t o o k his formal
studies u n d e r Gauss, a full-scale revolution swept Germany. T h e working
class, long suffering u n d e r i n h u m a n living conditions, rose up against
the government, with workers in scores of cities t h r o u g h o u t Germany
taking up arms. T h e demonstrations and uprisings in early 1848 inspired
the writings of a n o t h e r G e r m a n , Karl Marx, a n d deeply affected the
course of revolutionary movements t h r o u g h o u t E u r o p e for the next 50
years.
With all of Germany swept up in turmoil, Riemann's studies were
interrupted. He was inducted into the student corps, where he h a d the
dubious h o n o r of s p e n d i n g 16 weary h o u r s protecting s o m e o n e even
m o r e terrified than h e : the king, who was quivering with fear in his
royal palace in Berlin, trying to hide from the wrath of the working
class.
Beyond Euclidean Geometry
Not only in Germany, but in mathematics, too, fierce revolutionary winds
were blowing. T h e p r o b l e m that riveted Riemann's interest was the
i m p e n d i n g collapse of yet a n o t h e r bastion of authority, Euclidean geom-
33
etry, which holds that space is three dimensional. F u r t h e r m o r e , this
three-dimensional space is "flat" (in flat space, t h e shortest distance
between two points is a straight line; this omits the possibility that space
can be curved, as on a s p h e r e ) .
In fact, after the Bible, Euclid's Elements was probably the most influential book of all time. For 2 millennia, the keenest minds of Western
civilization have marveled at its elegance a n d the beauty of its geometry.
T h o u s a n d s of the finest cathedrals in E u r o p e were erected according to
its principles. In retrospect, perhaps it was too successful. Over the centuries, it b e c a m e something of a religion; anyone who dared to propose
curved space or higher dimensions was relegated to the ranks of crackpots or heretics. For untold generations, schoolchildren have wrestled
with the t h e o r e m s of Euclid's geometry: that the circumference of a
circle is pi times the diameter, a n d that the angles within a triangle a d d
up to 180 degrees. However, try as they might, the finest mathematical
minds for several centuries could not prove these deceptively simple
propositions. In fact, the mathematicians of Europe began to realize that
even Euclid's Elements, which h a d b e e n revered for 2,300 years, was
incomplete. Euclid's geometry was still viable if o n e stayed within the
confines of flat surfaces, but if o n e strayed into t h e world of curved
surfaces, it was actually incorrect.
To Riemann, Euclid's geometry was particularly sterile when comp a r e d with the rich diversity of the world. Nowhere in the natural world
do we see the flat, idealized geometric figures of Euclid. Mountain
ranges, ocean waves, clouds, a n d whirlpools are n o t perfect circles, triangles, a n d squares, but are curved objects that b e n d a n d twist in infinite
diversity.
T h e time was ripe for a revolution, b u t who would lead it a n d what
would replace the old geometry?
The Rise of Riemannian Geometry
Riemann rebelled against the a p p a r e n t mathematical precision of Greek
geometry, whose foundation, he discovered, ultimately was based on the
shifting sand of c o m m o n sense a n d intuition, n o t the firm g r o u n d of
logic.
It is obvious, said Euclid, that a point has no dimension at all. A line
has o n e dimension: length. A plane has two dimensions: length a n d
breadth. A solid has t h r e e dimensions: length, breadth, a n d height. And
t h e r e it stops. N o t h i n g has four dimensions. These sentiments were ech-
34
oed by the philosopher Aristotle, who apparently was the first person to
state categorically that the fourth spatial dimension is impossible. In On
Heaven, he wrote, " T h e line has m a g n i t u d e in o n e way, the plane in two
ways, a n d the solid in three ways, a n d beyond these there is no o t h e r
m a g n i t u d e because t h e three are all." F u r t h e r m o r e , in A . D . 150, the
astronomer Ptolemy from Alexandria went beyond Aristotle a n d offered,
in his book On Distance, the first ingenious " p r o o f that t h e fourth
dimension is impossible.
First, he said, draw three mutually perpendicular lines. For example,
the corner of a cube consists of three mutually perpendicular lines.
T h e n , he argued, try to draw a fourth line that is perpendicular to the
o t h e r three lines. No matter how o n e tries, he reasoned, four mutually
perpendicular lines are impossible to draw. Ptolemy claimed that a
fourth perpendicular line is "entirely without measure a n d without definition." T h u s the fourth dimension is impossible.
What Ptolemy actually proved was that it is impossible to visualize the
fourth dimension with o u r three-dimensional brains. (In fact, today we
know that many objects in mathematics c a n n o t be visualized b u t can be
shown to exist.) Ptolemy may go down in history as the m a n who opposed
two great ideas in science: the sun-centered solar system a n d t h e fourth
dimension.
Over the centuries, in fact, some mathematicians went o u t of their
way to d e n o u n c e the fourth dimension. In 1685, the mathematician
J o h n Wallis polemicized against the concept, calling it a " M o n s t e r in
Nature, less possible than a Chimera or Centaure. . . . Length, Breadth,
a n d Thickness, take up the whole of Space. N o r can Fansie imagine how
there should be a Fourth Local Dimension beyond these T h r e e . " For
several thousand years, mathematicians would repeat this simple b u t
fatal mistake, that the fourth dimension c a n n o t exist because we cannot
picture it in o u r minds.
3
The Unity of Ail Physical Law
T h e decisive break with Euclidean geometry came w h e n Gauss asked his
student Riemann to p r e p a r e an oral presentation on the "foundation
of geometry." Gauss was keenly interested in seeing if his student could
develop an alternative to Euclidean geometry. (Decades before, Gauss
h a d privately expressed d e e p a n d extensive reservations a b o u t Euclidean
geometry. He even spoke to his colleagues of hypothetical " b o o k w o r m s "
that might live entirely on a two-dimensional surface. He spoke of gen-
35
eralizing this to the geometry of higher-dimensional space. However,
being a deeply conservative m a n , he never published any of his work on
h i g h e r dimensions because of the outrage it would create a m o n g the
narrow-minded, conservative old guard. He derisively called t h e m
" B o e o t i a n s " after a mentally retarded Greek tribe. )
4
Riemann, however, was terrified. This timid man, terrified of public
speaking, was being asked by his m e n t o r to prepare a lecture before the
entire faculty on the most difficult mathematical p r o b l e m of the century.
Over t h e n e x t several m o n t h s , Riemann began painfully developing
the theory of higher dimensions, straining his health to the point of a
nervous breakdown. His stamina further deteriorated because of his dismal financial situation. He was forced to take low-paying tutoring j o b s
to provide for his family. F u r t h e r m o r e , he was b e c o m i n g sidetracked
trying to explain problems of physics. In particular, he was helping
a n o t h e r professor, Wilhelm Weber, c o n d u c t experiments in a fascinating new field of research, electricity.
Electricity, of course, h a d b e e n known to the ancients in the form of
lightning a n d sparks. But in the early n i n e t e e n t h century, this p h e n o m e n o n became the central focus of physics research. In particular, the
discovery that passing a c u r r e n t of wire across a compass needle can
m a k e the n e e d l e spin riveted t h e attention of the physics community.
Conversely, moving a bar m a g n e t across a wire can induce an electric
c u r r e n t in the wire. (This is called Faraday's Law, a n d today all electric
generators a n d transformers—and h e n c e m u c h of the foundation of
m o d e r n technology—are based on this principle.)
To Riemann, this p h e n o m e n o n indicated that electricity a n d magnetism are somehow manifestations of t h e same force. R i e m a n n was
excited by the new discoveries a n d was convinced that he could give a
mathematical explanation that would unify electricity a n d magnetism.
He immersed himself in Weber's laboratory, convinced that the new
mathematics would yield a comprehensive u n d e r s t a n d i n g of these
forces.
Now, b u r d e n e d with having to p r e p a r e a major public lecture on t h e
"foundation of geometry," to support his family, a n d to c o n d u c t scientific experiments, his health finally collapsed a n d he suffered a nervous breakdown in 1854. Later, he wrote to his father, "I b e c a m e so
absorbed in my investigation of the unity of all physical laws that when
the subject of the trial lecture was given m e , I could n o t tear myself away
from my research. T h e n , partly as a result of b r o o d i n g on it, partly from
staying indoors too m u c h in this vile weather, I fell ill." This letter is
significant, for it clearly shows that, even d u r i n g m o n t h s of illness,
5
36
Riemann firmly believed that he would discover the "unity of all physical
laws" a n d that mathematics would eventually pave t h e way for this unification.
Force = Geometry
Eventually, despite his frequent illnesses, R i e m a n n developed a startling
new picture of the m e a n i n g of a " f o r c e . " Ever since Newton, scientists
h a d considered a force to be an instantaneous interaction between two
distant bodies. Physicists called it action-at-a-distance, which m e a n t that
a body could influence the motions of distant bodies instantaneously.
Newtonian mechanics u n d o u b t e d l y could describe t h e motions of the
planets. However, over the centuries, critics argued that action-at-a-distance was unnatural, because it m e a n t that o n e body could change the
direction of a n o t h e r without even t o u c h i n g it.
Riemann developed a radically new physical picture. Like Gauss's
" b o o k w o r m s , " Riemann imagined a race of two-dimensional creatures
living on a sheet of paper. But the decisive break he m a d e was to p u t
these bookworms on a crumpled sheet of paper. What would these bookworms think a b o u t their world? Riemann realized t h a t they would conclude that their world was still perfectly flat. Because their bodies would
also be crumpled, these bookworms would never notice that their world
was distorted. However, R i e m a n n argued that if these bookworms tried
to move across the c r u m p l e d sheet of paper, they would feel a mysterious, unseen " f o r c e " that prevented them from moving in a straight line.
They would be p u s h e d left a n d right every time their bodies moved over
a wrinkle on the sheet.
T h u s R i e m a n n m a d e the first m o m e n t o u s break with Newton in 200
years, banishing the action-at-a-distance principle. To Riemann, "force"
was a consequence of geometry.
Riemann t h e n replaced the two-dimensional sheet with o u r threedimensional world c r u m p l e d in the fourth dimension. It would n o t be
obvious to us that o u r universe was warped. However, we would immediately realize that something was amiss when we tried to walk in a
straight line. We would walk like a d r u n k a r d , as t h o u g h an unseen force
were tugging at us, p u s h i n g us left a n d right.
Riemann concluded that electricity, magnetism, a n d gravity are
caused by the crumpling of o u r three-dimensional universe in the
unseen fourth dimension. T h u s a " f o r c e " has no i n d e p e n d e n t life of its
own; it is only the a p p a r e n t effect caused by the distortion of geometry.
6
37
By introducing the fourth spatial dimension, Riemann accidentally stumbled on what would b e c o m e o n e of the d o m i n a n t themes in m o d e r n
theoretical physics, that the laws of n a t u r e a p p e a r simple when expressed
in higher-dimensional space. He then set about developing a mathematical language in which this idea could be expressed.
Riemann's Metric Tensor: A New Pythagorean Theorem
Riemann spent several m o n t h s recovering from his nervous breakdown.
Finally, when he delivered his oral presentation in 1854, the reception
was enthusiastic. In retrospect, this was, without question, o n e of t h e
most i m p o r t a n t public lectures in the history of mathematics. Word
spread quickly t h r o u g h o u t E u r o p e that Riemann h a d decisively broken
o u t of the confines of Euclidean geometry that had ruled mathematics
for 2 millennia. News of the lecture soon spread t h r o u g h o u t all the
centers of learning in E u r o p e , a n d his contributions to mathematics
were being hailed t h r o u g h o u t the academic world. His talk was translated into several languages and created quite a sensation in mathematics. T h e r e was no t u r n i n g back to the work of Euclid.
Like many of the greatest works in physics a n d mathematics, t h e
essential kernel underlying Riemann's great p a p e r is simple to understand. Riemann began with the famous Pythagorean T h e o r e m , o n e of
the Greeks' greatest discoveries in mathematics. T h e t h e o r e m establishes
the relationship between the lengths of the three sides of a right triangle:
It states that the sum of the squares of the smaller sides equals the square
of the longest side, the hypotenuse; that is, if a a n d b are the lengths of
the two short sides, a n d c is the length of the hypotenuse, t h e n a + b
= c . (The Pythagorean T h e o r e m , of course, is the foundation of all
architecture; every structure built on this p l a n e t is based on it.)
2
2
2
For three-dimensional space, the t h e o r u m can easily be generalized.
It states that the sum of the squares of three adjacent sides of a cube is
equal to the square of the diagonal; so if a, b, a n d c represent the sides
of a cube, a n d d is its diagonal length, t h e n a + b + c = d (Figure
2.1).
It is now simple to generalize this to the case of N - d i m e n s i o n s . Imagine an N-dimensional cube. If a,b,c, . . . are the lengths of the sides of a
" h y p e r c u b e , " a n d z is the length of the diagonal, t h e n a + b + c +
d + . . . = z . Remarkably, even t h o u g h o u r brains c a n n o t visualize an
N-dimensional cube, it is easy to write down the formula for its sides.
(This is a c o m m o n feature of working in hyperspace. Mathematically
2
2
2
2
2
2
2
2
2
38
Figure 2.1. The length of a diagonal of a cube is given by a three-dimensional
version of the Pythagorean Theorem: a + b + c = d . By simply adding more
terms to the Pythagorean Theorem, this equation easily generalizes to the diagonal
of a hypercube in N dimensions. Thus although higher dimensions cannot be
visualized, it is easy to represent N dimensions mathematically.
2
2
2
2
manipulating N-dimensional space is no m o r e difficult than manipulating three-dimensional space. It is n o t h i n g short of amazing that on a
plain sheet of paper, you can mathematically describe the properties of
higher-dimensional objects that cannot be visualized by o u r brains.)
Riemann t h e n generalized these equations for spaces of arbitrary
dimension. These spaces can be either flat or curved. If flat, then the
usual axioms of Euclid apply: T h e shortest distance between two points
is a straight line, parallel lines never meet, a n d the sum of the interior
angles of a triangle add to 180 degrees. But Riemann also found that
surfaces can have "positive curvature," as in the surface of a sphere,
where parallel lines always m e e t a n d where the sum of the angles of a
triangle can exceed 180 degrees. Surfaces can also have "negative cur-
39
v a t u r e , " as in a saddle-shaped or a trumpet-shaped surface. On these
surfaces, the sum of the interior angles of a triangle a d d to less than 180
degrees. Given a line a n d a p o i n t off that line, t h e r e are an infinite
n u m b e r of parallel lines o n e can draw t h r o u g h that p o i n t (Figure 2.2).
R i e m a n n ' s aim was to i n t r o d u c e a new object in mathematics that
would enable h i m to describe all surfaces, no matter how complicated.
This inevitably led h i m to reintroduce Faraday's concept of the field.
Faraday's field, we recall, was like a farmer's field, which occupies a
region of two-dimensional space. Faraday's field occupies a region of
three-dimensional space; at any point in space, we assign a collection of
n u m b e r s that describes the magnetic or electric force at that point. Riem a n n ' s idea was to i n t r o d u c e a collection of n u m b e r s at every p o i n t in
space that would describe how m u c h it was b e n t or curved.
For example, for an ordinary two-dimensional surface, R i e m a n n
i n t r o d u c e d a collection of three n u m b e r s at every p o i n t that completely
describe the b e n d i n g of that surface. R i e m a n n found that in four spatial
dimensions, o n e needs a collection of ten n u m b e r s at each point to
describe its properties. No matter how c r u m p l e d or distorted the space,
this collection of ten n u m b e r s at each p o i n t is sufficient to e n c o d e all
the information a b o u t that space. Let us label these ten n u m b e r s by t h e
symbols g , g , g . . , . (When analyzing a four-dimensional space, the
lower index can range from o n e to four.) T h e n R i e m a n n ' s collection of
ten n u m b e r s can be symmetrically a r r a n g e d as in Figure 2 . 3 . (It appears
as t h o u g h t h e r e are 16 c o m p o n e n t s . However, g = g , g = g a n d
so on, so there are actually only ten i n d e p e n d e n t components.) Today,
this collection of n u m b e r s is called the R i e m a n n metric tensor. Roughly
speaking, the greater the value of the metric tensor, t h e greater t h e
crumpling of the sheet. No matter how c r u m p l e d the sheet of paper,
the metric tensor gives us a simple m e a n s of measuring its curvature at
any point. If we flattened the c r u m p l e d sheet completely, t h e n we would
retrieve t h e formula of Pythagoras.
u
l2
13
7
12
21
l3
31
R i e m a n n ' s metric tensor allowed him to erect a powerful apparatus
for describing spaces of any dimension with arbitrary curvature. To his
surprise, he found that all these spaces are well defined a n d self-consistent. Previously, it was t h o u g h t that terrible contradictions would arise
when investigating the forbidden world of higher dimensions. To his
surprise, Riemann found n o n e . In fact, it was almost trivial to extend his
work to N-dimensional space. T h e metric tensor would now resemble
the squares of a checker b o a r d that was N X N in size. This will have
p r o f o u n d physical implications when we discuss the unification of all
forces in the n e x t several chapters.
Zero
curvature
Positive
curvature
Negative
curvature
Figure 2.2. A plane has zero curvature. In Euclidean geometry, the interior angles
of a triangle sum to 180 degrees, and parallel lines never meet. In non-Euclidean
geometry, a sphere has positive curvature. A triangle's interior angles sum to
greater than 180 degrees and parallel lines always meet. (Parallel lines include
arcs whose centers coincide with the center of the sphere. This rules out latitudinal
lines.) A saddle has negative curvature. The interior angles sum to less than 180
degrees. There are an infinite number of lines parallel to a given line that go
through a fixed point.
40
41
Figure 2.3. Riemann's metric tensor contains all the information necessary to
describe mathematically a curved space in N dimensions. It takes 16 numbers to
describe the metric tensor for each point in four-dimensional space. These numbers
can be arranged in a square array (six of these numbers are actually redundant;
so the metric tensor has ten independent numbers).
(The secret of unification, we will see, lies in e x p a n d i n g R i e m a n n ' s
metric to N-dimensional space a n d t h e n c h o p p i n g it up into rectangular
pieces. Each rectangular piece corresponds to a different force. In this
way, we can describe the various forces of n a t u r e by slotting t h e m into
the metric tensor like pieces of a puzzle. This is the mathematical expression of the principle that higher-dimensional space unifies t h e laws of
n a t u r e , that there is " e n o u g h r o o m " to unite t h e m in N-dimensional
space. More precisely, there is " e n o u g h r o o m " in R i e m a n n ' s metric to
unite the forces of nature.)
Riemann anticipated a n o t h e r development in physics; he was o n e of
the first to discuss multiply c o n n e c t e d spaces, or wormholes. To visualize
this concept, take two sheets of p a p e r a n d place o n e on top of the other.
Make a short cut on each sheet with scissors. T h e n glue the two sheets
together along the two cuts (Figure 2.4). (This is topologically the same
as Figure 1.1, except that the neck of the wormhole has length zero.)
If a bug lives on the top sheet, he may o n e day accidentally walk into
the cut a n d find himself on the b o t t o m sheet. He will be puzzled because
everything is in the wrong place. After m u c h experimentation, the b u g
42
Figure 2.4. Riemann's cut, with two sheets are connected together along a line.
If we walk around the cut, we stay within the same space. But if we walk through
the cut, we pass from one sheet to the next. This is a multiply connected surface.
will find that he can re-emerge in his usual world by re-entering the cut.
If he walks a r o u n d the cut, t h e n his world looks normal; b u t when he
tries to take a short-cut t h r o u g h the cut, he has a problem.
R i e m a n n ' s cuts are an example of a w o r m h o l e (except that it has
zero length) connecting two spaces. Riemann's cuts were used with great
effect by the mathematician Lewis Carroll in his b o o k Through the Looking-Glass. R i e m a n n ' s cut, c o n n e c t i n g England with W o n d e r l a n d , is the
looking glass. Today, R i e m a n n ' s cuts survive in two forms. First, they are
cited in every graduate mathematics course in the world when applied
to the theory of electrostatics or conformal mapping. Second, R i e m a n n ' s
cuts can be found in episodes of " T h e Twilight Z o n e . " (It should be
stressed that Riemann himself did n o t view his cuts as a m o d e of travel
between universes.)
Riemann's Legacy
R i e m a n n persisted with his work in physics. In 1858, he even a n n o u n c e d
that he h a d finally succeeded in a unified description of light a n d electricity. He wrote, "I am fully convinced that my theory is the correct o n e ,
a n d that in a few years it will be recognized as s u c h . " Although his
metric tensor gave him a powerful way to describe any curved space in
any dimension, he did n o t know the precise equations that the metric
tensor obeyed; that is, he did n o t know what m a d e the sheet crumple.
8
43
Unfortunately, R i e m a n n ' s efforts to solve this p r o b l e m were continually thwarted by grinding poverty. His successes did not translate into
money. He suffered a n o t h e r nervous breakdown in 1857. After many
years, he was finally a p p o i n t e d to Gauss's coveted position at Gottingen,
b u t it was too late. A life of poverty h a d b r o k e n his health, a n d like many
of the greatest mathematicians t h r o u g h o u t history, he died prematurely
of consumption at the age of 39, before he could complete his geometric
theory of gravity a n d electricity a n d magnetism.
In summary, Riemann did m u c h m o r e than lay the foundation of
the mathematics of hyperspace. In retrospect, we see that R i e m a n n anticipated some of the major t h e m e s in m o d e r n physics. Specifically,
1. He used higher-dimensional space to simplify t h e laws of nature;
that is, to him, electricity a n d magnetism as well as gravity were
just effects caused by the crumpling or warping of hyperspace.
2. He anticipated the concept of wormholes. R i e m a n n ' s cuts are
the simplest examples of multiply c o n n e c t e d spaces.
3. He expressed gravity as a field. T h e metric tensor, because it
describes the force of gravity (via curvature) at every point in
space, is precisely Faraday's field c o n c e p t when applied to
gravity.
Riemann was unable to complete his work on force fields because he
lacked the field equations that electricity a n d magnetism a n d gravity
obey. In o t h e r words, he did n o t know precisely how the universe would
be c r u m p l e d in o r d e r to yield the force of gravity. He tried to discover
the field equations for electricity a n d magnetism, b u t he died before he
could finish that project. At his death, he still h a d no way of calculating
how m u c h crumpling would be necessary to describe the forces. These
crucial developments would be left to Maxwell a n d Einstein.
Living in a Space Warp
T h e spell was finally b r o k e n .
Riemann, in his short life, lifted the spell cast by Euclid m o r e than
2,000 years before. R i e m a n n ' s metric tensor was the weapon with which
young mathematicians could defy the Boeotians, who howled at any
m e n t i o n of higher dimensions. Those who followed in R i e m a n n ' s footsteps found it easier to speak of unseen worlds.
Soon, research b l o o m e d all over E u r o p e . P r o m i n e n t scientists began
44
Figure 2.5. A two-dimensional being cannot eat. Its digestive tract necessarily
divides it into two distinct pieces, and the being falls apart.
popularizing t h e idea for the general public. H e r m a n n von Helmholtz,
perhaps the most famous G e r m a n physicist of his generation, was deeply
affected by Riemann's work a n d wrote a n d spoke extensively to the general public a b o u t the mathematics of intelligent beings living on a ball
or sphere.
According to Helmholtz, these creatures, with reasoning powers similar to o u r own, would independently discover that all of Euclid's pos-
45
tulates a n d theorems were useless. On a sphere, for example, the sums
of the interior angles of a triangle do n o t add up to 180 degrees. T h e
" b o o k w o r m s " first talked a b o u t by Gauss now found themselves inhabiting Helmholtz's two-dimensional spheres. Helmholtz wrote that "geometrical axioms must vary according to the kind of space inhabited by
beings whose powers of reasoning are quite in conformity with o u r s . "
However, in his Popular Lectures of Scientific Subjects (1881), Helmholtz
warned his readers that it is impossible for us to visualize the fourth
dimension. In fact, he said "such a 'representation' is as impossible as
the 'representation' of colours would be to o n e b o r n b l i n d . "
9
10
Some scientists, marveling at the elegance of Riemann's work, tried
to find physical applications for such a powerful a p p a r a t u s . While some
scientists were exploring the applications of higher dimension, o t h e r
scientists asked m o r e practical, m u n d a n e questions, such as: How does
a two-dimensional being eat? In o r d e r for Gauss's two-dimensional people to eat, their m o u t h s would have to face to the side. But if we now
draw their digestive tract, we notice that this passageway completely
bisects their bodies (Figure 2.5). Thus if they eat, their bodies will split
into two pieces. In fact, any tube that connects two openings in their
bodies will separate t h e m into two unattached pieces. This presents us
with a difficult choice. Either these people eat like we do a n d their bodies break apart, or they obey different laws of biology. '
11
Unfortunately, the advanced mathematics of Riemann outstripped
the relatively backward u n d e r s t a n d i n g of physics in the n i n e t e e n t h century. T h e r e was no physical principle to guide further research. We
would have to wait a n o t h e r century for the physicists to catch up with
the mathematicians. But this did n o t stop nineteenth-century scientists
from speculating endlessly about what beings from the fourth dimension
would look like. Soon, they realized that such a fourth-dimensional
being would have almost God-like powers.
To Be a God
Imagine being able to walk t h r o u g h walls.
You wouldn't have to b o t h e r with o p e n i n g doors; you could pass right
t h r o u g h t h e m . You w o u l d n ' t have to go a r o u n d buildings; you could
e n t e r t h e m t h r o u g h their walls a n d pillars a n d o u t t h r o u g h the back
wall. You w o u l d n ' t have to d e t o u r a r o u n d mountains; you could step
right into t h e m . W h e n hungry, you could simply reach t h r o u g h the
46
refrigerator d o o r without o p e n i n g it. You could never be accidentally
locked outside your car; you could simply step t h r o u g h the car door.
Imagine b e i n g able to disappear or r e a p p e a r at will. Instead of driving to school or work, you would j u s t vanish a n d rematerialize in your
classroom or office. You w o u l d n ' t n e e d an airplane to visit far-away
places, you could j u s t vanish a n d rematerialize where you wanted. You
would never be stuck in city traffic d u r i n g rush hours; you a n d your car
would simply disappear a n d rematerialize at your destination.
Imagine having x-ray eyes. You would be able to see accidents happ e n i n g from a distance. After vanishing a n d rematerializing at the site
of any accident, you could see exactly where the victims were, even if
they were buried u n d e r debris.
Imagine being able to reach into an object without o p e n i n g it. You
could extract the sections from an orange without peeling or cutting it.
You would be hailed as a master surgeon, with the ability to repair the
internal organs of patients without ever cutting the skin, thereby greatly
reducing pain a n d the risk of infection. You would simply reach into the
person's body, passing directly t h r o u g h the skin, a n d perform the delicate operation.
Imagine what a criminal could do with these powers. He could e n t e r
the most heavily g u a r d e d bank. He could see t h r o u g h the massive doors
of the vault for the valuables a n d cash a n d reach inside a n d pull t h e m
out. He could t h e n stroll outside as the bullets from the guards passed
right t h r o u g h him. With these powers, no prison could hold a criminal.
No secrets could be kept from us. No treasures could be h i d d e n from
us. No obstructions could stop us. We would truly be miracle workers,
performing feats beyond the c o m p r e h e n s i o n of mortals. We would also
be o m n i p o t e n t .
What being could possess such God-like power? T h e answer: a being
from a higher-dimensional world. Of course, these feats are beyond the
capability of any three-dimensional person. For us, walls are solid a n d
prison bars are unbreakable. Attempting to walk t h r o u g h walls will only
give us a painful, bloody nose. But for a four-dimensional being, these
feats would be child's play.
To u n d e r s t a n d how these miraculous feats can be performed, consider again Gauss's mythical two-dimensional beings, living on a twodimensional table top. To jail a criminal, the Flatlanders simply draw a
circle a r o u n d him. No matter which way the criminal moves, he hits the
impenetrable circle. However, it is a trivial task for us to spring the priso n e r from jail. We just reach down, grab the Flatlander, peel him off
the two-dimensional world, a n d redeposit h i m elsewhere on his world (Fig-
47
Figure 2.6. In Flatland, a "jail" is a circle drawn around a person. Escape from
this circle is impossible in two dimensions. However, a three-dimensional person
can yank a Flatlander out of jail into the third dimension. To a jailer, it appears
as though the prisoner has mysteriously vanished into thin air.
u r e 2.6). This feat, which is quite ordinary in three dimensions, appears
fantastic in two dimensions.
To his jailer, the prisoner has suddenly disappeared from an escapeproof prison, vanishing into thin air. T h e n just as suddenly, he reappears
somewhere else. If you explain to the jailer that the prisoner was moved
" u p " a n d off Flatland, he would n o t u n d e r s t a n d what you were saying.
T h e word up does n o t exist in the Flatlander's vocabulary, n o r can he
visualize the concept.
T h e o t h e r feats can be similarly explained. For example, notice that
the internal organs (like the stomach or heart) of a Flatlander are completely visible to us, in the same way that we can see the internal structure
of cells on a microscope slide. It's now trivial to reach inside a Flatlander
a n d perform surgery without cutting the skin. We can also peel the Flat-
48
Figure 2.7. If we peel a Flatlander from his world and flip him over in three
dimensions, his heart now appears on the right-hand side. All his internal organs
have been reversed. This transformation is a medical impossibility to someone who
lives strictly in Flatland.
lander off his world, flip h i m a r o u n d , a n d p u t him back down. Notice
that his left a n d right organs are now reversed, so that his heart is on
the right side (Figure 2.7).
Viewing Flatland, notice also that we are o m n i p o t e n t . Even if the
Flatlander hides inside a house or u n d e r the g r o u n d , we can see him
perfectly. He would regard our powers as magical; we, however, would
know that n o t magic, but simply a m o r e advantageous perspective, is at
work. (Although such feats of " m a g i c " are, in principle, possible within
t h e realm of hyperspace physics, we should caution, o n c e again, that the
technology necessary to manipulate space-time far exceeds anything
possible on the earth, at least for h u n d r e d s of years. T h e ability to manipulate s p a c e - t i m e may be within the d o m a i n of only s o m e extraterrestrial
life in the universe far in advance of anything found on the earth, with
the technology to master energy on a scale a quadrillion times larger
than o u r most powerful machines.)
Although R i e m a n n ' s famous lecture was popularized by the work of
Helmholtz a n d many others, the lay public could make little sense of
this or the eating habits of two-dimensional creatures. For the average
49
person, the question was m o r e direct: What kind of beings can walk
t h r o u g h walls, see t h r o u g h steel, a n d perform miracles? W h a t kind of
beings are o m n i p o t e n t a n d obey a set of laws different from ours?
Why ghosts, of course!
In the absence of any physical principle motivating the introduction
of higher dimensions, the theory of the fourth dimension suddenly took
an u n e x p e c t e d turn. We will now begin a strange b u t i m p o r t a n t d e t o u r
in the history of hyperspace, examining its u n e x p e c t e d but p r o f o u n d
impact on the arts a n d philosophy. This tour t h r o u g h p o p u l a r culture
will show how the mystics gave us clever ways in which to "visualize"
higher-dimensional space.
Ghosts from the Fourth Dimension
T h e fourth dimension p e n e t r a t e d the public's consciousness in 1877,
when a scandalous trial in L o n d o n gave it an international notoriety.
T h e L o n d o n newspapers widely publicized the sensational claims
a n d bizarre trial of psychic Henry Slade. T h e raucous proceedings drew
in some of the most p r o m i n e n t physicists of the day. As a result of all
the publicity, talk of the fourth dimension left the blackboards of
abstract mathematicians a n d burst into polite society, turning up in dinner-table conversations t h r o u g h o u t L o n d o n . T h e " n o t o r i o u s fourth
d i m e n s i o n " was now the talk of the town.
It all began, innocently e n o u g h , when Slade, a psychic from the
U n i t e d States, visited L o n d o n a n d held seances with p r o m i n e n t townspeople. He was subsequently arrested for fraud a n d charged with "using
subtle crafts a n d devices, by palmistry a n d otherwise," to deceive his
clients. Normally, this trial might have gone unnoticed. But L o n d o n
society was scandalized a n d amused when e m i n e n t physicists came to his
defense, claiming that his psychic feats actually proved that he could
s u m m o n spirits living in the fourth dimension. This scandal was fueled
by the fact that Slade's defenders were n o t ordinary British scientists,
but rather some of the greatest physicists in the world. Many went on to
win the Nobel Prize in physics.
12
Playing a leading role in stirring up this scandal was J o h a n n Zollner,
a professor of physics a n d astronomy at the University of Leipzig. It was
Zollner who marshaled a galaxy of leading physicists to c o m e to Slade's
defense.
T h a t mystics could perform parlor tricks for the royal court a n d
p r o p e r society, of course, was n o t h i n g new. For centuries, they h a d
50
claimed that they could s u m m o n spirits to read the writing within closed
envelopes, pull objects from closed bottles, reseal b r o k e n match sticks,
a n d intertwine rings. T h e strange twist to this trial was that leading scientists claimed these feats were possible by manipulating objects in the
fourth dimension. In the process, they gave the public its first understanding of how to perform these miraculous feats via the fourth dimension.
Zollner enlisted t h e h e l p of internationally p r o m i n e n t physicists who
participated in the Society for Psychical Research a n d who even rose to
lead the organization, including some of the most distinguished names
of nineteenth-century physics: William Crookes, inventor of the cathode
ray tube, which today is used in every television set a n d c o m p u t e r monitor in the world; Wilhelm Weber, Gauss's collaborator a n d the m e n t o r
of Riemann (today, the international unit of magnetism is officially
n a m e d the " w e b e r " after h i m ) ; J. J. T h o m p s o n , w h o won t h e Nobel
Prize in 1906 for the discovery of the electron; a n d Lord Rayleigh, recognized by historians as o n e of the greatest classical physicists of the late
n i n e t e e n t h century a n d winner of the Nobel Prize in physics in 1904.
Crookes, Weber, a n d Zollner, in particular, took a special interest in
the work of Slade, who was eventually convicted of fraud by the court.
However, he insisted that he could prove his i n n o c e n c e by duplicating
his feats before a scientific body. Intrigued, Zollner took up the challenge. A n u m b e r of controlled experiments were c o n d u c t e d in 1877 to
test Slade's ability to send objects t h r o u g h the fourth dimension. Several
distinguished scientists were invited by Zollner to evaluate Slade's abilities.
First, Slade was given two separate, u n b r o k e n w o o d e n rings. Could
he push o n e wooden ring past the other, so that they were intertwined
without breaking? If Slade succeeded, Zollner wrote, it would " r e p r e s e n t
a miracle, that is, a p h e n o m e n o n which o u r conceptions heretofore of
physical and organic processes would be absolutely i n c o m p e t e n t to
explain."
Second, he was given the shell of a sea snail, which twisted either to
the right or to the left. Could Slade transform a right-handed shell into
a left-handed shell a n d vice versa?
Third, he was given a closed loop of r o p e m a d e of dried animal gut.
Could he m a k e a k n o t in the circular r o p e without cutting it?
Slade was also given variations of these tests. For example, a rope was
tied into a right-handed k n o t a n d its e n d s were sealed with wax a n d
impressed with Zollner's personal seal. Slade was asked to untie the knot,
without breaking the wax seal, a n d retie the rope in a left-handed knot.
13
14
51
Since knots can always be untied in the fourth dimension, this feat
should be easy for a fourth-dimensional person. Slade was also asked to
remove the contents of a sealed bottle without breaking the bottle.
Could Slade demonstrate this astounding ability?
Magic in the Fourth Dimension
Today we realize that the manipulation of higher-dimensional space, as
claimed by Slade, would require a technology far in advance of anything
possible on this planet for the conceivable future. However, what is interesting a b o u t this notorious case is that Zollner correctly c o n c l u d e d that
Slade's feats of wizardry could be explained if o n e could somehow move
objects t h r o u g h the fourth dimension. T h u s for pedagogical reasons,
t h e experiments of Zollner are compelling a n d worth discussing.
For example, in three dimensions, separate rings c a n n o t be p u s h e d
t h r o u g h each o t h e r until they intertwine without breaking them. Similarly, closed, circular pieces of r o p e c a n n o t be twisted into knots without
cutting them. Any Boy or Girl Scout who has struggled with knots for
his or h e r merit badges knows that knots in a circular loop of r o p e
c a n n o t be removed. However, in h i g h e r dimensions, knots a r e easily
unraveled a n d rings can be intertwined. This is because there is " m o r e
r o o m " in which to move ropes past each o t h e r a n d rings into each other.
If the fourth dimension existed, ropes a n d rings could be lifted off o u r
universe, intertwined, a n d t h e n r e t u r n e d to o u r world. In fact, in the
fourth dimension, knots can never remain tied. They can always be
unraveled without cutting the r o p e . This feat is impossible in three
dimensions, b u t trivial in the fourth. T h e third dimension, as it turns
out, is the only dimension in which knots stay knotted. (The proof of
this r a t h e r u n e x p e c t e d result is given in the n o t e s . )
15
Similarly, in three dimensions it is impossible to convert a rigid lefth a n d e d object into a right-handed o n e . H u m a n s are b o r n with hearts
on their left side, a n d no surgeon, no matter now skilled, can reverse
h u m a n internal organs. This is possible (as first pointed out by mathematician August Mobius in 1827) only if we lift the body o u t of o u r
universe, rotate it in the fourth dimension, a n d t h e n reinsert it back into
o u r universe. Two of these tricks are depicted in Figure 2.8; they can be
performed only if objects can be moved in the fourth dimension.
Polarizing the Scientific Community
Zollner sparked a storm of controversy when, publishing in both the
Quarterly Journal of Science a n d Transcendental Physics, he claimed that
Figure 2.8. The mystic Henry Slade claimed to be able to change right-handed
snail shells into left-handed ones, and to remove objects from sealed bottles. These
feats are impossible in three dimensions, but are trivial if one can move objects
through the fourth dimension.
52
S3
Slade amazed his audiences with these " m i r a c u l o u s " feats d u r i n g
seances in the presence of distinguished scientists. (However, Slade also
flunked some of the tests that were c o n d u c t e d u n d e r controlled conditions.)
Zollner's spirited defense of Slade's feats was sensationalized
t h r o u g h o u t L o n d o n society. (In fact, this was actually o n e of several
highly publicized incidents involving spiritualists a n d m e d i u m s in the
late n i n e t e e n t h century. Victorian England was apparently fascinated
with the occult.) Scientists, as well as the general public, quickly took
sides in the matter. Supporting Zollner's claims was his circle of reputable scientists, including W e b e r a n d Crookes. These were n o t average
scientists, b u t masters of the art of science a n d seasoned observers of
experiment. They h a d spent a lifetime working with natural p h e n o m e n a ,
a n d now before their eyes, Slade was performing feats that were possible
only if spirits lived in the fourth dimension.
But detractors of Zollner pointed out that scientists, because they are
trained to trust their senses, are the worst possible people to evaluate a
magician. A magician is trained specifically to distract, deceive, a n d confuse those very senses. A scientist may carefully observe the magician's
right h a n d , but it is the left h a n d that secretly performs the trick. Critics
also p o i n t e d out that only a n o t h e r magician is clever e n o u g h to detect
the sleight-of-hand tricks of a fellow magician. Only a thief can catch a
thief.
O n e particularly savage piece of criticism, published in the science
quarterly magazine Bedrock, was m a d e against two o t h e r p r o m i n e n t physicists, Sir W. F. Barrett a n d Sir Oliver Lodge, a n d their work on telepathy.
T h e article was merciless:
It is not necessary either to regard the p h e n o m e n a of so-called telepathy
a s i n e x p l i c a b l e o r t o r e g a r d t h e m e n t a l c o n d i t i o n o f Sir W . F . B a r r e t t a n d
Sir O l i v e r L o d g e as i n d i s t i n g u i s h a b l e f r o m i d i o c y . T h e r e is a t h i r d possibility. The will to believe h a s m a d e t h e m r e a d y to a c c e p t e v i d e n c e o b t a i n e d
u n d e r conditions which they would recognize to be u n s o u n d if they had
b e e n trained in experimental psychology.
Over a century later, precisely the same arguments, p r o a n d con,
would be used in the debate over the feats of the Israeli psychic Uri
Geller, who convinced two reputable scientists at the Stanford Research
Institute in California that he could b e n d keys by mental power alone
a n d perform o t h e r miracles. ( C o m m e n t i n g on this, some scientists have
repeated a saying that dates back to the Romans: " P o p u l u s vult decipi,
54
e r g o d e c i p i a t u r " [People want to be deceived, therefore let t h e m be
deceived].)
T h e passions raging within the British scientific community t o u c h e d
off a lively d e b a t e that quickly spread across the English C h a n n e l . Unfortunately, in the decades following R i e m a n n ' s death, scientists lost sight
of his original goal, to simplify the laws of n a t u r e t h r o u g h higher dimensions. As a consequence, the theory of higher dimensions w a n d e r e d into
many interesting b u t questionable directions. This is an i m p o r t a n t lesson. Without a clear physical motivation or a guiding physical picture,
p u r e mathematical concepts sometimes drift into speculation.
These decades were n o t a complete loss, however, because mathematicians a n d mystics like Charles H i n t o n would invent ingenious ways
in which to " s e e " t h e fourth dimension. Eventually, the pervasive influence of the fourth dimension would c o m e full circle a n d cross-pollinate
the world of physics once again.
The Man Who "Saw"
the Fourth Dimension
[ T ] h e fourth d i m e n s i o n h a d b e c o m e almost a h o u s e h o l d
w o r d by 1 9 1 0 . . . . R a n g i n g f r o m an i d e a l P l a t o n i c or K a n t i a n
r e a l i t y — o r e v e n H e a v e n — t h e a n s w e r t o all o f t h e p r o b l e m s
puzzling contemporary science, the fourth d i m e n s i o n c o u l d
b e all t h i n g s t o all p e o p l e .
Linda Dalrymple Henderson
W
ITH the passions aroused by the trial of the " n o t o r i o u s Mr.
Slade," it was p e r h a p s inevitable that the controversy would eventually spawn a best-selling novel.
In 1884, after a decade of acrimonious debate, clergyman Edwin
Abbot, headmaster of the City of L o n d o n School, wrote the surprisingly
successful a n d e n d u r i n g novel Flatland: A Romance of Many Dimensions by
a Square* Because of the intense public fascination with higher dimen*It wasn't surprising that a clergyman wrote the novel, since theologians of the Church
of England were a m o n g the first to j u m p into the fray created by the sensationalized trial.
For uncounted centuries, clergymen had skillfully dodged such perennial questions as
Where are heaven and hell? and Where do angels live? Now, they found a convenient
resting place for these heavenly bodies: the fourth dimension. T h e Christian spiritualist A.
T. Schofield, in his 1888 book Another World, argued at length that God and the spirits
resided in the fourth dimension. Not to be outdone, in 1893 the theologian Arthur Willink
wrote The World of the Unseen, in which he claimed that it was unworthy of God to reside in
the lowly fourth dimension. Willink claimed that the only domain magnificent e n o u g h for
God was infinite-dimensional space.'
1
2
55
56
sions, the book was an instant success in England, with n i n e successive
reprintings by the year 1915, a n d editions too n u m e r o u s to c o u n t today.
W h a t was surprising a b o u t the novel Flatland was that Abbott, for the
first time, used the controversy s u r r o u n d i n g the fourth dimension as a
vehicle for biting social criticism a n d satire. Abbot took a playful swipe
at the rigid, pious individuals who refused to admit the possibility of
o t h e r worlds. T h e " b o o k w o r m s " of Gauss b e c a m e the Flatlanders. T h e
Boeotians w h o m Gauss so feared became the High Priests, who would
persecute—with the vigor a n d impartiality of the Spanish Inquisition—
anyone who d a r e d m e n t i o n the u n s e e n third dimension.
Abbot's Flatland is a thinly disguised criticism of the subtle bigotry
a n d suffocating prejudice prevalent in Victorian England. T h e h e r o of
the novel is Mr. Square, a conservative gentleman who lives in a socially
stratified, two-dimensional land where everyone is a geometric object.
W o m e n , occupying the lowest rank in the social hierarchy, are m e r e
lines, the nobility are polygons, while the High Priests are circles. T h e
m o r e sides p e o p l e have, the higher their social rank.
Discussion of the third dimension is strictly forbidden. Anyone mentioning it is sentenced to severe p u n i s h m e n t . Mr. Square is a smug, selfrighteous person who would never think of challenging the Establishm e n t for its injustices. O n e day, however, his life is permanently t u r n e d
upside down when he is visited by a mysterious Lord Sphere, a threedimensional sphere. Lord Sphere appears to Mr. Square as a circle that
can magically change size (Figure 3.1)
Lord Sphere patiently tries to explain that he comes from a n o t h e r
world called Spaceland, where all objects have three dimensions. However, Mr. Square remains unconvinced; he stubbornly resists t h e idea
that a third dimension can exist. Frustrated, Lord Sphere decides to
resort to deeds, n o t m e r e words. He t h e n peels Mr. Square off the twodimensional Flatland a n d hurls him into Spaceland. It is a fantastic,
almost mystical experience that changes Mr. Square's life.
As the flat Mr. Square floats in the third dimension like a sheet of
p a p e r drifting in the wind, he can visualize only two-dimensional slices
of Spaceland. Mr. Square, seeing only the cross sections of three-dimensional objects, views a fantastic world where objects c h a n g e shape a n d
even a p p e a r a n d disappear into thin air. However, when he tries to tell
his fellow Flatlanders of the marvels he saw in his visit to the third dimension, the High Priests consider h i m a blabbering, seditious maniac. Mr.
Square becomes a threat to the High Priests because he dares to challenge their authority a n d their sacred belief that only two dimensions
can possibly exist.
Figure 3.1. In Flatland, Mr. Square encounters Lord Sphere. As Lord Sphere
passes through Flatland, he appears to be a circle that becomes successivley larger
and then smaller. Thus Flatlanders cannot visualize three-dimensional beings,
but can understand their cross sections.
57
58
The
he did,
Square
solitary
b o o k ends on a pessimistic note. Although he is convinced that
i n d e e d , visit the third-dimensional world of Spaceland, Mr.
is sent to jail a n d c o n d e m n e d to spend the rest of his days in
confinement.
A Dinner Party in the Fourth Dimension
Abbot's novel is i m p o r t a n t because it was the first widely read popularization of a visit to a higher-dimensional world. His description of Mr.
Square's psychedelic trip into Spaceland is mathematically correct. In
p o p u l a r accounts a n d the movies, interdimensional travel t h r o u g h
hyperspace is often pictured with blinking lights a n d dark, swirling
clouds. However, the mathematics of higher-dimensional travel is m u c h
m o r e interesting t h a n the imagination of fiction writers. To visualize
what an interdimensional trip would look like, imagine peeling Mr.
Square off Flatland a n d throwing him into the air. As he floats t h r o u g h
o u r three-dimensional world, let's say that he comes across a h u m a n
being. What do we look like to Mr. Square?
Because his two-dimensional eyes can see only flat slices of o u r world,
a h u m a n would look like a singularly ugly a n d frightening object. First,
he might see two leather circles hovering in front of h i m (our shoes).
As he drifts upward, these two circles change color a n d turn into cloth
(our pants). T h e n these two circles coalesce into o n e circle (our waist)
a n d split into three circles of cloth a n d change color again (our shirt
a n d o u r arms). As he continues to float upward, these three circles of
cloth merge into o n e smaller circle of flesh (our neck a n d h e a d ) . Finally,
this circle of flesh turns into a mass of hair, a n d then abruptly disappears
as Mr. Square floats above o u r heads. To Mr. Square, these mysterious
" h u m a n s " are a nightmarish, maddeningly confusing collection of constantly changing circles m a d e of leather, cloth, flesh, a n d hair.
Similarly, if we were peeled off o u r three-dimensional universe a n d
hurled into the fourth dimension, we would find that c o m m o n sense
becomes useless. As we drift t h r o u g h the fourth dimension, blobs appear
from nowhere in front of o u r eyes. They constantly c h a n g e in color, size,
a n d composition, defying all the rules of logic of o u r three-dimensional
world. And they disappear into thin air, to be replaced by o t h e r hovering
blobs.
If we were invited to a d i n n e r party in the fourth dimension, how
would we tell the creatures apart? We would have to recognize t h e m by
The Man Who "Saw" the Fourth Dimension
59
the differences in how these blobs change. Each person in higher dimensions would have his or h e r own characteristic sequences of c h a n g i n g
blobs. Over a period of time, we would learn to tell these creatures
apart by recognizing their distinctive patterns of c h a n g i n g blobs a n d
colors. Attending d i n n e r parties in hyperspace m i g h t be a trying experience.
Class Struggle in the Fourth Dimension
T h e concept of the fourth dimension h a d so pervasively infected the
intellectual climate by the late n i n e t e e n t h century that even playwrights
p o k e d fun at it. In 1891, Oscar Wilde wrote a spoof on these ghost stories,
" T h e Canterville Ghost," which lampoons the exploits of a certain gullible "Psychical Society" (a thinly veiled reference to Crookes's Society
for Psychical Research). Wilde wrote of a long-suffering ghost who
e n c o u n t e r s the newly arrived American tenants of Canterville. Wilde
wrote, " T h e r e was evidently no time to be lost, so hastily adopting the
Fourth Dimension of Space as a means of escape, he [the ghost] vanished t h r o u g h the wainscoting a n d the house b e c a m e quiet."
A m o r e serious contribution to the literature of the fourth dimension
was the work of H. G. Wells. Although he is principally r e m e m b e r e d for
his works in science fiction, he was a d o m i n a n t figure in the intellectual
life of L o n d o n society, n o t e d for his literary criticism, reviews, a n d piercing wit. In his 1894 novel, The Time Machine, he c o m b i n e d several mathematical, philosophical, a n d political themes. He popularized a new idea
in science—that the fourth dimension might also be viewed as time, n o t
necessarily space:*
C l e a r l y . . . a n y real b o d y m u s t h a v e e x t e n s i o n i n
four d i r e c t i o n s :
it m u s t
have L e n g t h , Breadth, Thickness, a n d — D u r a t i o n . But t h r o u g h a natural
infirmity o f t h e flesh . . . w e i n c l i n e t o o v e r l o o k this fact. T h e r e a r e really
f o u r d i m e n s i o n s , t h r e e w h i c h w e call t h e t h r e e l a n e s o f S p a c e , a n d a
F o u r t h , T i m e . T h e r e is, h o w e v e r , a t e n d e n c y t o d r a w a n u n r e a l d i s t i n c t i o n
b e t w e e n t h e f o r m e r t h r e e d i m e n s i o n s a n d t h e latter, b e c a u s e i t h a p p e n s
that our consciousness m o v e s intermittently in o n e direction a l o n g the
latter f r o m t h e b e g i n n i n g t o t h e e n d o f o u r l i v e s .
3
*Wells was not the first to speculate that time could be viewed as a new type of fourth
dimension, different from a spatial one. Jean d'Alembert had considered time as the fourth
dimension in his 1754 article "Dimension."
60
Like Flatland before it, what makes The Time Machine so e n d u r i n g ,
even a century after its conception, is its s h a r p political a n d social critique. E n g l a n d in the year 802,701, Wells's protagonist finds, is n o t the
gleaming citadel of m o d e r n scientific marvels that t h e positivists foretold. Instead, the future England is a land where the class struggle went
awry. T h e working class was cruelly forced to live u n d e r g r o u n d , until
the workers m u t a t e d into a new, brutish species of h u m a n , the Morlocks,
while t h e ruling class, with its u n b r i d l e d debauchery, deteriorated a n d
evolved into t h e useless race of elflike creatures, the Eloi.
Wells, a p r o m i n e n t Fabian socialist, was using the fourth dimension
to reveal the ultimate irony of the class struggle. T h e social contract
between the p o o r a n d t h e rich h a d g o n e completely m a d . T h e useless
Eloi are fed a n d clothed by the hard-working Morlocks, b u t the workers
get the final revenge: T h e Morlocks eat the Eloi. T h e fourth dimension,
in o t h e r words, b e c a m e a foil for a Marxist critique of m o d e r n society,
b u t with a novel twist: T h e working class will n o t break the chains of the
rich, as Marx predicted. They will eat the rich.
In a short story, " T h e Plattner Story," Wells even toyed with t h e
p a r a d o x of h a n d e d n e s s . Gottfried Plattner, a science teacher, is performing an elaborate chemical e x p e r i m e n t , b u t his e x p e r i m e n t blows up a n d
sends him into a n o t h e r universe. W h e n he r e t u r n s from the netherworld
to the real world, he discovers that his body has b e e n altered in a curious
fashion: His h e a r t is now on his right side, a n d he is now left h a n d e d .
W h e n they e x a m i n e him, his doctors are s t u n n e d to find that P l a n n e r ' s
entire body has b e e n reversed, a biological impossibility in o u r threedimensional world: " [ T ] h e curious inversion of P l a n n e r ' s right a n d left
sides is proof that he has moved o u t of o u r space into what is called the
F o u r t h Dimension, a n d that he has r e t u r n e d again to o u r world." However, Plattner resists t h e idea of a p o s t m o r t e m dissection after his death,
thereby p o s t p o n i n g " p e r h a p s forever, the positive proof that his entire
body h a d h a d its left a n d right sides transposed."
Wells was well aware that t h e r e are two ways to visualize how lefth a n d e d objects can be transformed into right-handed objects. A Flatlander, for example, can be lifted o u t of his world, flipped over, a n d
t h e n placed back in Flatland, thereby reversing his organs. Or the Flatl a n d e r may live on a Mobius strip, created by twisting a strip of p a p e r
180 degrees a n d t h e n gluing t h e e n d s together. If a Flatlander walks
completely a r o u n d the Mobius strip a n d returns, he finds that his organs
have b e e n reversed (Figure 3.2). Mobius strips have o t h e r remarkable
properties that have fascinated scientists over the past century. For example, if you walk completely a r o u n d the surface, you will find t h a t it has
61
Figure 3.2. A Mobius strip is a strip with only one side. Its outside and inside
are identical. If a Flatlander wanders around a Mobius strip, his internal organs
will be reversed.
only o n e side. Also, if you cut it in half along the center strip, it remains
in o n e piece. This has given rise to the mathematicians' limerick:
A mathematician confided
That a Mobius band is one-sided
And you'll get quite a laugh
If you cut it in half,
For it stays in one piece when divided.
In his classic The Invisible Man, Wells speculated that a m a n might
even b e c o m e invisible by some trick involving "a formula, a geometrical
expression involving four dimensions." Wells knew that a Flatlander disappears if he is peeled off his two-dimensional universe; similarly, a m a n
could b e c o m e invisible if he could somehow leap into the fourth dimension.
In the short story " T h e Remarkable Case of Davidson's Eyes," Wells
explored the idea that a " k i n k in s p a c e " m i g h t enable an individual to
62
see across vast distances. Davidson, the h e r o of the story, o n e day finds
he has the disturbing power of being able to see events transpiring on
a distant South Sea island. This "kink in s p a c e " is a space warp whereby
light from the South Seas goes t h r o u g h hyperspace a n d enters his eyes
in England. T h u s Wells used R i e m a n n ' s wormholes as a literary device
in his fiction.
In The Wonderful Visit, Wells explored the possibility that heaven exists
in a parallel world or dimension. T h e plot revolves a r o u n d the predica m e n t of an angel who accidentally falls from heaven a n d lands in an
English country village.
T h e popularity of Wells's work o p e n e d up a new g e n r e of fiction.
George McDonald, a friend of mathematician Lewis Carroll, also speculated a b o u t the possibility of heaven being located in the fourth dimension. In McDonald's fantasy Lilith, written in 1895, the h e r o creates a
dimensional window between o u r universe a n d o t h e r worlds by manipulating mirror reflections. And in the 1901 story The Inheritors by J o s e p h
C o n r a d a n d Ford Madox Ford, a race of s u p e r m e n from the fourth
dimension enters into o u r world. Cruel a n d unfeeling, these s u p e r m e n
begin to take over the world.
The Fourth Dimension as Art
T h e years 1890 to 1910 may be considered the Golden Years of the
Fourth Dimension. It was a time d u r i n g which the ideas originated by
Gauss a n d Riemann p e r m e a t e d literary circles, the avant garde, a n d the
thoughts of the general public, affecting trends in art, literature, a n d
philosophy. T h e new b r a n c h of philosophy, called Theosophy, was
deeply influenced by higher dimensions.
On the o n e h a n d , serious scientists regretted this development
because the rigorous results of Riemann were now being dragged
t h r o u g h tabloid headlines. On the o t h e r h a n d , the popularization of the
fourth dimension h a d a positive side. Not only did it make the advances
in mathematics available to the general public, b u t it also served as a
m e t a p h o r that could enrich a n d cross-fertilize cultural currents.
Art historian Linda Dalrymple H e n d e r s o n , writing in The Fourth
Dimension and Non-Euclidean Geometry in Modern Art, elaborates on this
a n d argues that the fourth dimension crucially influenced the developm e n t of Cubism a n d Expressionism in the art world. She writes that " i t
was a m o n g the Cubists that the first a n d most c o h e r e n t art theory based
on the new geometries was d e v e l o p e d . " To the avant garde, the fourth
4
63
Figure 3.3. One scene in the Bayeux Tapestry depicts frightened English troops
pointing to an apparition in the sky (Halley's comet). The figures are flat, as in
most of the art done in the Middle Ages. This signified that God was omnipotent.
Pictures were thus drawn two dimensionally. (Giraudon/Art Resource)
dimension symbolized the revolt against the excesses of capitalism. They
saw its oppressive positivism a n d vulgar materialism as stifling creative
expression. T h e Cubists, for example, rebelled against the insufferable
arrogance of the zealots of science w h o m they perceived as d e h u m a n izing the creative process.
T h e avant garde seized on the fourth dimension as their vehicle. On
the o n e h a n d , the fourth dimension p u s h e d the boundaries of m o d e r n
science to their limit. It was m o r e scientific than the scientists. On the
o t h e r h a n d , it was mysterious. And flaunting the fourth dimension
tweaked the noses of the stiff, know-it-all positivists. In particular, this
took the form of an artistic revolt against the laws of perspective.
In the Middle Ages, religious art was distinctive for its deliberate lack
of perspective. Serfs, peasants, a n d kings were depicted as t h o u g h they
were flat, m u c h in the way children draw people. These paintings largely
reflected the c h u r c h ' s view that God was o m n i p o t e n t and could therefore see all parts of o u r world equally. Art h a d to reflect his point of
view, so the world was painted two dimensionally. For example, the
famous Bayeux Tapestry (Figure 3.3) depicts the superstitious soldiers
of King Harold II of England p o i n t i n g in frightened w o n d e r at an omi-
64
Figure 3.4. During the Renaissance, painters discovered the third dimension.
Pictures were painted with perspective and were viewed from the vantage point of
a single eye, not God's eye. Note that all the lines in Leonardo da Vinci's fresco
T h e Last Supper converge to a point at the horizon. (Bettmann Archive)
n o u s comet soaring overhead in April 1066, convinced that it is an o m e n
of i m p e n d i n g defeat. (Six centuries later, the same comet would be
christened Halley's comet.) Harold subsequently lost the crucial Battle
of Hastings to William t h e C o n q u e r o r , who was crowned the king of
England, a n d a new c h a p t e r in English history began. However, the
Bayeux Tapestry, like o t h e r medieval works of art, depicts H a r o l d ' s soldiers' arms a n d faces as flat, as t h o u g h a p l a n e of glass h a d b e e n placed
over their bodies, compressing t h e m against the tapestry.
Renaissance art was a revolt against this flat God-centered perspective, a n d man-centered art began to flourish, with sweeping landscapes
a n d realistic, three-dimensional people painted from the point of view
of a person's eye. In L e o n a r d o da Vinci's powerful studies on perspective, we see the lines in his sketches vanishing into a single p o i n t on the
horizon. Renaissance art reflected the way the eye viewed t h e world,
from the singular point of view of the observer. In Michelangelo's frescoes or in da Vinci's sketch book, we see bold, imposing figures j u m p i n g
out of the second dimension. In o t h e r words, Renaissance art discovered
t h e third dimension (Figure 3.4).
With the b e g i n n i n g of the m a c h i n e age a n d capitalism, the artistic
world revolted against t h e cold materialism that s e e m e d to d o m i n a t e
65
industrial society. To the Cubists, positivism was a straitjacket that confined us to what could be measured in the laboratory, suppressing the
fruits of o u r imagination. They asked: Why must art be clinically "realistic"? This Cubist "revolt against perspective" seized the fourth dimension because it t o u c h e d the third dimension from all possible perspectives. Simply put, Cubist art e m b r a c e d the fourth dimension.
Picasso's paintings are a splendid example, showing a clear rejection
of the perspective, with w o m e n ' s faces viewed simultaneously from several angles. Instead of a single point of view, Picasso's paintings show
multiple perspectives, as though they were painted by s o m e o n e from the
fourth dimension, able to see all perspectives simultaneously (Figure
3.5).
Picasso was once accosted on a train by a stranger who recognized
him. T h e stranger complained: Why c o u l d n ' t he draw pictures of people
the way they actually were? Why did he have to distort the way people
looked? Picasso then asked the m a n to show him pictures of his family.
After gazing at the snapshot, Picasso replied, " O h , is your wife really
that small a n d flat?" To Picasso, any picture, no matter how "realistic,"
d e p e n d e d on the perspective of the observer.
Abstract painters tried n o t only to visualize people's faces as though
painted by a four-dimensional person, but also to treat time as the fourth
dimension. In Marcel D u c h a m p ' s painting Nude Descending a Staircase,
we see a blurred representation of a woman, with an infinite n u m b e r of
h e r images superimposed over time as she walks down the stairs. This is
how a four-dimensional person would see people, viewing all time
sequences at once, if time were the fourth dimension.
In 1937, art critic Meyer Schapiro summarized the influence of these
new geometries on the art world when he wrote, "Just as the discovery
of non-Euclidean geometry gave a powerful impetus to the view that
mathematics was i n d e p e n d e n t of existence, so abstract painting cut at
the roots of the classic ideas of artistic imitation." Or, as art historian
Linda H e n d e r s o n has said, " t h e fourth dimension a n d non-Euclidean
geometry e m e r g e as a m o n g the most i m p o r t a n t themes unifying m u c h
of m o d e r n art a n d t h e o r y . "
5
Bolsheviks and the Fourth Dimension
T h e fourth dimension also crossed over into Czarist Russia via the writings of the mystic P. D. Ouspensky, who introduced Russian intellectuals
to its mysteries. His influence was so p r o n o u n c e d that even Fyodor Dos-
Figure 3.5. Cubism was heavily influenced by the fourth dimension. For example,
it tried to view reality through the eyes of a fourth-dimensional person. Such a
being, looking at a human face, would see all angles simultaneously. Hence, both
eyes would be seen at once by a fourth-dimensional being, as in Picasso's painting
Portrait of Dora Maar. (Giraudon/Art Resource. ® 1993. Ars, New York/
Spadem, Paris)
66
67
toyevsky, in The Brothers Karamazov, h a d his protagonist Ivan Karamazov
speculate on the existence of h i g h e r dimensions a n d non-Euclidean
geometries d u r i n g a discussion on the existence of God.
Because of the historic events unfolding in Russia, the fourth dimension was to play a curious role in the Bolshevik Revolution. Today, this
strange interlude in the history of science is i m p o r t a n t because Vladimir
Lenin would j o i n the debate over the fourth dimension, which would
eventually exert a powerful influence on the science of the former Soviet
U n i o n for the next 70 years. (Russian physicists, of course, have played
key roles in developing the present-day ten-dimensional theory.)
6
After the Czar brutally crushed the 1905 revolution, a faction called
the Otzovists, or "God-builders," developed within the Bolshevik party.
They argued that the peasants w e r e n ' t ready for socialism; to p r e p a r e
t h e m , Bolsheviks should appeal to them t h r o u g h religion a n d spiritualism. To bolster their heretical views, the God-builders quoted from the
work of the G e r m a n physicist a n d philosopher Ernst Mach, who h a d
written eloquently a b o u t the fourth dimension a n d the recent discovery
of a new, unearthly property of matter called radioactivity. T h e Godbuilders p o i n t e d out that the discovery of radioactivity by the French
scientist H e n r i Becquerel in 1896 a n d the discovery of radium by Marie
Curie in 1896 h a d ignited a furious philosophical debate in French a n d
G e r m a n literary circles. It a p p e a r e d that matter could slowly disintegrate
a n d that energy (in the form of radiation) could reappear.
Without question, the new experiments on radiation showed that the
foundation of Newtonian physics was crumbling. Matter, t h o u g h t by the
Greeks to be eternal a n d immutable, was now disintegrating before o u r
very eyes. U r a n i u m a n d radium, c o n f o u n d i n g accepted belief, were
mutating in the laboratory. To some, Mach was the p r o p h e t who would
lead t h e m out of the wilderness. However, he p o i n t e d in the wrong
direction, rejecting materialism a n d declaring that space a n d time were
products of o u r sensations. In vain, he wrote, "I h o p e that nobody will
defend ghost-stories with the h e l p of what I have said a n d written on this
subject."
7
A split developed within the Bolsheviks. Their leader, Vladimir
Lenin, was horrified. Are ghosts a n d d e m o n s compatible with socialism?
In exile in Geneva in 1908, he wrote a m a m m o t h philosophical tome,
Materialism and Empirio-Criticism, defending dialectical materialism from
the onslaught of mysticism a n d metaphysics. To Lenin, the mysterious
disappearance of matter a n d energy did n o t prove the existence of spirits. He argued that this m e a n t instead that a new dialectic was emerging,
which would embrace both matter a n d energy. No longer could they be
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viewed as separate entities, as Newton h a d d o n e . They must now be
viewed as two poles of a dialectical unity. A new conservation principle
was n e e d e d . (Unknown to Lenin, Einstein h a d p r o p o s e d the correct
principle 3 years earlier, in 1905.) F u r t h e r m o r e , Lenin questioned
Mach's easy e m b r a c e of t h e fourth dimension. First, L e n i n praised Mach,
who " h a s raised the very i m p o r t a n t a n d useful question of a space of n
dimensions as a conceivable s p a c e . " T h e n he took Mach to task for
failing to emphasize that only the three dimensions of space could be
verified experimentally. Mathematics may explore t h e fourth dimension
a n d the world of what is possible, a n d this is good, wrote Lenin, but the
Czar can be overthrown only in t h e third d i m e n s i o n !
Fighting on the b a t t l e g r o u n d of the fourth dimension a n d the new
theory of radiation, Lenin n e e d e d years to r o o t o u t Otzovism from the
Bolshevik party. Nevertheless, he won the battle shortly before the outbreak of the 1917 O c t o b e r Revolution.
8
Bigamists and the Fourth Dimension
Eventually, the ideas of the fourth dimension crossed the Atlantic a n d
c a m e to America. T h e i r messenger was a colorful English mathematician
n a m e d Charles Howard H i n t o n . While Albert Einstein was toiling at his
desk j o b in the Swiss p a t e n t office in 1905, discovering the laws of relativity, H i n t o n was working at the United States Patent Office in Washington, D.C. Although they probably never met, their paths would cross
in several interesting ways.
H i n t o n spent his entire adult life obsessed with the n o t i o n of popularizing a n d visualizing the fourth dimension. He would go down in the
history of science as the m a n who "saw" the fourth dimension.
H i n t o n was the son of J a m e s Hinton, a r e n o w n e d British ear surgeon
of liberal persuasion. Over the years, the charismatic elder H i n t o n
evolved into a religious philosopher, an outspoken advocate of free love
a n d o p e n polygamy, a n d finally the leader of an influential cult in
England. He was s u r r o u n d e d by a fiercely loyal a n d devoted circle of
free-thinking followers. O n e of his best-known remarks was "Christ was
the Savior of m e n , b u t I am the savior of women, a n d I d o n ' t envy H i m
a bit!"
His son Charles, however, seemed d o o m e d to lead a respectable,
boring life as a mathematician. He was fascinated n o t by polygamy, b u t
by polygons! Having g r a d u a t e d from Oxford in 1877, he b e c a m e a
respectable master at the U p p i n g h a m School while working on his mas9
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ter's degree in mathematics. At Oxford, H i n t o n b e c a m e intrigued with
trying to visualize the fourth dimension. As a mathematician, he knew
that o n e c a n n o t visualize a four-dimensional object in its entirety. However, it is possible, he reasoned, to visualize the cross section or the
unraveling of a four-dimensional object.
H i n t o n published his notions in the p o p u l a r press. He wrote the
influential article " W h a t is the Fourth D i m e n s i o n ? " for the Dublin University Magazine a n d the Cheltenham Ladies' College Magazine, reprinted in
1884 with the catchy subtitle "Ghosts Explained."
H i n t o n ' s life as a comfortable academic, however, took a sharp turn
for the worse in 1885 when he was arrested for bigamy a n d p u t on trial.
Earlier, H i n t o n h a d married Mary Everest Boole, the d a u g h t e r of a memb e r of his father's circle, a n d widow of the great mathematician George
Boole (founder of Boolean algebra). However, he was also the father of
twins b o r n to a certain Maude Weldon.
T h e headmaster at U p p i n g h a m , noticing Hinton in the presence of
his wife, Mary, a n d his mistress, Maude, h a d assumed that Maude was
H i n t o n ' s sister. All was going well for H i n t o n , until he m a d e the mistake
of marrying Maude as well. W h e n the headmaster learned that H i n t o n
was a bigamist, it set off a scandal. He was promptly fired from his j o b
at U p p i n g h a m a n d placed on trial for bigamy. He was imprisoned for 3
days, b u t Mary H i n t o n declined to press charges a n d together they left
England for the United States.
H i n t o n was hired as an instructor in the mathematics d e p a r t m e n t at
Princeton University, where his obsession with the fourth dimension was
temporarily sidetracked when he invented the baseball m a c h i n e . T h e
Princeton baseball team benefited from H i n t o n ' s machine, which could
fire baseballs at 70 miles p e r h o u r . T h e descendants of H i n t o n ' s creation
can now be found on every major baseball field in the world.
H i n t o n was eventually fired from Princeton, b u t m a n a g e d to get a
j o b at the United States Naval Observatory t h r o u g h the influence of its
director, a devout advocate of the fourth dimension. T h e n , in 1902, he
took a j o b at the Patent Office in Washington.
Hinton's Cubes
H i n t o n spent years developing ingenious m e t h o d s by which the average
person a n d a growing legion of followers, n o t only professional mathematicians, could " s e e " four-dimensional objects. Eventually, he perfected special cubes that, if o n e tried h a r d e n o u g h , could allow o n e to
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visualize hypercubes, or cubes in four dimensions. These would eventually be called H i n t o n ' s cubes. H i n t o n even coined the official n a m e
for an unraveled hypercube, a tesseract, which found its way into the
English language.
H i n t o n ' s cubes were widely advertised in w o m e n ' s magazines a n d
were even used in seances, where they soon b e c a m e objects of mystical
importance. By mediating on H i n t o n ' s cubes, it was claimed by m e m b e r s
of high society, you could catch glimpses of the fourth dimension a n d
h e n c e the n e t h e r world of ghosts a n d the dearly d e p a r t e d . His disciples
spent hours contemplating a n d meditating on these cubes, until they
attained the ability to mentally rearrange a n d reassemble these cubes
via the fourth dimension into a hypercube. Those who could perform
this mental feat, it was said, would attain the highest state of nirvana.
As an analogy, take a three-dimensional cube. Although a Flatlander
c a n n o t visualize a cube in its entirety, it is possible for us to unravel the
c u b e in three dimensions, so that we have a series of six squares making
a cross. Of course, a Flatlander cannot reassemble the squares to m a k e
a cube. In the second dimension, the joints between each square are
rigid a n d c a n n o t be moved. However, these joints are easy to b e n d in
the third dimension. A Flatlander witnessing this event would see the
squares disappear, leaving only o n e square in his universe (Figure 3.6).
Likewise, a hypercube in four dimensions c a n n o t be visualized. But
o n e can unravel a hypercube into its lower c o m p o n e n t s , which are ordinary three-dimensional cubes. These cubes, in turn, can be arranged in
a three-dimensional cross—a tesseract. It is impossible for us to visualize
how to wrap up these cubes to form a hypercube. However, a higherdimensional person can "lift" each cube off o u r universe a n d t h e n wrap
up the cube to form a hypercube. ( O u r three-dimensional eyes, witnessing this spectacular event, would only see the o t h e r cubes disappear,
leaving only o n e cube in o u r universe.) So pervasive was H i n t o n ' s influe n c e that Salvadore Dali used H i n t o n ' s tesseract in his famous painting
Christus Hypercubus, on display at the Metropolitan Museum of Art in
New York, which depicts Christ b e i n g crucified on a four-dimensional
cross (Figure 3.7).
H i n t o n also knew of a second way to visualize higher-dimensional
objects: by looking at the shadows they cast in lower dimensions. For
example, a Flatlander can visualize a c u b e by looking at its two-dimensional shadow. A cube looks like two squares j o i n e d together. Similarly,
a hypercube's shadow cast on t h e third dimension b e c o m e s a c u b e
within a cube (Figure 3.8).
In addition to visualizing unravelings of hypercubes a n d e x a m i n i n g
their shadows, H i n t o n was aware of a third way to conceptualize the
Figure 3.6. Flatlanders cannot visualize a cube, but they can conceptualize a
three-dimensional cube by unraveling it. To a Flatlander, a cube, when unfolded,
resembles a cross, consisting of six squares. Similarly, we cannot visualize a fourdimensional hypercube, but if we unfold it we have a series of cubes arranged in
a crosslike tesseract. Although the cubes of a tesseract appear immobile, a fourdimensional person can "wrap up" the cubes into a hypercube.
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Figure 3.7. In Christus Hypercubus, Salvador Dali depicted Christ as being
crucified on a tesseract, an unraveled hypercube. (The Metropolitan Museum of
Art. Gift of Chester Dale, Collection, 1955. © 1993. Ars, New York/Demart Pro
Arte, Geneva)
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Figure 3.8. A Flatlander can visualize a cube by examining its shadow, which
appears as a square within a square. If the cube is rotated, the squares execute
motions that appear impossible to a Flatlander. Similarly, the shadow of a hypercube is a cube within a cube. If the hypercube is rotated in four dimensions, the
cubes execute motions that appear impossible to our three-dimensional brains.
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74
fourth dimension: by cross sections. For example, when Mr. Square is
sent into the third dimension, his eyes can see only two-dimensional
cross sections of the third dimension. T h u s he can see only circles
appear, get larger, c h a n g e color, a n d t h e n suddenly disappear. If Mr.
Square moved past an apple, he would see a red circle materialize o u t
of nowhere, gradually e x p a n d , t h e n contract, t h e n turn into a small
brown circle (the stem), a n d finally disappear. Likewise, H i n t o n knew
that if we were h u r l e d into the fourth dimension, we would see strange
objects suddenly a p p e a r o u t of n o w h e r e , get larger, c h a n g e color,
change shape, get smaller, a n d finally disappear.
In summary, H i n t o n ' s contribution may be his popularization of
higher-dimensional figures using three m e t h o d s : by examining their
shadows, their cross sections, a n d their unravellings. Even today, these
t h r e e m e t h o d s are the chief ways in which professional mathematicians
a n d physicists conceptualize higher-dimensional objects in their work.
T h e scientists whose diagrams a p p e a r in today's physics j o u r n a l s owe a
small d e b t of gratitude to H i n t o n ' s work.
The Contest on the Fourth Dimension
In his articles, H i n t o n h a d answers for all possible questions. W h e n people asked h i m to n a m e the fourth dimension, he would reply that the
words ana a n d kata described moving in the fourth dimension a n d were
the counterparts of the terms up a n d down, or left a n d right. W h e n asked
where the fourth dimension was, he also h a d a ready answer.
For t h e m o m e n t , consider the motion of cigarette smoke in a closed
r o o m . Because the atoms of the smoke, by the laws of thermodynamics,
spread a n d diffuse into all possible locations in the r o o m , we can determ i n e if t h e r e are any regions of ordinary three-dimensional space that
the smoke molecules miss. However, experimental observations show
that there are no such h i d d e n regions. Therefore, the fourth spatial
dimension is possible only if it is smaller than the smoke particles. T h u s
if the fourth dimension actually exists, it must be incredibly small, even
smaller t h a n an atom. This is the philosophy that H i n t o n a d o p t e d , that
all objects in o u r three-dimensional universe exist in the fourth dimension, b u t that the fourth dimension is so small that it evades any experimental observation. (We will find that physicists today a d o p t essentially
the same philosophy as H i n t o n a n d conclude that t h e h i g h e r dimensions
are too small to be experimentally seen. W h e n asked, " W h a t is light?"
he also h a d a ready answer. Following Riemann, H i n t o n believed that
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light is a vibration of the unseen fourth dimension, which is essentially
the viewpoint taken today by many theoretical physicists.)
In the United States, H i n t o n single-handedly sparked an e n o r m o u s
public interest in the fourth dimension. Popular magazines like Harper's
Weekly, McClure's, Current Literature, Popular Science Monthly, a n d Science all
devoted pages to the blossoming interest in the fourth dimension. But
what probably e n s u r e d H i n t o n ' s fame in America was the famous contest
sponsored by Scientific American in 1909. This unusual contest offered a
$500 prize (a considerable a m o u n t of m o n e y in 1909) to " t h e best popular explanation of the Fourth Dimension." T h e magazine's editors
were pleasantly surprised by the deluge of letters that p o u r e d into their
offices, including entries from as far away as Turkey, Austria, Holland,
India, Australia, France, a n d Germany.
T h e object of the contest was to "set forth in an essay n o t longer
t h a n twenty-five h u n d r e d words the m e a n i n g of the term so that the
ordinary lay reader could u n d e r s t a n d it." It drew a large n u m b e r of
serious essays. Some l a m e n t e d the fact that people like Zollner a n d Slade
h a d besmirched the reputation of the fourth dimension by confusing it
with spiritualism. However, many of the essays recognized H i n t o n ' s pion e e r i n g work on the fourth dimension. (Surprisingly, n o t o n e essay mentioned the work of Einstein. In 1909, it was still far from clear that Einstein h a d uncovered the secret of space a n d time. In fact, the idea of
time as the fourth dimension did n o t a p p e a r in a single essay.)
Without experimental verification, the Scientific American contest
could not, of course, resolve the question of the existence of higher
dimensions. However, the contest did address the question of what
higher-dimensional objects might look like.
Monsters from the Fourth Dimension
What would it be like to m e e t a creature from a higher dimension?
Perhaps the best way to explain the wonder a n d excitement of a
hypothetical visit to o t h e r dimensions is t h r o u g h science fiction, where
writers have tried to grapple with this question.
In " T h e Monster from N o w h e r e , " writer Nelson Bond tried to imagine what would h a p p e n if an explorer in the jungles of Latin America
e n c o u n t e r e d a beast from a higher dimension.
O u r h e r o is Burch Patterson, adventurer, b o n vivant, a n d soldier of
fortune, who hits on the idea of capturing wild animals in the towering
m o u n t a i n s of Peru. T h e expedition will be paid for by various zoos,
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which p u t up the m o n e y for the trip in r e t u r n for whatever animals
Patterson can find. With m u c h hoopla a n d fanfare, the press covers the
progress of the expedition as it journeys i n t o u n e x p l o r e d territory. But
after a few weeks, the expedition loses contact with the outside world
a n d mysteriously disappears without a trace. After a long a n d futile
search, the authorities reluctantly give the explorers up for dead.
Two years later, Burch Patterson abruptly reappears. He meets
secretly with reporters a n d tells t h e m an astonishing story of tragedy a n d
heroism. Just before the expedition disappeared, it e n c o u n t e r e d a fantastic animal in the Maratan Plateau of u p p e r Peru, an unearthly bloblike
creature that was constantly c h a n g i n g shape in the most bizarre fashion.
T h e s e black blobs hovered in midair, disappearing a n d r e a p p e a r i n g a n d
changing shape a n d size. T h e blobs t h e n unexpectedly attacked the
expedition, killing most of the m e n . T h e blobs hoisted some of the
remaining m e n off the g r o u n d ; they screamed a n d t h e n disappeared
into thin air.
Only Burch escaped the rout. Dazed a n d frightened, he nonetheless
studied these blobs from a distance a n d gradually formed a theory about
what they were a n d how to c a p t u r e t h e m . He h a d read Flatland years
before, a n d imagined that anyone sticking his fingers into a n d out of
Flatland would startle the two-dimensional inhabitants. T h e Flatlanders
would see pulsating rings of flesh hovering in midair (our fingers poking
t h r o u g h Flatland), constantly changing size. Likewise, reasoned Patterson, any higher-dimensional creature sticking his foot or arms t h r o u g h
o u r universe would appear as three-dimensional, pulsating blobs of
flesh, a p p e a r i n g o u t of nowhere a n d constantly changing shape and size.
T h a t would also explain why his team m e m b e r s h a d disappeared into
thin air: They h a d b e e n dragged into a higher-dimensional universe.
But o n e question still plagued him: How do you capture a higherdimensional being? If a Flatlander, seeing our finger p o k e its way
t h r o u g h his two-dimensional universe, tried to capture o u r finger, he
would be at a loss. If he tried to lasso our finger, we could simply remove
o u r finger a n d disappear. Similarly, Patterson reasoned, he could p u t a
n e t a r o u n d o n e of these blobs, b u t then the higher-dimensional creature
could simply pull his " f i n g e r " or " l e g " o u t of our universe, a n d the net
would collapse.
Suddenly, the answer came to him: If a Flatlander were to try to
capture o u r finger as it poked its way into Flatland, the Flatlander could
stick a n e e d l e through our finger, painfully impaling it to the two-dimensional universe. T h u s Patterson's strategy was to drive a spike t h r o u g h
o n e of the blobs a n d impale the creature in our universe!
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After m o n t h s of observing the creature, Patterson identified what
looked like the creature's " f o o t " a n d drove a spike right t h r o u g h it. It
took h i m 2 years to c a p t u r e the creature a n d ship t h e writhing, struggling blob back to New Jersey.
Finally, Patterson a n n o u n c e s a major press conference where he will
unveil a fantastic creature caught in Peru. Journalists a n d scientists alike
gasp in h o r r o r when the creature is unveiled, writhing a n d struggling
against a large steel rod. Like a scene from King Kong, o n e newspaperman, against the rules, takes flash pictures of the creature. T h e flash
enrages the creature, which t h e n struggles so hard against the rod that
its flesh begins to tear. Suddenly, the m o n s t e r is free, a n d p a n d e m o n i u m
breaks out. People are torn to shreds, a n d Patterson a n d others are
g r a b b e d by the creature a n d t h e n disappear into the fourth dimension.
In the aftermath of the tragedy, o n e of the survivors of the massacre
decides to b u r n all evidence of the creature. Better to leave this mystery
forever unsolved.
Building a Four-Dimensional House
In t h e previous section, the question of what h a p p e n s when we e n c o u n ter a higher-dimensional being was explored. But what h a p p e n s in the
reverse situation, when we visit a higher-dimensional universe? As we
have seen, a Flatlander c a n n o t possibly visualize a three-dimensional universe in its entirety. However, there are, as H i n t o n showed, several ways
in which the Flatlander can c o m p r e h e n d revealing fragments of higherdimensional universes.
In his classic short story " . . . And He Built a Crooked H o u s e
Robert Heinlein explored the many possibilities of living in an unraveled
hypercube.
Quintus Teal is a brash, flamboyant architect whose ambition is to
build a house in a truly revolutionary shape: a tesseract, a hypercube
that has b e e n unraveled in the third dimension. He cons his friends Mr.
a n d Mrs. Bailey into buying the house.
Built in Los Angeles, the tesseract is a series of eight u l t r a m o d e r n
cubes stacked on top of o n e a n o t h e r in the shape of a cross. Unfortunately, just as Teal is a b o u t to show off his new creation to the Baileys,
an e a r t h q u a k e strikes southern California, a n d the h o u s e collapses into
itself. T h e cubes begin to topple, b u t strangely only a single c u b e is left
standing. T h e o t h e r cubes have mysteriously disappeared. W h e n Teal
a n d the Baileys cautiously enter the house, now just a single cube, they
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are amazed that the o t h e r missing rooms are clearly visible t h r o u g h the
windows of the first floor. But that is impossible. T h e h o u s e is now only
a single cube. How can the interior of a single cube be c o n n e c t e d to a
series of o t h e r cubes that c a n n o t be seen from the outside?
They climb the stairs a n d find the master b e d r o o m above the entryway. Instead of finding the third floor, however, they find themselves
back on the g r o u n d floor. T h i n k i n g t h e h o u s e is h a u n t e d , the frightened
Baileys race to the front door. Instead of leading to the outside, the front
d o o r just leads to a n o t h e r r o o m . Mrs. Bailey faints.
As they explore the house, they find that each r o o m is c o n n e c t e d to
an impossible series of o t h e r rooms. In the original house, each cube
h a d windows to view the outside. Now, all windows face o t h e r rooms.
T h e r e is no outside!
Scared o u t of their wits, they slowly try all the doors of t h e house,
only to wind up in o t h e r rooms. Finally, in the study they decide to o p e n
the four Venetian blinds a n d look outside. W h e n they o p e n the first
Venetian blind, they find that they are p e e r i n g down at the Empire State
Building. Apparently, that window o p e n e d up to a " w i n d o w " in space
j u s t above t h e spire of t h e tower. W h e n they o p e n the second Venetian
blind, they find themselves staring at a vast ocean, except it is upside
down. O p e n i n g the third Venetian blind, they find themselves looking
at Nothing. N o t empty space. Not inky blackness. Just Nothing. Finally,
o p e n i n g up the last Venetian blind, they find themselves gazing at a
bleak desert landscape, probably a scene from Mars.
After a harrowing tour t h r o u g h the rooms of the house, with each
r o o m impossibly c o n n e c t e d to the o t h e r rooms, Teal finally figures it all
out. T h e earthquake, he reasons, must have collapsed the joints of various cubes a n d folded t h e h o u s e in the fourth dimension.
On the outside, Teal's house originally looked like an ordinary
sequence of cubes. T h e h o u s e did n o t collapse because the j o i n t s
between the cubes were rigid a n d stable in three dimensions. However,
viewed from the fourth dimension, Teal's house is an unraveled hyperc u b e that can be reassembled or folded back into a hypercube. T h u s
when the house was shaken by the earthquake, it somehow folded up in
four dimensions, leaving only a single c u b e dangling in o u r third dimension. Anyone walking into the single remaining cube would view a series
of rooms c o n n e c t e d in a seemingly impossible fashion. By racing
t h r o u g h the various rooms, Teal has moved t h r o u g h the fourth dimension without noticing it.
Although our protagonists seem d o o m e d to s p e n d their lives fruitlessly wandering in circles inside a hypercube, a n o t h e r violent earth-
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quake shakes the tesseract. H o l d i n g their breath, Teal a n d the terrified
Baileys leap o u t the nearest window. W h e n they land, they find themselves in J o s h u a T r e e National M o n u m e n t , miles from Los Angeles.
H o u r s later, hitching a ride back to the city, they r e t u r n to the house,
only to find that the last r e m a i n i n g cube has vanished. W h e r e did the
tesseract go? It is probably drifting somewhere in the fourth dimension.
The Useless Fourth Dimension
In retrospect, R i e m a n n ' s famous lecture was popularized to a wide audie n c e via mystics, philosophers, a n d artists, b u t did little to further o u r
u n d e r s t a n d i n g of n a t u r e . From the perspective of m o d e r n physics, we
can also see why the years 1860 to 1905 did n o t p r o d u c e any fundamental
b r e a k t h r o u g h s in o u r u n d e r s t a n d i n g of hyperspace.
First, t h e r e was no attempt to use hyperspace to simplify the laws of
n a t u r e . Without R i e m a n n ' s original guiding principle—that the laws of
n a t u r e b e c o m e simple in h i g h e r dimensions—scientists d u r i n g this
p e r i o d were g r o p i n g in the dark. R i e m a n n ' s seminal idea of using geometry—that is, c r u m p l e d hyperspace—to explain the essence of a " f o r c e "
was forgotten d u r i n g those years.
Second, t h e r e was no a t t e m p t to exploit Faraday's field concept or
R i e m a n n ' s metric tensor to find the field equations obeyed by hyperspace. T h e mathematical apparatus developed by Riemann became a
province of p u r e mathematics, contrary to R i e m a n n ' s original intentions. Without field theory, you c a n n o t make any predictions with hyperspace.
T h u s by the t u r n of the century, the cynics claimed (with justification) that there was no experimental confirmation of the fourth dimension. Worse, they claimed, t h e r e was no physical motivation for introd u c i n g t h e fourth dimension, o t h e r t h a n to titillate the general public
with ghost stories. This deplorable situation would soon change, however. Within a few decades, the theory of the fourth dimension (of time)
would forever c h a n g e the course of h u m a n history. It would give us t h e
atomic b o m b a n d the theory of Creation itself. And the m a n who would
do it would be an obscure physicist n a m e d Albert Einstein.
4
The Secret of Light:
Vibrations in
the Fifth Dimension
If [relativity] s h o u l d p r o v e to be c o r r e c t , as I e x p e c t it will, he
will b e c o n s i d e r e d t h e C o p e r n i c u s o f t h e t w e n t i e t h c e n t u r y .
Max Planck on Albert Einstein
T
HE life of Albert Einstein a p p e a r e d to be o n e l o n g series of failures
a n d disappointments. Even his m o t h e r was distressed at how slowly
he l e a r n e d to talk. His elementary-school teachers t h o u g h t h i m a foolish
d r e a m e r . They complained that he was constantly disrupting classroom
discipline with his silly questions. O n e teacher even told the boy bluntly
that he would prefer that Einstein d r o p o u t of his class.
He h a d few friends in school. Losing interest in his courses, he
d r o p p e d o u t of high school. Without a high-school diploma, he h a d to
take special exams to e n t e r college, but he did n o t pass t h e m a n d h a d
to take t h e m a second time. He even failed the exam for the Swiss military because he h a d flat feet.
After graduation, he could n o t get a j o b . He was an unemployed
physicist w h o was passed over for a teaching position at the university
a n d was rejected for j o b s everywhere he applied. He e a r n e d barely 3
francs an h o u r — a pittance—by tutoring students. He told his friend
Maurice Solovine that " a n easier way of earning a living would be to
play the violin in public places."
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The Secret of Light: Vibrations in the Fifth Dimension
81
Einstein was a m a n who rejected the things most m e n chase after,
such as power a n d money. However, he o n c e n o t e d pessimistically, "By
t h e m e r e existence of his stomach, everyone is c o n d e m n e d to participate
in that c h a s e . " Finally, t h r o u g h the influence of a friend, he landed a
lowly j o b as a clerk at the Swiss p a t e n t office in Bern, e a r n i n g j u s t e n o u g h
m o n e y so his p a r e n t s would n o t have to s u p p o r t him. On his m e a g e r
salary, he s u p p o r t e d his young wife a n d their n e w b o r n baby.
Lacking financial resources or connections with the scientific establishment, Einstein b e g a n to work in solitude at t h e p a t e n t office. In
between p a t e n t applications, his m i n d drifted to problems that h a d
intrigued him as a youth. He then u n d e r t o o k a task that would eventually
c h a n g e t h e course of h u m a n history. His tool was the fourth
dimension.
Children's Questions
W h e r e i n lies the essence of Einstein's genius? In The Ascent of Man, J a c o b
Bronowski wrote: " T h e genius of m e n like Newton a n d Einstein lies in
that: they ask transparent, i n n o c e n t questions which turn o u t to have
catastrophic answers. Einstein was a m a n w h o could ask immensely simple questions." As a child, Einstein asked himself the simple question:
W h a t would a light b e a m look like if you could catch up with one? Would
you see a stationary wave, frozen in time? This question set h i m on a 50year j o u r n e y t h r o u g h the mysteries of space and time.
1
Imagine trying to overtake a train in a speeding car. If we hit the gas
pedal, o u r car races neck-and-neck with the train. We can p e e r inside
the train, which now appears to be at rest. We can see the seats a n d t h e
p e o p l e , who are acting as t h o u g h the train weren't moving. Similarly,
Einstein as a child imagined traveling alongside a light beam.
He t h o u g h t that the light beam should resemble a series of
stationary waves, frozen in time; that is, the light b e a m should a p p e a r
motionless.
W h e n Einstein was 16 years old, he spotted the flaw in this a r g u m e n t .
He recalled later,
After ten years of reflection such a principle resulted f r o m a p a r a d o x u p o n
w h i c h I h a d already hit at the a g e of sixteen: If I pursue a b e a m of light
with t h e v e l o c i t y c ( v e l o c i t y of l i g h t in a v a c u u m ) I s h o u l d o b s e r v e s u c h a
b e a m o f l i g h t a s a spatially oscillatory e l e c t r o m a g n e t i c f i e l d a t rest. H o w -
82
e v e r , t h e r e s e e m s t o b e n o s u c h t h i n g , w h e t h e r o n t h e basis o f e x p e r i e n c e
or according to Maxwell's equations.
2
In college, Einstein confirmed his suspicions. He learned that light
can be expressed in terms of Faraday's electric a n d magnetic fields, a n d
that these fields obey the field equations found by J a m e s Clerk Maxwell.
As he suspected, he found that stationary, frozen waves are n o t allowed
by Maxwell's field equations. In fact, Einstein showed that a light b e a m
travels at the same velocity c, no matter how hard you try to catch
up with it.
At first, this seemed absurd. This m e a n t that we could never overtake
the train (light b e a m ) . Worse, no matter how fast we drove o u r car, the
train would always seem to be traveling ahead of us at the same velocity.
In o t h e r words, a light b e a m is like the "ghost s h i p " that old sailors love
to spin tall tales about. It is a p h a n t o m vessel that can never be caught.
No matter how fast we sail, the ghost ship always eludes us, taunting us.
In 1905, with plenty of time on his h a n d s at the p a t e n t office, Einstein
carefully analyzed the field equations of Maxwell and was led to postulate
the principle of special relativity: T h e speed of light is the same in all
constantly moving frames. This innocent-sounding principle is o n e of
the greatest achievements of the h u m a n spirit. Some have said that it
ranks with Newton's law of gravitation as o n e of the greatest scientific
creations of the h u m a n m i n d in the 2 million years o u r species has been
evolving on this planet. From it, we can logically unlock the secret of the
vast energies released by the stars a n d galaxies.
To see how this simple statement can lead to such profound conclusions, let us return to the analogy of the car trying to overtake the train.
Let us say that a pedestrian on the sidewalk clocks o u r car traveling at
99 miles p e r hour, a n d the train traveling at 100 miles per h o u r . Naturally, from o u r point of view in the car, we see the train moving ahead
of us at 1 mile per h o u r . This is because velocities can be a d d e d a n d
subtracted, j u s t like ordinary n u m b e r s .
Now let us replace the train by a light b e a m , but keep the velocity of
light at j u s t 100 miles p e r hour. T h e pedestrian still clocks o u r car traveling at 99 miles p e r h o u r in h o t pursuit of the light beam traveling at
100 miles per hour. According to the pedestrian, we should be closing
in on the light beam. However, according to relativity, we in the car
actually see the light b e a m n o t traveling ahead of us at 1 mile per hour,
as expected, b u t speeding ahead of us at 100 miles p e r h o u r . Remarkably, we see the light b e a m racing ahead of us as t h o u g h we were at rest.
Not believing o u r own eyes, we slam on the gas pedal until the pedestrian
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clocks o u r car racing a h e a d at 99.99999 miles p e r h o u r . Surely, we think,
we must be a b o u t to overtake the light beam. However, when we look
o u t the window, we see the light b e a m still speeding a h e a d of us at 100
miles p e r h o u r .
Uneasily, we reach several bizarre, disturbing conclusions. First, no
matter how m u c h we gun the engines of o u r car, the pedestrian tells us
that we can approach b u t never exceed 100 miles p e r hour. This seems
to be the top velocity of the car. Second, no matter how close we come
to 100 miles p e r h o u r , we still see the light b e a m speeding a h e a d of us
at 100 miles per h o u r , as t h o u g h we weren't moving at all.
But this is absurd. How can b o t h people in the speeding car a n d the
stationary person measure the velocity of the light b e a m to be the same?
Ordinarily, this is impossible. It appears to be nature's colossal j o k e .
T h e r e is only o n e way out of this paradox. Inexorably, we are led to
the astonishing conclusion that shook Einstein to the core when he first
conceived of it. T h e only solution to this puzzle is that time slows down
for us in the car. If the pedestrian takes a telescope a n d peers into o u r
car, he sees everyone in the car moving exceptionally slowly. However,
we in the car never notice that time is slowing down because o u r brains,
too, have slowed down, a n d everything seems n o r m a l to us. Furtherm o r e , he sees that the car has b e c o m e flattened in the direction of
m o t i o n . T h e car has s h r u n k like an accordion. However, we never feel
this effect because o u r bodies, too, have shrunk.
Space a n d time play tricks on us. In actual experiments, scientists
have shown that the speed of light is always c, no matter how fast we
travel. This is because the faster we travel, the slower o u r clocks tick a n d
the shorter o u r rulers become. In fact, o u r clocks slow down a n d o u r
rulers shrink j u s t e n o u g h so that whenever we measure the speed of
light, it comes o u t the same.
But why c a n ' t we see or feel this effect? Since o u r brains are thinking
m o r e slowly, a n d our bodies are also getting t h i n n e r as we a p p r o a c h the
speed of light, we are blissfully unaware that we are t u r n i n g into slowwitted pancakes.
These relativistic effects, of course, are too small to be seen in everyday life because the speed of light is so great. Being a New Yorker, however, I am constantly r e m i n d e d of these fantastic distortions of space
a n d time whenever I ride the subway. W h e n I am on the subway platform
with n o t h i n g to do except wait for the next subway train, I sometimes
let my imagination drift a n d w o n d e r what it would be like if the speed
of light were only, say, 30 miles per h o u r , the speed of a subway train.
T h e n when the train finally roars into the station, it appears squashed,
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like an accordion. T h e train, I imagine, would be a flattened slab of
metal 1 foot thick, barreling down the tracks. And everyone inside the
subway cars would be as thin as paper. They would also be virtually frozen
in time, as t h o u g h they were motionless statues. However, as the train
comes to a grinding halt, it suddenly expands, until this slab of metal
gradually fills the entire station.
As absurd as these distortions might appear, the passengers inside
the train would be totally oblivious to these changes. T h e i r bodies a n d
space itself would be compressed along the direction of motion of the
train; everything would a p p e a r to have its normal shape. F u r t h e r m o r e ,
their brains would have slowed down, so that everyone inside the train
would act normally. T h e n when the subway train finally comes to a halt,
they are totally unaware that their train, to s o m e o n e on the platform,
appears to miraculously e x p a n d until it fills up the entire platform.
W h e n the passengers d e p a r t from the train, they are totally oblivious to
the profound changes d e m a n d e d by special relativity.*
The Fourth Dimension and High-School Reunions
T h e r e have b e e n , of course, h u n d r e d s of p o p u l a r accounts of Einstein's
theory, stressing different aspects of his work. However, few accounts
capture the essence b e h i n d the theory of special relativity, which is that
time is the fourth dimension a n d that the laws of n a t u r e are simplified
a n d unified in higher dimensions. I n t r o d u c i n g time as the fourth dimension overthrew the c o n c e p t of time dating all the way back to Aristotle.
Space a n d time would now be forever dialectically linked by special relativity. (Zollner a n d H i n t o n h a d assumed that the next dimension to be
discovered would be the fourth spatial dimension. In this respect, they
were wrong a n d H. G. Wells was correct. T h e next dimension to be
discovered would be time, a fourth temporal dimension. Progress in
u n d e r s t a n d i n g the fourth spatial dimension would have to wait several
m o r e decades.)
To see how higher dimensions simplify the laws of n a t u r e , we recall
that any object has length, width, a n d d e p t h . Since we have the freedom
•Similarly, passengers riding in the train would think that the train was at rest and that
the subway station was coming toward the train. They would see the platform and everyone
standing on it compressed like an accordian. T h e n this leads us to a contradiction, that
people on the train and in the station each think that the other has b e e n compressed. T h e
resolution of this paradox is a bit delicate.
3
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to rotate an object by 90 degrees, we can t u r n its length into width a n d
its width into d e p t h . By a simple rotation, we can interchange any of the
three spatial dimensions. Now if time is the fourth dimension, t h e n it is
possible to make " r o t a t i o n s " that convert space into time a n d vice versa.
These four-dimensional " r o t a t i o n s " are precisely the distortions of
space a n d time d e m a n d e d by special relativity. In o t h e r words, space a n d
time have mixed in an essential way, governed by relativity. T h e m e a n i n g
of time as being the fourth dimension is that time a n d space can rotate
into each o t h e r in a mathematically precise way. From now on, they must
be treated as two aspects of the same quantity: space-time. Thus a d d i n g
a higher dimension h e l p e d to unify the laws of n a t u r e .
Newton, writing 300 years ago, t h o u g h t that time beat at the same
rate everywhere in the universe. W h e t h e r we sat on the earth, on Mars,
or on a distant star, clocks were expected to tick at the same rate. T h e r e
was t h o u g h t to be an absolute, uniform rhythm to the passage of time
t h r o u g h o u t the entire universe. Rotations between time a n d space were
inconceivable. Time a n d space were two distinct quantities with no relationship between t h e m . Unifying t h e m into a single quantity was
unthinkable. However, according to special relativity, time can beat at
different rates, d e p e n d i n g on how fast o n e is moving. Time being the
fourth dimension m e a n s that time is intrinsically linked with m o v e m e n t
in space. How fast a clock ticks d e p e n d s on how fast it is moving in space.
Elaborate experiments d o n e with atomic clocks sent into orbit a r o u n d
the earth have confirmed that a clock on the earth a n d a clock rocketing
in o u t e r space tick at different rates.
I was graphically r e m i n d e d of the relativity principle when I was
invited to my twentieth high-school r e u n i o n . Although I h a d n ' t seen
most of my classmates since graduation, I assumed that all of t h e m would
show the same telltale signs of aging. As expected, most of us at the
r e u n i o n were relieved to find that the aging process was universal: It
seemed that all of us sported graying temples, e x p a n d i n g waistlines, a n d
a few wrinkles. Although we were separated across space a n d time by
several thousand miles a n d 20 years, each of us had assumed that time
h a d beat uniformly for all. We automatically assumed that each of us
would age at the same rate.
T h e n my m i n d wandered, a n d I imagined what would h a p p e n if a
classmate walked into the r e u n i o n hall looking exactly as he h a d on graduation day. At first, he would probably draw stares from his classmates.
Was this the same person we knew 20 years ago? W h e n people realized
that he was, a panic would surge t h r o u g h the hall.
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We would be jolted by this e n c o u n t e r because we tacitly assume that
clocks beat the same everywhere, even if they are separated by vast distances. However, if time is the fourth dimension, then space and time
can rotate into each other and clocks can beat at different rates, depending on how fast they move. This classmate, for example, may have
entered a rocket traveling at near-light speeds. For us, the rocket trip
may have lasted for 20 years. However, for him, because time slowed
down in the speeding rocket, he aged only a few m o m e n t s from graduation day. To him, he just e n t e r e d the rocket, sped into outer space
for a few minutes, a n d then landed back on earth in time for his twentieth high-school r e u n i o n after a short, pleasant journey, still looking
youthful amid a field of graying hair.
I am also r e m i n d e d that the fourth dimension simplifies the laws of
nature whenever I think back to my first e n c o u n t e r with Maxwell's field
equations. Every u n d e r g r a d u a t e student learning the theory of electricity and magnetism toils for several years to master these eight abstract
equations, which are exceptionally ugly a n d very opaque. Maxwell's
eight equations are clumsy and difficult to memorize because time and
space are treated separately. (To this day, I have to look t h e m up in a
book to make sure that I get all the signs and symbols correct.) I still
r e m e m b e r the relief I felt when I learned that these equations collapse
into o n e trivial-looking equation when time is treated as the fourth
dimension. In o n e masterful stroke, the fourth dimension simplifies
these equations in a beautiful, transparent fashion. Written in this way,
the equations possess a higher symmetry; that is, space and time can turn
into each other. Like a beautiful snowflake that remains the same when
we rotate it a r o u n d its axis, Maxwell's field equations, written in relativistic form, remain the same when we rotate space into time.
4
Remarkably, this o n e simple equation, written in a relativistic fashion,
contains the same physical c o n t e n t as the eight equations originally written down by Maxwell over 100 years ago. This o n e equation, in turn,
governs the properties of dynamos, radar, radio, television, lasers, household appliances, and the cornucopia of consumer electronics that
appear in everyone's living r o o m . This was o n e of my first exposures to
the concept of beauty in physics—that is, that the symmetry of fourdimensional space can explain a vast ocean of physical knowledge that
would fill an engineering library.
O n c e again, this demonstrates o n e of the main themes of this book,
that the addition of higher dimensions helps to simplify a n d unify the
laws of nature.
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Matter as Condensed Energy
This discussion of unifying the laws of n a t u r e , so far, has b e e n rather
abstract, a n d would have r e m a i n e d so h a d Einstein n o t taken the next
fateful step. He realized that if space a n d time can be unified into a
single entity, called space-time, t h e n p e r h a p s matter a n d energy can
also be united into a dialectical relationship. If rulers can shrink a n d
clocks slow down, he reasoned, then everything that we measure with
rulers a n d clocks must also change. However, almost everything in a
physicist's laboratory is measured by rulers a n d clocks. This m e a n t that
physicists h a d to recalibrate all the laboratory quantities they once took
for granted to be constant.
Specifically, energy is a quantity that d e p e n d s on how we measure
distances a n d time intervals. A speeding test car slamming into a brick
wall obviously has energy. If the speeding car approaches the speed of
light, however, its properties b e c o m e distorted. It shrinks like an accordion a n d clocks in it slow down.
More important, Einstein found that the mass of the car also
increases as it speeds u p . But where did this excess mass c o m e from?
Einstein concluded that it came from the energy.
This h a d disturbing consequences. Two of the great discoveries of
nineteenth-century physics were the conservation of mass a n d the conservation of energy; that is, the total mass a n d total energy of a closed
system, taken separately, do n o t change. For example, if the speeding
car hits the brick wall, the energy of the car does n o t vanish, but is
converted into the s o u n d energy of the crash, the kinetic energy of the
flying brick fragments, heat energy, a n d so on. T h e total energy (and
total mass) before a n d after the crash is the same.
However, Einstein now said that the energy of the car could be converted into mass—a new conservation principle that said that the sum
total of the mass a d d e d to energy must always remain the same. Matter
does not suddenly disappear, n o r does energy spring out of nothing. In
this regard, the God-builders were wrong a n d Lenin was right. Matter
disappears only to unleash e n o r m o u s quantities of energy, or vice versa.
W h e n Einstein was 26 years old, he calculated precisely how energy
must c h a n g e if the relativity principle was correct, a n d he discovered the
relation E = mc . Since the speed of light squared (c ) is an astronomically large n u m b e r , a small a m o u n t of matter can release a vast a m o u n t
of energy. Locked within the smallest particles of matter is a storehouse
of energy, m o r e than 1 million times the energy released in a chemical
2
2
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explosion. Matter, in some sense, can be seen as an almost inexhaustible
storehouse of energy; that is, matter is c o n d e n s e d energy.
In this respect, we see the p r o f o u n d difference between the work of
the mathematician (Charles H i n t o n ) a n d that of the physicist (Albert
Einstein). H i n t o n spent most of his adult years trying to visualize higher
spatial dimensions. He h a d no interest in finding a physical interpretation for the fourth dimension. Einstein saw, however, that the fourth
dimension can be taken as a temporal o n e . He was guided by a conviction
a n d physical intuition that higher dimensions have a purpose: to unify
the principles of n a t u r e . By a d d i n g higher dimensions, he could unite
physical concepts that, in a three-dimensional world, have no connection, such as matter a n d energy.
From t h e n on, the concept of matter a n d energy would be taken as
a single unit: m a t t e r - e n e r g y . T h e direct impact of Einstein's work on
the fourth dimension was, of course, the hydrogen b o m b , which has
proved to be the most powerful creation of twentieth-century science.
"The Happiest Thought of My Life"
Einstein, however, wasn't satisfied. His special theory of relativity alone
would have g u a r a n t e e d h i m a place a m o n g the giants of physics. But
there was something missing.
Einstein's key insight was to use the fourth dimension to unite the
laws of n a t u r e by introducing two new concepts: space-time a n d m a t t e r energy. Although he h a d unlocked some of the deepest secrets of
n a t u r e , he realized there were several gaping holes in his theory. What
was the relationship between these two new concepts? More specifically,
what a b o u t accelerations, which are ignored in special relativity? And
what about gravitation?
His friend Max Planck, the founder of the q u a n t u m theory, advised
the young Einstein that the p r o b l e m of gravitation was too difficult.
Planck told h i m that he was too ambitious: "As an older friend I must
advise you against it for in the first place you will n o t succeed; a n d even
if you succeed, no o n e will believe y o u . " Einstein, however, plunged
a h e a d to unravel the mystery of gravitation. O n c e again, the key to his
m o m e n t o u s discovery was to ask questions that only children ask.
W h e n children ride in an elevator, they sometimes nervously ask,
" W h a t h a p p e n s if the rope breaks?" T h e answer is that you b e c o m e
weightless a n d float inside the elevator, as t h o u g h in o u t e r space,
because b o t h you a n d the elevator are falling at the same rate. Even
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t h o u g h b o t h you a n d the elevator are accelerating in the e a r t h ' s gravitational field, the acceleration is the same for both, a n d h e n c e it appears
that you are weightless in the elevator (at least until you reach the bottom of the shaft).
In 1907, Einstein realized that a person floating in the elevator might
think that s o m e o n e h a d mysteriously t u r n e d off gravity. Einstein once
recalled, "I was sitting in a chair in the p a t e n t office at Bern when all of
a sudden a t h o u g h t occurred to me: 'If a person falls freely he will n o t
feel his own weight.' I was startled. This simple t h o u g h t m a d e a d e e p
impression on m e . It impelled me toward a theory of gravitation." Einstein would call it " t h e happiest t h o u g h t of my life."
6
Reversing the situation, he knew that s o m e o n e in an accelerating
rocket will feel a force pushing h i m into his seat, as t h o u g h there were
a gravitational pull on him. (In fact, the force of acceleration felt by o u r
astronauts is routinely measured in g's—that is, multiples of the force of
the earth's gravitation.) T h e conclusion he reached was that s o m e o n e
accelerating in a speeding rocket may think that these forces were caused
by gravity.
From this children's question, Einstein grasped the fundamental
n a t u r e of gravitation: The laws of nature in an accelerating frame are equivalent to the laws in a gravitational field. This simple statement, called the
equivalence principle, may n o t m e a n m u c h to the average person, b u t once
again, in the h a n d s of Einstein, it became the foundation of a theory of
t h e cosmos.
(The equivalence principle also gives simple answers to complex
physics questions. For example, if we are holding a helium balloon while
riding in a car, a n d the car suddenly swerves to the left, o u r bodies will
be jolted to the right, b u t which way will the balloon move? C o m m o n
sense tells us that the balloon, like o u r bodies, will move to the right.
However, the correct resolution of this subtle question has s t u m p e d even
experienced physicists. T h e answer is to use the equivalence principle.
Imagine a gravitational field pulling on the car from the right. Gravity
will make us lurch us to the right, so the helium balloon, which is lighter
than air a n d always floats " u p , " opposite the pull of gravity, must float
to the left, into the direction of the swerve, defying c o m m o n sense.)
Einstein exploited the equivalence principle to solve the long-standing p r o b l e m of w h e t h e r a light b e a m is affected by gravity. Ordinarily,
this is a highly nontrivial question. T h r o u g h the equivalence principle,
however, t h e answer becomes obvious. If we shine a flashlight inside an
accelerating rocket, the light b e a m will b e n d downward toward the floor
(because the rocket has accelerated b e n e a t h the light beam d u r i n g the
90
time it takes for the light b e a m to move across the r o o m ) . Therefore,
argued Einstein, a gravitational field will also b e n d the path of light.
Einstein knew that a fundamental principle of physics is that a light
beam will take the path requiring the least a m o u n t of time between two
points. (This is called Fermat's least-time principle.) Ordinarily, the path
with the smallest time between two points is a straight line, so light beams
are straight. (Even when light b e n d s u p o n entering glass, it still obeys
t h e least-time principle. This is because light slows down in glass, a n d
the p a t h with the least time t h r o u g h a combination of air a n d glass is
now a b e n t line. This is called refraction, which is the principle b e h i n d
microscopes a n d telescopes.)*
However, if light takes the p a t h with the least time between two
points, and light beams b e n d u n d e r the influence of gravity, t h e n the
shortest distance between two points is a curved line. Einstein was
shocked by this conclusion: If light could be observed traveling in a
curved line, it would m e a n that space itself is curved.
Space Warps
At the core of Einstein's belief was the idea that " f o r c e " could be
explained using p u r e geometry. For example, think of riding on a merrygo-round. Everyone knows that if we change horses on a merry-go-round,
we feel a " f o r c e " tugging at us as we walk across the platform. Because
the outer rim of the merry-go-round moves faster t h a n the center, the
o u t e r rim of the merry-go-round must shrink, according to special relativity. However, if the platform of the merry-go-round now has a
s h r u n k e n rim or circumference, the platform as a whole must be curved.
To s o m e o n e on the platform, light no longer travels in a straight line,
as t h o u g h a " f o r c e " were pulling it toward the rim. T h e usual t h e o r e m s
of geometry no longer hold. T h u s the " f o r c e " we feel while walking
between horses on a merry-go-round can be explained as the curving of
space itself.
Einstein i n d e p e n d e n t l y discovered R i e m a n n ' s original p r o g r a m , to
give a purely geometric explanation of the c o n c e p t of " f o r c e . " We recall
*For example, imagine being a lifeguard on a beach, at some distance from the water;
out of the corner of your eye, you spy s o m e o n e drowning in the ocean far off at an angle.
Assume that you can run very slowly in the soft sand, but can swim swiftly in the water. A
straight path to the victim will spend too m u c h time on the sand. T h e path with the least
time is a bent line, o n e that reduces the time spent running on the sand and maximizes
the time spent swimming in the water.
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that R i e m a n n used the analogy of Flatlanders living on a c r u m p l e d sheet
of paper. To us, it is obvious that Flatlanders moving over a wrinkled
surface will be incapable of walking in a straight line. Whichever way
they walk, they will experience a " f o r c e " that tugs at t h e m from left a n d
right. To Riemann, the b e n d i n g or warping of space causes the appearance of a force. Thus forces do n o t really exist; what is actually h a p p e n ing is that space itself is being b e n t o u t of shape.
T h e p r o b l e m with Riemann's approach, however, was that he
no idea specifically how gravity or electricity and magnetism caused
warping of space. His approach was purely mathematical, without
concrete physical picture of precisely how the b e n d i n g of space
accomplished. H e r e Einstein succeeded where R i e m a n n failed.
had
the
any
was
Imagine, for example, a rock placed on a stretched bedsheet. Obviously the rock will sink into the sheet, creating a s m o o t h depression. A
small marble shot o n t o the bedsheet will t h e n follow a circular or an
elliptical p a t h a r o u n d the rock. S o m e o n e looking from a distance at the
marble orbiting a r o u n d the rock may say that there is an "instantaneous
f o r c e " emanating from the rock that alters the path of the marble. However, on close inspection it is easy to see what is really h a p p e n i n g : T h e
rock has warped the bedsheet, a n d h e n c e the path of the marble.
By analogy, if the planets orbit a r o u n d the sun, it is because they are
moving in space that has been curved by the presence of the sun. T h u s
the reason we are standing on the earth, rather than being h u r l e d into
the vacuum of outer space, is that the earth is constantly warping the
space a r o u n d us (Figure 4.1).
Einstein noticed that the presence of the sun warps the path of light
from t h e distant stars. This simple physical picture therefore gave a way
in which the theory could be tested experimentally. First, we measure
t h e position of t h e stars at night, when the sun is absent. T h e n , d u r i n g
an eclipse of the sun, we measure the position of the stars, when the sun
is present (but d o e s n ' t overwhelm the light from the stars). According
to Einstein, the a p p a r e n t relative position of the stars should change
when the sun is present, because the sun's gravitational field will have
b e n t the path of the light of those stars on its way to the earth. By comp a r i n g the p h o t o g r a p h s of the stars at night a n d the stars d u r i n g an
eclipse, o n e should be able to test this theory.
This picture can be summarized by what is called Mach's principle,
the guide Einstein used to create his general theory of relativity. We
recall that the warping of the bedsheet was d e t e r m i n e d by the presence
of the rock. Einstein summarized this analogy by stating: T h e presence
of m a t t e r - e n e r g y determines the curvature of the space-time surrounding it. This is the essence of the physical principle t h a t Riemann failed
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Figure 4.1. To Einstein, "gravity" was an illusion caused by the bending of
space. He predicted that starlight moving around the sun would be bent, and
hence the relative positions of the stars should appear distored in the presence of
the sun. This has been verified by repeated experiments.
to discover, that the b e n d i n g of space is directly related to t h e a m o u n t
of energy a n d matter contained within that space.
This, in turn, can be summarized by Einstein's famous e q u a t i o n ,
7
which essentially states:
M a t t e r - e n e r g y —» curvature of space-time
where the arrow m e a n s " d e t e r m i n e s . " This deceptively short equation
is o n e of the greatest triumphs of the h u m a n mind. From it e m e r g e the
principles b e h i n d the motions of stars a n d galaxies, black holes, the Big
Bang, a n d p e r h a p s the fate of t h e universe itself.
93
Nevertheless, Einstein was still missing a piece of t h e puzzle. He h a d
discovered the correct physical principle, b u t lacked a rigorous mathematical formalism powerful e n o u g h to express this principle. He lacked
a version of Faraday's fields for gravity. Ironically, R i e m a n n h a d the
mathematical apparatus, b u t n o t the guiding physical principle. Einstein, by contrast, discovered the physical principle, b u t lacked the mathematical apparatus.
Field Theory of Gravity
Because Einstein formulated this physical principle without knowing of
Riemann, he did not have the mathematical language or skill with which
to express his principle. He spent 3 long, frustrating years, from 1912 to
1915, in a desperate search for a mathematical formalism powerful
e n o u g h to express the principle. Einstein wrote a desperate letter to his
close friend, mathematician Marcel Grossman, pleading, "Grossman,
you must h e l p me or else I'll go crazy!"
8
Fortunately, Grossman, when combing t h r o u g h the library for clues
to Einstein's problem, accidentally stumbled on the work of Riemann.
Grossman showed Einstein the work of Riemann and his metric tensor,
which had b e e n ignored by physicists for 60 years. Einstein would later
recall that Grossman " c h e c k e d t h r o u g h the literature a n d soon discovered that the mathematical p r o b l e m h a d already b e e n solved by Riem a n n , Ricci, a n d Levi-Civita. . . . Riemann's achievement was the greatest o n e . "
To his shock, Einstein found Riemann's celebrated 1854 lecture to
be the key to the problem. He found that he could incorporate the
entire body of Riemann's work in the reformulation of his principle.
Almost line for line, the great work of Riemann found its true h o m e in
Einstein's principle. This was Einstein's p r o u d e s t piece of work, even
m o r e than his celebrated equation E = mc . T h e physical reinterpretation of Riemann's famous 1854 lecture is now called general relativity, a n d
Einstein's field equations rank a m o n g the most profound ideas in scientific history.
2
Riemann's great contribution, we recall, was that he introduced the
c o n c e p t of the metric tensor, a field that is defined at all points in space.
T h e metric tensor is n o t a single n u m b e r . At each point in space, it
consists of a collection of ten n u m b e r s . Einstein's strategy was to follow
Maxwell a n d write down the field theory of gravity. T h e object of his
search for a field to describe gravity was found practically on the first
94
page of Riemann's lecture. In fact, R i e m a n n ' s metric tensor was precisely
the Faraday field for gravity!
W h e n Einstein's equations are fully expressed in terms of R i e m a n n ' s
metric tensor, they assume an elegance never before seen in physics.
Nobel laureate Subrahmanyan C h a n d r a s e k h a r o n c e called it " t h e most
beautiful theory there ever was." (In fact, Einstein's theory is so simple
yet so powerful that physicists are sometimes puzzled as to why it works
so well. MIT physicist Victor Weisskopf o n c e said, "It's like the peasant
who asks the engineer how the steam engine works. T h e engineer
explains to the peasant exactly where the steam goes a n d how it moves
through the engine a n d so on. And then the peasant says: 'Yes, I understand all that, but where is the horse?' That's how I feel a b o u t general
relativity. I know all the details, I u n d e r s t a n d where the steam goes, b u t
I'm still n o t sure I know where the horse is." )
In retrospect, we now see how close Riemann came to discovering
the theory of gravity 60 years before Einstein. T h e entire mathematical
apparatus was in place in 1854. His equations were powerful e n o u g h to
describe the most complicated twisting of space-time in any dimension.
However, he lacked the physical picture (that m a t t e r - e n e r g y determines
t h e curvature of space-time) a n d t h e keen physical insight that Einstein
provided.
9
Living in Curved Space
I once a t t e n d e d a hockey game in Boston. All the action, of course, was
concentrated on the hockey players as they glided on the ice rink.
Because the puck was being rapidly battered back a n d forth between the
various players, it r e m i n d e d me of how atoms exchange electrons when
they form chemical elements or molecules. I noticed that the skating
rink, of course, did n o t participate in the g a m e . It only m a r k e d the
various boundaries; it was a passive a r e n a on which the hockey players
scored points.
Next, I imagined what it must be like if the skating rink actively participated in the game: What would h a p p e n if the players were forced to
play on an ice rink whose surface was curved, with rolling hills a n d steep
valleys?
T h e hockey game would suddenly became m o r e interesting. T h e
players would have to skate along a curved surface. T h e rink's curvature
would distort their motion, acting like a " f o r c e " pulling the players o n e
\
95
way or another. T h e puck would move in a curved line like a snake,
making the g a m e m u c h m o r e difficult.
T h e n I imagined taking this o n e step further; I imagined that the
players were forced to play on a skating rink shaped like a cylinder. If
the players could generate e n o u g h speed, they could skate upside down
a n d move entirely a r o u n d the cylinder. New strategies could be devised,
such as a m b u s h i n g an opposing player by skating upside down a r o u n d
the cylinder a n d catching him unawares. O n c e the ice rink was b e n t in
the shape of a circle, space would b e c o m e the decisive factor in explaining the motion of matter on its surface.
Another, m o r e relevant example for o u r universe might be living in
a curved space given by a hypersphere, a sphere in four d i m e n s i o n s .
If we look ahead, light will circle completely a r o u n d the small p e r i m e t e r
of the hypersphere a n d return to o u r eyes. Thus we will see s o m e o n e
standing in front of us, with his back facing us, a person who is wearing
the same clothes as we are. We look disapprovingly at the unruly,
u n k e m p t mass of hair on this person's h e a d , a n d t h e n r e m e m b e r that
we forgot to c o m b o u r hair that day.
10
Is this person a fake image created by mirrors? To find out, we stretch
o u t o u r h a n d a n d p u t it on his shoulder. We find that the person in
front of us is a real person, not j u s t a fake. If we look into the distance,
in fact, we see an infinite n u m b e r of identical p e o p l e , each facing forward, each with his h a n d on the shoulder of the person in front.
But what is most shocking is that we feel s o m e o n e ' s h a n d sneaking
up from b e h i n d , which t h e n grabs o u r shoulder. Alarmed, we look back,
a n d see a n o t h e r infinite sequence of identical p e o p l e b e h i n d us, with
their faces t u r n e d the o t h e r way.
What's really h a p p e n i n g ? We, of course, are the only person living
in this hypersphere. T h e person in front of us is really ourself. We are
staring at the back of o u r own h e a d . By placing o u r h a n d in front of us,
we are really stretching o u r h a n d a r o u n d the hypersphere, until we place
o u r h a n d on o u r own shoulder.
T h e counterintuitive stunts that are possible in a hypersphere are
physically interesting because many cosmologists believe that o u r universe is actually a large hypersphere. T h e r e are also o t h e r equally strange
topologies, like h y p e r d o u g h n u t s a n d Mobius strips. Although they may
ultimately have no practical application, they help to illustrate many of
the features of living in hyperspace.
For example, let us assume that we are living on a h y p e r d o u g h n u t .
If we look to o u r left a n d right, we see, m u c h to o u r surprise, a person
on either side. Light circles completely a r o u n d the larger perimeter of
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ENTERING
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the d o u g h n u t , a n d returns to its starting point. T h u s if we turn o u r heads
a n d look to the left, we see the right side of someone's body. By t u r n i n g
o u r heads the o t h e r way, we see someone's left side. No matter how fast
we t u r n o u r heads, the people ahead of us a n d to o u r sides turn their
heads just as fast, a n d we can never see their faces.
Now imagine stretching o u r arms to either side. Both the person on
the left a n d the o n e on the right will also stretch their arms. In fact, if
you are close e n o u g h , you can grab the left a n d right h a n d s of the persons to either side. If you look carefully in either direction, you can see
an infinitely long, straight line of people all holding h a n d s . If you look
ahead, there is a n o t h e r infinite sequence of people standing before you,
arranged in a straight line, all holding hands.
What's actually happening? In reality o u r arms are long e n o u g h to
reach a r o u n d the d o u g h n u t , until the arms have touched. T h u s we have
actually grabbed o u r own h a n d s (Figure 4.2)!
Now we find ourselves tiring of this charade. These people seem to
be taunting us; they are copy-cats, doing exactly what we do. We get
annoyed—so we get a gun a n d point it at the person in front of us. Just
before we pull the trigger, we ask ourselves: Is this person a fake mirror
image? If so, t h e n the bullet will go right t h r o u g h him. But if not, t h e n
the bullet will go completely a r o u n d the universe a n d hit us in the back.
Maybe firing a gun in this universe is not such a good idea!
For an even m o r e bizarre universe, imagine living on a Mobius strip,
which is like a long strip of p a p e r twisted 180 degrees a n d then reglued
back together into a circular strip. W h e n a right-handed Flatlander
moves completely a r o u n d the Mobius strip, he finds that he has b e c o m e
left-handed. Orientations are reversed when traveling a r o u n d the universe. This is like H. G. Wells's " T h e Planner Story," in which the h e r o
returns to earth after an accident to find that his body is completely
reversed; for example, his heart is on his right side.
If we lived on a hyper-Mobius strip, a n d we p e e r e d in front of us, we
would see the back of s o m e o n e ' s head. At first, we w o u l d n ' t think it
could be o u r head, because the p a r t of the hair would be on the wrong
side. If we reached out a n d placed o u r right h a n d on his shoulder, then
he would lift up his left h a n d a n d place it on the shoulder of the person
ahead of him. In fact, we would see an infinite chain of people with
h a n d s on each other's shoulders, except the hands would alternate from
the left to the right shoulders.
If we left some of o u r friends at o n e spot a n d walked completely
a r o u n d this universe, we would find that we h a d r e t u r n e d to o u r original
spot. But o u r friends would be shocked to find that o u r body was
The Secret of Light: Vibrations in the Fifth Dimension 97
Figure 4.2. If we lived in a hyperdoughnut, we would see an infinite succession
of ourselves repeated in front of us, to the back of us, and to our sides. This is
because there are two ways that light can travel around the doughnut. If we hold
hands with the people to our sides, we are actually holding our own hands; that
is, our arms are actually encircling the doughnut.
reversed. T h e part in o u r hair a n d the rings on o u r fingers would be on
the wrong side, a n d o u r internal organs would have been reversed. O u r
friends would be amazed at the reversal of o u r body, a n d would ask if
we felt well. In fact, we would feel completely normal; to us, it would be
o u r friends who h a d b e e n completely t u r n e d a r o u n d ! An a r g u m e n t
would now ensue over who was really reversed.
98
These a n d o t h e r interesting possibilities o p e n up when we live in a
universe where space a n d time are curved. No longer a passive arena,
space becomes an active player in the d r a m a unfolding in o u r universe.
In summary, we see that Einstein fulfilled the p r o g r a m initiated by
R i e m a n n 60 years earlier, to use h i g h e r dimensions to simplify the laws
of n a t u r e . Einstein, however, went beyond Riemann in several ways. Like
Riemann before him, Einstein i n d e p e n d e n t l y realized that " f o r c e " is a
c o n s e q u e n c e of geometry, b u t unlike Riemann, Einstein was able to find
the physical principle b e h i n d this geometry, that the curvature of s p a c e time is d u e to the presence of matter-energy. Einstein, also like Riem a n n , knew that gravitation can be described by a field, the metric tensor, b u t Einstein was able to find the precise field equations that these
fields obey.
A Universe Made of Marble
By the mid-1920s, with the development of b o t h special a n d general
relativity, Einstein's place in the history of science was assured. In 1921,
astronomers h a d verified that starlight indeed b e n d s as it travels a r o u n d
the sun, precisely as Einstein h a d predicted. By then, Einstein was being
celebrated as the successor to Isaac Newton.
However, Einstein still was n o t satisfied. He would try o n e last time
to p r o d u c e a n o t h e r world-class theory. But on his third try, he failed.
His third a n d final theory was to have been the crowning achievement
of his lifetime. He was searching for the " t h e o r y of everything," a theory
that would explain all the familiar forces found in n a t u r e , including light
a n d gravity. He coined this theory the unified field theory. Alas, his search
for a unified theory of light a n d gravity was fruitless. W h e n he died, he
left only the unfinished ideas of various manuscripts on his desk.
Ironically, the source of Einstein's frustration was the structure of his
own equation. For 30 years, he was disturbed by a fundamental flaw in
this formulation. On o n e side of the equation was the curvature of
space-time, which he likened to " m a r b l e " because of its beautiful geometric structure. To Einstein, the curvature of space-time was like the
epitome of Greek architecture, beautiful a n d serene. However, he hated
the o t h e r side of this equation, describing matter-energy, which he considered to be ugly a n d which he c o m p a r e d to " w o o d . " While the "marb l e " of space-time was clean a n d elegant, the " w o o d " of m a t t e r - e n e r g y
was a horrible j u m b l e of confused, seemingly r a n d o m forms, from subatomic particles, atoms, polymers, a n d crystals to rocks, trees, planets,
99
and stars. But in the 1920s a n d 1930s, when Einstein was actively working
on the unified field theory, the true nature of matter r e m a i n e d an unsolved mystery.
Einstein's grand strategy was to turn wood into marble—that is, to
give a completely geometric origin to matter. But without m o r e physical
clues a n d a d e e p e r physical understanding of the wood, this was impossible. By analogy, think of a magnificent, gnarled tree growing in the
middle of a park. Architects have s u r r o u n d e d this grizzled tree with a
plaza m a d e of beautiful pieces of the purest marble. T h e architects have
carefully assembled the marble pieces to resemble a dazzling floral pattern with vines and roots emanating from the tree. To paraphrase
Mach's principle: T h e presence of the tree determines the pattern of
the marble s u r r o u n d i n g it. But Einstein hated this dichotomy between
wood, which seemed to be ugly and complicated, and marble, which was
simple and p u r e . His d r e a m was to turn the tree into marble; he would have
liked to have a plaza completely m a d e of marble, with a beautiful, symmetrical marble statue of a tree at its center.
In retrospect, we can probably spot Einstein's error. We recall that
the laws of nature simplify and unify in higher dimensions. Einstein
correctly applied this principle twice, in special and general relativity.
However, on his third try, he a b a n d o n e d this fundamental principle.
Very little was known about the structure of atomic a n d nuclear matter
in his time; consequently, it was n o t clear how to use higher-dimensional
space as a unifying principle.
Einstein blindly tried a n u m b e r of purely mathematical approaches.
He apparently t h o u g h t that " m a t t e r " could be viewed as kinks, vibrations, or distortions of space-time. In this picture, matter was a concentrated distortion of space. In other words, everything we see a r o u n d us,
from the trees and clouds to the stars in the heavens, was probably an
illusion, some form of crumpling of hyperspace. However, without any
m o r e solid leads or experimental data, this idea led to a blind alley.
It would be left to an obscure mathematician to take the next step,
which would lead us to the fifth dimension.
The Birth of Kaluza-Klein Theory
In April 1919, Einstein received a letter that left him speechless.
It was from an unknown mathematician, T h e o d r Kaluza, at the University of Konigsberg in Germany, in what is Kaliningrad in the former
Soviet Union. In a short article, only a few pages long, this obscure math-
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ematician was proposing a solution to o n e of the greatest problems of
the century. In just a few lines, Kaluza was uniting Einstein's theory of
gravity with Maxwell's theory of light by introducing the fifth dimension
(that is, four dimensions of space a n d o n e dimension of time).
In essence, he was resurrecting the old "fourth d i m e n s i o n " of Hinton a n d Zollner a n d incorporating it into Einstein's theory in a fresh
fashion as the fifth dimension. Like R i e m a n n before him, Kaluza
assumed that light is a disturbance caused by the rippling of this higher
dimension. T h e key difference separating this new work from Riem a n n ' s , H i n t o n ' s , a n d Zollner's was that Kaluza was proposing a g e n u i n e
field theory.
In this short note, Kaluza began, innocently e n o u g h , by writing down
Einstein's field equations for gravity in five dimensions, n o t the usual
four. (Riemann's metric tensor, we recall, can be formulated in any n u m b e r of dimensions.) T h e n he p r o c e e d e d to show that these five-dimensional equations contained within them Einstein's earlier four-dimensional theory (which was to be expected) with an additional piece. But
what shocked Einstein was that this additional piece was precisely Maxwell's theory of light. In o t h e r words, this unknown scientist was proposing to combine, in o n e stroke, the two greatest field theories known
to science, Maxwell's a n d Einstein's, by mixing t h e m in the fifth dimension. This was a theory m a d e of p u r e marble—that is, p u r e geometry.
Kaluza h a d found the first i m p o r t a n t clue in turning wood into marble. In the analogy of the park, we recall that the marble plaza is two
dimensional. Kaluza's observation was that we could build a " t r e e " of
marble if we could move the pieces of marble up into the third dimension.
To the average layman, light a n d gravity have n o t h i n g in c o m m o n .
After all, light is a familiar force that comes in a spectacular variety of
colors a n d forms, while gravity is invisible a n d m o r e distant. On the
earth, it is the electromagnetic force, not gravity, that has h e l p e d us tame
nature; it is the electromagnetic force that powers o u r machines, electrifies o u r cities, lights o u r n e o n signs, a n d brightens o u r television sets.
Gravity, by contrast, operates on a larger scale; it is the force that guides
the planets a n d keeps the sun from exploding. It is a cosmic force that
permeates the universe a n d binds the solar system. (Along with Weber
a n d Riemann, o n e of the first scientists to search actively for a link
between light and gravity in the laboratory was Faraday himself. T h e
actual experimental apparatus used by Faraday to measure the link
between these two forces can still be found in the Royal Institution in
Piccadilly, L o n d o n . Although he failed experimentally to find any con-
101
nection at all between the two forces, Faraday was confident of the power
of unification. He wrote, "If the h o p e [of unification] should prove well
founded, how great a n d mighty a n d sublime in its h i t h e r t o unchangeable character is the force I am trying to deal with, a n d how large may
be the new d o m a i n of knowledge that may be o p e n e d to the m i n d of
man." )
Even mathematically, light and gravity are like oil and water. Maxwell's field theory of light requires four fields, while Einstein's metric
theory of gravity requires ten. Yet Kaluza's p a p e r was so elegant and
compelling that Einstein could n o t reject it.
11
At first, it seemed like a c h e a p mathematical trick simply to e x p a n d
the n u m b e r or dimensions of space and time from four to five. This was
because, as we recall, there was no experimental evidence for the fourth
spatial dimension. What astonished Einstein was that once the fivedimensional field theory was b r o k e n down to a four-dimensional field
theory, both Maxwell's and Einstein's equations remained. In o t h e r
words, Kaluza succeeded in j o i n i n g the two pieces of the jigsaw puzzle
because both of t h e m were part of a larger whole, a five-dimensional
space.
" L i g h t " was emerging as the warping of the geometry of higherdimensional space. This was the theory that seemed to fulfill Riemann's
old d r e a m of explaining forces as the crumpling of a sheet of paper. In
his article, Kaluza claimed that his theory, which synthesized the two
most important theories up to that time, possessed "virtually unsurpassed formal unity." He furthermore insisted that the sheer simplicity and
beauty of his theory could n o t " a m o u n t to the mere alluring play of a
capricious a c c i d e n t . " What shook Einstein was the audacity and simplicity of the article. Like all great ideas, Kaluza's essential a r g u m e n t was
elegant and compact.
12
T h e analogy with piecing together the parts of a jigsaw puzzle is a
meaningful one. Recall that the basis of Riemann's a n d Einstein's work
is the metric tensor—that is, a collection of ten n u m b e r s defined at each
point in space. This was a natural generalization of Faraday's field concept. In Figure 2.2, we saw how these ten n u m b e r s can be arranged as
in the pieces of a checker board with dimensions 4 X 4 . We can d e n o t e
these ten n u m b e r s as g , g , . . . . F u r t h e r m o r e , the field of Maxwell is
a collection of four n u m b e r s defined at each point in space. These four
n u m b e r s can be represented by the symbols A , A. , A , A .
To u n d e r s t a n d Kaluza's trick, let us now begin with R i e m a n n ' s theory
in five dimensions. T h e n the metric tensor can be arranged in a 5 X 5
checkerboard. Now, by definition, we will r e n a m e the c o m p o n e n t s of
u
12
1
2
s
4
Figure 4.3. Kaluza's brilliant idea was to unite down the Riemann metric in five
dimensions. The fifth column and row are identified as the electromagnetic field
of Maxwell, while the remaining 4X4 block is the old four-dimensional metric
of Einstein. In one stroke, Kaluza unified the theory of gravity with light simply
by adding another dimension.
102
Kaluza's field, so that some of t h e m b e c o m e Einstein's original field a n d
some of t h e m b e c o m e Maxwell's field (Figure 4.3). This is the essence
of Kaluza's trick, which caught Einstein totally by surprise. By simply
a d d i n g Maxwell's field to Einstein's, Kaluza was able to reassemble both
of t h e m into a five-dimensional field.
Notice that there is " e n o u g h r o o m " within the 15 c o m p o n e n t s of
R i e m a n n ' s five-dimensional gravity to fit b o t h the ten c o m p o n e n t s of
Einstein's field a n d the four c o m p o n e n t s of Maxwell's field! T h u s Kaluza's brilliant idea can be crudely summarized as
15 = 10 + 4 + 1
(the leftover c o m p o n e n t is a scalar particle, which is u n i m p o r t a n t for
o u r discussion). W h e n carefully analyzing the full five-dimensional theory, we find that Maxwell's field is nicely included within the R i e m a n n
metric tensor, j u s t as Kaluza claimed. This innocent-looking equation
thus summarized o n e of the seminal ideas of the century.
In summary, the five-dimensional metric tensor included both Maxwell's field a n d Einstein's metric tensor. It seemed incredible to Einstein
that such a simple idea could explain the two most fundamental forces
of nature: gravity a n d light.
Was it j u s t a parlor trick? Or numerology? Or black magic? Einstein
was deeply shaken by Kaluza's letter and, in fact, refused to respond to
the article. He mulled over the letter for 2 years, an unusually long time
for s o m e o n e to hold up publication of an important article. Finally,
convinced that this article was potentially important, he submitted it for
publication in the Sitzungsberichte Preussische Akademie der Wissenschaften.
It b o r e the imposing title " O n the Unity Problem of Physics."
In the history of physics, no o n e h a d found any use for the fourth
spatial dimension. Ever since Riemann, it was known that the mathematics of h i g h e r dimensions was o n e of breathtaking beauty, b u t without
physical application. For the first time, s o m e o n e h a d found a use for the
fourth spatial dimension: to unite the laws of physics! In some sense,
Kaluza was proposing that the four dimensions of Einstein were " t o o
small" to a c c o m m o d a t e both the electromagnetic a n d gravitational
forces.
We can also see historically that Kaluza's work was not totally unexpected. Most historians of science, when they m e n t i o n Kaluza's work at
all, say that the idea of a fifth dimension was a bolt out of the blue, totally
u n e x p e c t e d a n d original. Given the continuity of physics research, these
historians are startled to find a new avenue of science o p e n i n g up with-
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o u t any historical precedent. But their a m a z e m e n t is probably d u e to
their unfamiliarity with the nonscientific work of the mystics, literati,
a n d avante garde. A closer look at the cultural and historical setting
shows that Kaluza's work was n o t such an u n e x p e c t e d development. As
we have seen, because of H i n t o n , Zollner, a n d others, the possible existence of higher dimensions was p e r h a p s the single most popular quasiscientific idea circulating within the arts. From this larger cultural point
of view, it was only a matter of time before some physicist took seriously
H i n t o n ' s widely known idea that light is a vibration of the fourth spatial
dimension. In this sense, the work of Riemann pollinated the world of
arts and letters via H i n t o n and Zollner, a n d then probably cross-pollinated back into the world of science t h r o u g h the work of Kaluza. (In
support of this thesis, it was recently revealed by F r e u n d that Kaluza was
actually n o t the first o n e to propose a five-dimensional theory of gravity.
G u n n a r Nordstrom, a rival of Einstein, actually published the first fivedimensional field theory, b u t it was too primitive to include both Einstein's and Maxwell's theories. T h e fact that both Kaluza and Nordstrom
independently tried to exploit the fifth dimension indicates that the
concepts widely circulating within popular culture affected their thinking. )
l3
The Fifth Dimension
Every physicist receives quite a j o l t when confronting the fifth dimension
for the first time. Peter F r e u n d r e m e m b e r s clearly the precise m o m e n t
when he first e n c o u n t e r e d the fifth and higher dimensions. It was an
event that left a d e e p impression on his thinking.
It was 1953 in Romania, the country of F r e u n d ' s birth. J o s e p h Stalin
had just died, an i m p o r t a n t event that led to a considerable relaxation
of tensions. F r e u n d was a precocious college freshman that year, and he
a t t e n d e d a talk by George Vranceanu. He vividly r e m e m b e r s hearing
Vranceanu discuss the i m p o r t a n t question: Why should light and gravity
be so disparate? T h e n the lecturer m e n t i o n e d an old theory that could
contain both the theory of light and Einstein's equations of gravity. T h e
secret was to use Kaluza-Klein theory, which was formulated in five
dimensions.
F r e u n d was shocked. H e r e was a brilliant idea that took him completely by surprise. Although only a freshman, he had the audacity to
pose the obvious question: How does this Kaluza-Klein theory explain
the o t h e r forces? He asked, "Even if you achieve a unification of light
a n d gravity, you will n o t achieve anything because there is still the
nuclear force." He realized that the nuclear force was outside KaluzaKlein theory. (In fact, the hydrogen b o m b , which h u n g like a sword over
everyone on the planet at the height of the Cold War, was based on
unleashing the nuclear force, n o t electromagnetism or gravity.)
T h e lecturer had no answer. In his youthful enthusiasm, F r e u n d
blurted out, "What a b o u t a d d i n g m o r e dimensions?"
" B u t how many m o r e dimensions?" asked the lecturer.
F r e u n d was caught off guard. He did n o t want to give a low n u m b e r
of dimensions, only to be scooped by s o m e o n e else. So he p r o p o s e d a
n u m b e r that no o n e could possibly top: an infinite n u m b e r of dimensions! (Unfortunately for this precocious physicist, an infinite n u m b e r
of dimensions does n o t seem to be physically possible.)
14
Life on a Cylinder
After the initial shock of confronting the fifth dimension, most physicists
invariably begin to ask questions. In fact, Kaluza's theory raised m o r e
questions t h a n it answered. T h e obvious question to ask Kaluza was:
W h e r e is the fifth dimension? Since all earthly experiments showed conclusively that we live in a universe with three dimensions of space a n d
o n e of time, the embarrassing question still remained.
Kaluza h a d a clever response. His solution was essentially the same
as that p r o p o s e d by H i n t o n years before, that the higher dimension,
which was n o t observable by experiment, was different from the o t h e r
dimensions. It had, in fact, collapsed down to a circle so small that even
atoms could n o t fit inside it. T h u s the fifth dimension was not a mathematical trick i n t r o d u c e d to manipulate electromagnetism a n d gravity,
b u t a physical dimension that provided the glue to unite these two fund a m e n t a l forces into o n e force, but was j u s t too small to measure.
Anyone walking in the direction of the fifth dimension would eventually find himself back where he started. This is because the fifth dimension is topologically identical to a circle, a n d the universe is topologically
identical to a cylinder.
F r e u n d explains it this way:
T h i n k o f s o m e i m a g i n a r y p e o p l e living i n L i n e l a n d , w h i c h c o n s i s t s o f a
s i n g l e l i n e . T h r o u g h o u t t h e i r history, t h e y b e l i e v e d t h a t t h e i r w o r l d was
j u s t a s i n g l e l i n e . T h e n , a s c i e n t i s t i n L i n e l a n d p r o p o s e d that t h e i r w o r l d
was n o t j u s t a o n e - d i m e n s i o n a l l i n e , b u t a t w o - d i m e n s i o n a l w o r l d . W h e n
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asked where this mysterious and unobservable second dimension was, he
would reply that the second dimension was curled up into a small ball.
Thus, the line people actually live on the surface of a long, but very thin,
cylinder. The radius of the cylinder is too small to be measured; it is so
small, in fact, that it appears that the world is just a line.
15
If the radius of the cylinder were larger, the line people could move
off their universe a n d move perpendicular to their line world. In other
words, they could perform interdimensional travel. As they moved perpendicular to Lineland, they would e n c o u n t e r an infinite n u m b e r of
parallel line worlds that coexisted with their universe. As they moved
farther into the second dimension, they would eventually r e t u r n to their
own line world.
Now think of Flatlanders living on a plane. Likewise, a scientist on
Flatland may make the outrageous claim that traveling t h r o u g h the third
dimension is possible. In principle, a Flatlander could rise off the surface
of Flatland. As this Flatlander slowly floated upward in the third dimension, his " e y e s " would see an incredible sequence of different parallel
universes, each coexisting with his universe. Because his eyes would be
able to see only parallel to the surface of Flatland, he would see different
Flatland universes appearing before him. If the Flatlander drifted too
far above the plane, eventually he would r e t u r n to his original Flatland
universe.
Now, imagine that o u r present three-dimensional world actually has
a n o t h e r dimension that has curled up into a circle. For the sake of argum e n t , assume that the fifth dimension is 10 feet long. By leaping into
the fifth dimension, we simply disappear instantly from o u r present universe. O n c e we move in the fifth dimension, we find that, after moving
10 feet, we are back where we started from. But why did the fifth dimension curl up into a circle in the first place? In 1926, the mathematician
Oskar Klein m a d e several improvements on the theory, stating that perhaps the q u a n t u m theory could explain why the fifth dimension rolled
u p . On this basis, he calculated that the size of the fifth dimension
should be 1 0
centimeters (the Planck length), which is m u c h too
small for any earthly e x p e r i m e n t to detect its presence. (This is the same
a r g u m e n t used today to justify the ten-dimensional theory.)
- 3 3
On the o n e h a n d , this m e a n t that the theory was in a g r e e m e n t with
e x p e r i m e n t because the fifth dimension was too small to be measured.
On the o t h e r h a n d , it also m e a n t that the fifth dimension was so fantastically small that o n e could never build machines powerful e n o u g h to
prove the theory was really correct. (The q u a n t u m physicist Wolfgang
Pauli, in his usual caustic way, would dismiss theories he d i d n ' t like by
saying, "It isn't even w r o n g . " In o t h e r words, they were so half-baked
that o n e could n o t even d e t e r m i n e if they were correct. Given the fact
that Kaluza's theory could n o t be tested, o n e could also say that it wasn't
even wrong.)
The Death of Kaluza-Klein Theory
As promising as Kaluza-Klein theory was for giving a purely geometric
foundation to t h e forces of n a t u r e , by the 1930s the theory was dead.
O n the o n e h a n d , physicists w e r e n ' t convinced that t h e f i f t h dimension
really existed. Klein's conjecture that the fifth dimension was curled up
into a tiny circle the size of the Planck length was untestable. T h e energy
necessary to p r o b e this tiny distance can be c o m p u t e d , a n d it is called
t h e Planck energy, or 1 0 billion electron volts. This fabulous energy is
almost beyond c o m p r e h e n s i o n . It is 100 billion billion times the energy
locked in a p r o t o n , an energy beyond anything we will be able to prod u c e within the next several centuries.
19
On the o t h e r h a n d , physicists left this area of research in droves
because of the discovery of a new theory that was revolutionizing the
world of science. T h e tidal wave unleashed by this theory of t h e subatomic world completely swamped research in Kaluza-Klein theory. T h e
new theory was called q u a n t u m mechanics, a n d it s o u n d e d the death
knell for Kaluza-Klein theory for the next 60 years. Worse, q u a n t u m
mechanics challenged the smooth, geometric interpretation of forces,
replacing it with discrete packets of energy.
Was the program initiated by Riemann a n d Einstein completely
wrong?
PART II
Unification in
Ten Dimensions
5
Quantum Heresy
Anyone w h o is not shocked by the quantum theory does not
u n d e r s t a n d it.
Niels B o h r
A Universe Made of Wood
I
N 1925, a new theory burst into existence. With dizzying, almost meteoric speed, this theory overthrew long-cherished notions a b o u t matter that had b e e n held since the time of t h e Greeks. Almost effortlessly,
it vanquished scores of long-standing fundamental p r o b l e m s that h a d
s t u m p e d physicists for centuries. What is matter m a d e of? What holds it
together? Why does it c o m e in an infinite variety of forms, such as gases,
metals, rocks, liquids, crystals, ceramics, glasses, lightning bolts, stars,
a n d so on?
T h e new theory was christened quantum mechanics, a n d gave us the
first comprehensive formulation with which to pry o p e n the secrets of
the atom. T h e subatomic world, o n c e a forbidden realm for physicists,
now began to spill its secrets into the o p e n .
To u n d e r s t a n d the speed with which this revolution demolished its
rivals, we n o t e that in the early 1920s some scientists still held serious
reservations a b o u t the existence of " a t o m s . " What c o u l d n ' t be seen or
m e a s u r e d directly in the laboratory, they scoffed, d i d n ' t exist. But by
1925 a n d 1926, Erwin Schrodinger, W e r n e r Heisenberg, a n d others h a d
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developed an almost complete mathematical description of the hydrogen atom. With devastating precision, they could now explain nearly all
the properties of the hydrogen atom from p u r e mathematics. By 1930,
q u a n t u m physicists such as Paul A. M. Dirac were declaring that all of
chemistry could be derived from first principles. They even m a d e the
brash claim that, given e n o u g h time on a calculating machine, they
could predict all the chemical properties of matter found in the universe. To them, chemistry would no longer be a fundamental science.
From now on, it would be " a p p l i e d physics."
Not only did its dazzling rise include a definitive explanation of the
bizarre properties of the atomic world; b u t q u a n t u m mechanics also
eclipsed Einstein's work for many decades: O n e of the first casualties of
the q u a n t u m revolution was Einstein's geometric theory of the universe.
In the halls of the Institute for Advanced Study, young physicists began
to whisper that Einstein was over the hill, that the q u a n t u m revolution
had bypassed him completely. T h e younger generation rushed to read
the latest papers written about q u a n t u m theory, n o t those about the
theory of relativity. Even the director of the institute, J. Robert O p p e n heimer, confided privately to his close friends that Einstein's work was
hopelessly b e h i n d the times. Even Einstein began to think of himself as
an "old relic."
Einstein's dream, we recall, was to create a universe m a d e of "marb l e " — t h a t is, p u r e geometry. Einstein was repelled by the relative ugliness of matter, with its confusing, anarchistic j u m b l e of forms, which he
called " w o o d . " Einstein's goal was to banish this blemish from his theories forever, to turn wood into marble. His ultimate h o p e was to create
a theory of the universe based entirely on marble. To his horror, Einstein
realized that the q u a n t u m theory was a theory m a d e entirely of wood!
Ironically, it now a p p e a r e d that he had m a d e a m o n u m e n t a l blunder,
that the universe apparently preferred wood to marble.
In the analogy between wood and marble, we recall that Einstein
wanted to convert the tree in the marble plaza to a marble statue, creating a park completely m a d e of marble. T h e q u a n t u m physicists, however, a p p r o a c h e d the p r o b l e m from the opposite perspective. Their
d r e a m was to take a sledge h a m m e r a n d pulverize all the marble. After
removing the shattered marble pieces, they would cover the park completely with wood.
Q u a n t u m theory, in fact, t u r n e d Einstein on his head. In almost every
sense of the word, q u a n t u m theory is the opposite of Einstein's theory.
Einstein's general relativity is a theory of the cosmos, a theory of stars
Quantum Heresy
113
a n d galaxies held together via the smooth fabric of space a n d time.
Q u a n t u m theory, by contrast, is a theory of the microcosm, where subatomic particles are held together by particlelike forces dancing on the
sterile stage of space-time, which is viewed as an empty arena, devoid of
any content. T h u s the two theories are hostile opposites. In fact, the
tidal wave generated by the q u a n t u m revolution swamped all attempts
at a geometric u n d e r s t a n d i n g of forces for over a half-century.
T h r o u g h o u t this book, we have developed the t h e m e that the laws
of physics a p p e a r simple a n d unified in higher dimensions. However,
with the appearance of the q u a n t u m heresy after 1925, we see the first
serious challenge to this theme. In fact, for the next 60 years, until the
mid-1980s, the ideology of the q u a n t u m heretics would d o m i n a t e the
world of physics, almost burying the geometric ideas of Riemann a n d
Einstein u n d e r an avalanche of u n d e n i a b l e successes a n d stunning
experimental victories.
Fairly rapidly, q u a n t u m theory began to give us a comprehensive
framework in which to describe the visible universe: T h e material universe consists of atoms a n d its constituents. T h e r e are a b o u t 100 different
types of atoms, or elements, out of which we can build all the known
forms of matter found on earth a n d even in outer space. Atoms, in turn,
consist of electrons orbiting a r o u n d nuclei, which in turn are composed
of n e u t r o n s a n d protons. In essence, the key differences between Einstein's beautiful geometric theory and q u a n t u m theory can now be summarized as follows.
1. Forces are created by the exchange of discrete packets of energy,
called quanta.
In contrast to Einstein's geometric picture of a " f o r c e , " in q u a n t u m
theory light was to be c h o p p e d up into tiny pieces. These packets of light
were n a m e d photons, a n d they behave very m u c h like point particles.
W h e n two electrons b u m p into each other, they repel each o t h e r n o t
because of the curvature of space, b u t because they exchange a packet
of energy, the p h o t o n .
T h e energy of these p h o t o n s is measured in units of something called
Planck's constant (hbar ~ 1 0
erg sec). T h e almost infinitesimal size of
Planck's constant means that q u a n t u m theory gives tiny corrections to
Newton's laws. These are called quantum corrections, a n d can be neglected
when describing o u r familiar, macroscopic world. T h a t is why we can,
for the most part, forget about q u a n t u m theory when describing everyday p h e n o m e n a . However, when dealing with the microscopic sub- 2 7
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IN
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DIMENSIONS
atomic world, these q u a n t u m corrections begin to d o m i n a t e any physical
process, accounting for the bizarre, counterintuitive properties of subatomic particles.
2. Different forces are caused by the exchange of different quanta.
T h e weak force, for example, is caused by the exchange of a different
type of q u a n t u m , called a W particle (W stands for " w e a k " ) . Similarly,
the strong force holding the p r o t o n s a n d n e u t r o n s together within the
nucleus of the atom is caused by the exchange of subatomic particles
called pi mesons. Both W bosons a n d pi mesons have b e e n seen experimentally in the debris of atom smashers, thereby verifying the fundamental correctness of this a p p r o a c h . And finally, the subnuclear force
h o l d i n g the p r o t o n s a n d n e u t r o n s a n d even the pi mesons together are
called gluons.
In this way, we have a new "unifying p r i n c i p l e " for the laws of physics.
We can unite the laws of electromagnetism, the weak force, a n d the
strong force by postulating a variety of different q u a n t a that mediate
them. T h r e e of the four forces (excluding gravity) are therefore united
by q u a n t u m theory, giving us unification without geometry, which
appears to contradict the t h e m e of this book a n d everything we have
considered so far.
3. We can never know simultaneously the velocity a n d position of a
subatomic particle.
This is the Heisenberg Uncertainty Principle, which is by far the most
controversial aspect of the theory, b u t o n e that has resisted every challenge in the laboratory for half a century. T h e r e is no known experimental deviation to this rule.
T h e Uncertainty Principle m e a n s that we can never be sure where
an electron is or what its velocity is. T h e best we can do is to calculate
the probability that the electron will a p p e a r at a certain place with a
certain velocity. T h e situation is not as hopeless as o n e might suspect,
because we can calculate with mathematical rigor the probability of finding that electron. Although the electron is a point particle, it is accomp a n i e d by a wave that obeys a well-defined equation, the Schrodinger
wave equation. Roughly speaking, the larger the wave, the greater the
probability of finding the electron at that point.
T h u s q u a n t u m theory merges concepts of b o t h particle a n d wave into
a nice dialectic: T h e fundamental physical objects of n a t u r e are particles,
but the probability of finding a particle at any given place in space a n d
Quantum Heresy
115
time is given by a probability wave. This wave, in turn, obeys a welldefined mathematical equation given by Schrodinger.
What is so crazy about the q u a n t u m theory is that it reduces everyt h i n g to these baffling probabilities. We can predict with great precision
how many electrons in a b e a m will scatter when moving t h r o u g h a screen
with holes in it. However, we can never know precisely which electron
will scatter in which direction. This is n o t a matter of having c r u d e instruments; according to Heisenberg, it is a law of nature.
This formulation, of course, had unsettling philosophical implications. T h e Newtonian vision held that the universe was a gigantic clock,
wound at the beginning of time a n d ticking ever since because it obeyed
Newton's three laws of motion; this picture of the universe was now
replaced by uncertainty and chance. Q u a n t u m theory demolished, once
a n d for all, the Newtonian d r e a m of mathematically predicting the
m o t i o n of all the particles in the universe.
If q u a n t u m theory violates o u r c o m m o n sense, it is only because
n a t u r e does not seem to care m u c h about o u r c o m m o n sense. As alien
and disturbing as these ideas may seem, they can be readily verified in
the laboratory. This is illustrated by the celebrated double-slit experim e n t . Let us say we fire a beam of electrons at a screen with two small
slits. Behind the screen, there is sensitive photographic paper. According
to nineteenth-century classical physics, there should be two tiny spots
b u r n e d into the photographic p a p e r by the beam of electrons b e h i n d
each hole. However, when the e x p e r i m e n t is actually performed in the
laboratory, we find an interference pattern (a series of bright a n d dark
lines) on the p h o t o g r a p h i c paper, which is commonly associated with
wavelike, n o t particlelike, behavior (Figure 5.1). (The simplest way of
creating an interference pattern is to take a quiet bath and then rhythmically splash waves on the water's surface. T h e spiderweblike pattern
of waves criss-crossing the surface of the water is an interference pattern
caused by the collision of many wave fronts.) T h e pattern on the photographic sheet corresponds to a wave that has penetrated both holes
simultaneously a n d then interfered with itself b e h i n d the screen. Since
the interference pattern is created by the collective motion of many
individual electrons, a n d since the wave has gone t h r o u g h both holes
simultaneously, naively we come to the absurd conclusion that electrons
can somehow e n t e r both holes simultaneously. But how can an electron
be in two places at the same time? According to q u a n t u m theory, the
electron is i n d e e d a p o i n t particle that went t h r o u g h o n e or the o t h e r
hole, but the wave function of the electron spread o u t over space, went
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U N I F I C A T I O N IN TEN D I M E N S I O N S
Beam
of
electrons
Figure 5.1. A beam of electrons is shot through two small holes and exposes some
film. We expect to see two dots on the film. Instead, we find an undulating
interference pattern. How can this be? According to quantum theory, the electron
is indeed a pointlike particle and cannot go through both holes, but the Schrodinger wave associated with each electron can pass through both holes and interfere with itself.
t h r o u g h both holes, a n d t h e n interacted with itself. As unsettling as this
idea is, it has b e e n verified repeatedly by experiment. As physicist Sir
J a m e s J e a n s once said, " I t is probably as meaningless to discuss how
m u c h r o o m an electron takes up as it is to discuss how m u c h r o o m a
fear, an anxiety, or an uncertainty takes u p . " ' (A b u m p e r sticker I once
saw in Germany s u m m e d this up succinctly. It read, " H e i s e n b e r g may
have slept h e r e . " )
4. T h e r e is a finite probability that particles may " t u n n e l " t h r o u g h
or make a q u a n t u m leap t h r o u g h impenetrable barriers.
This is o n e of m o r e stunning predictions of q u a n t u m theory. On the
atomic level, this prediction has h a d n o t h i n g less than spectacular success. " T u n n e l i n g , " or q u a n t u m leaps t h r o u g h barriers, has survived
every experimental challenge. In fact, a world without t u n n e l i n g is now
unimaginable.
O n e simple e x p e r i m e n t that demonstrates the correctness of quant u m t u n n e l i n g starts by placing an electron in a box. Normally, the electron does n o t have e n o u g h energy to penetrate the walls of the box. If
Quantum Heresy
117
classical physics is correct, t h e n the electron would never leave t h e box.
However, according to q u a n t u m theory, the electron's probability wave
will spread t h r o u g h t h e box a n d seep into the outside world. T h e seepage t h r o u g h the wall can be calculated precisely with the Schrodinger
wave equation; that is, there is a small probability that the electron's
position is somewhere outside the box. A n o t h e r way of saying this is that
t h e r e is a finite b u t small probability that t h e electron will t u n n e l its way
t h r o u g h the barrier (the wall of the box) a n d e m e r g e from t h e box. In
the laboratory, when o n e measures the rate at which electrons tunnel
t h r o u g h these barriers, the n u m b e r s agree precisely with the q u a n t u m
theory.
This q u a n t u m tunneling is the secret b e h i n d the tunnel diode, which
is a purely quantum-mechanical device. Normally, electricity might n o t
have e n o u g h energy to p e n e t r a t e past the t u n n e l diode. However, the
wave function of these electrons can p e n e t r a t e t h r o u g h barriers in t h e
d i o d e , so there is a non-negligible probability that electricity will e m e r g e
on the o t h e r side of the barrier by tunneling t h r o u g h it. W h e n you listen
to the beautiful sounds of stereo music, r e m e m b e r t h a t you are listening
to the rhythms of trillions of electrons obeying this a n d o t h e r bizarre
laws of q u a n t u m mechanics.
But if q u a n t u m mechanics were incorrect, t h e n all of electronics,
including television sets, computers, radios, stereo, a n d so on, would
cease to function. (In fact, if q u a n t u m theory were incorrect, the atoms
in o u r bodies would collapse, a n d we would instantly disintegrate.
According to Maxwell's equations, the electrons spinning in an a t o m
should lose their energy within a microsecond a n d plunge into the
nucleus. This s u d d e n collapse is prevented by q u a n t u m theory. Thus the
fact that we exist is living proof of t h e correctness of q u a n t u m
mechanics.)
This also means that there is a finite, calculable probability that
"impossible" events will occur. For example, I can calculate the probability that I will unexpectedly disappear a n d tunnel t h r o u g h the earth
a n d r e a p p e a r in Hawaii. (The time we would have to wait for such an
event to occur, it should be p o i n t e d out, is longer t h a n the lifetime of
the universe. So we c a n n o t use q u a n t u m mechanics to t u n n e l to vacation
spots a r o u n d the world.)
The Yang-Mills Field, Successor to Maxwell
Q u a n t u m physics, after an initial flush of success in the 1930s a n d 1940s
u n p r e c e d e n t e d in the history of science, b e g a n to r u n o u t of steam by
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the 1960s. Powerful a t o m smashers built to break up the nucleus of the
atom found h u n d r e d s of mysterious particles a m o n g the debris. Physicists, in fact, were deluged by mountains of experimental data spewing
from these particle accelerators.
While Einstein guessed the entire framework of general relativity
with only physical intuition, particle physicists were drowning in a mass
of experimental data in the 1960s. As Enrico Fermi, o n e of the builders
of the atomic b o m b , confessed, "If I could r e m e m b e r the names of all
these particles, I would have b e c o m e a botanist." As h u n d r e d s of "elem e n t a r y " particles were discovered in the debris of smashed atoms, particle physicists would propose i n n u m e r a b l e schemes to explain them, all
without luck. So great were the n u m b e r of incorrect schemes that it was
sometimes said that the half-life of a theory of subatomic physics is only
2 years.
2
Looking back at all the blind alleys and false starts in particle physics
during that period, o n e is r e m i n d e d of the story of the scientist and the
flea.
A scientist once trained a flea to j u m p whenever he rang a bell. Using
a microscope, he then anesthetized o n e of the flea's legs a n d rang the
bell again. T h e flea still j u m p e d .
T h e scientist then anesthetized a n o t h e r leg and then rang the bell.
T h e flea still j u m p e d .
Eventually, the scientist anesthetized m o r e and m o r e legs, each time
ringing the bell, a n d each time recording that the flea j u m p e d .
Finally, the flea had only o n e leg left. W h e n the scientist anesthetized
the last leg a n d rang the bell, he found to his surprise that the flea no
longer j u m p e d .
T h e n the scientist solemnly declared his conclusion, based on irrefutable scientific data: Fleas h e a r t h r o u g h their legs!
Although high-energy physicists have often felt like the scientist in
that story, over the decades a consistent q u a n t u m theory of matter has
slowly emerged. In 1971, the key development that propelled a unified
description of three of the q u a n t u m forces (excluding gravity) a n d
changed the landscape of theoretical physics was m a d e by a Dutch graduate student, Gerard 't Hooft, who was still in his twenties.
Based on the analogy with photons, the quanta of light, physicists
believed that the weak and strong forces were caused by the exchange
of a q u a n t u m of energy, called the Yang-Mills field. Discovered by C. N.
Yang a n d his student R. L. Mills in 1954, the Yang-Mills field is a generalization of the Maxwell field introduced a century earlier to describe
light, except that the Yang-Mills field has many m o r e c o m p o n e n t s and
Quantum Heresy
119
can have an electrical charge (the p h o t o n carries no electrical charge).
For the weak interactions, the q u a n t u m c o r r e s p o n d i n g to the Yang-Mills
field is the W particle, which can have charge + 1 , 0, a n d — 1. For the
strong interactions, the q u a n t u m c o r r e s p o n d i n g to t h e Yang-Mills field,
the " g l u e " that holds the p r o t o n s a n d n e u t r o n s together, was christened
the gluon.
Although this general picture was compelling, the p r o b l e m that
bedeviled physicists in the 1950s a n d 1960s was that the Yang-Mills field
is n o t " r e n o r m a l i z a b l e " ; that is, it does n o t yield finite, meaningful quantities when applied to simple interactions. This r e n d e r e d q u a n t u m theory useless in describing the weak a n d strong interactions. Q u a n t u m
physics h a d hit a brick wall.
This p r o b l e m arose because physicists, when they calculate what happens when two particles b u m p into each other, use s o m e t h i n g called
perturbation theory, which is a fancy way of saying they use clever approximations. For example, in Figure 5.2(a), we see what h a p p e n s when an
electron b u m p s into a n o t h e r weakly interacting particle, the elusive neutrino. As a first guess, this interaction can be described by a diagram
(called a Feynman diagram) showing that a q u a n t u m of the weak interactions, the W particle, is e x c h a n g e d between the electron a n d the neutrino. To a first approximation, this gives us a crude b u t reasonable fit
to the experimental data.
But according to q u a n t u m theory, we must also a d d small q u a n t u m
corrections to o u r first guess. To make o u r calculation rigorous, we must
also a d d in the Feynman diagrams for all possible graphs, including ones
that have " l o o p s " in t h e m , as in Figure 5.2(b). Ideally, these q u a n t u m
corrections should be tiny. After all, as we m e n t i o n e d earlier, q u a n t u m
theory was m e a n t to give tiny q u a n t u m corrections to Newtonian physics.
But m u c h to the h o r r o r of physicists, these q u a n t u m corrections, or
" l o o p g r a p h s , " instead of b e i n g small, were infinite. No matter how
physicists tinkered with their equations or tried to disguise these infinite
quantities, these divergences were persistently found in any calculation
of q u a n t u m corrections.
F u r t h e r m o r e , the Yang-Mills field h a d a formidable reputation of
being devilishly h a r d to calculate with, c o m p a r e d with the simpler Maxwell field. T h e r e was a mythology s u r r o u n d i n g the Yang-Mills field that
held that it was simply too complicated for practical calculations. Perhaps it was fortunate that 't Hooft was only a graduate student a n d wasn't
influenced by the prejudices of m o r e " s e a s o n e d " physicists. Using techniques p i o n e e r e d by his thesis adviser, Martinus Veltman, 't Hooft
showed that whenever we have "symmetry b r e a k i n g " (which we will
a
b
Figure 5.2. (a) In quantum theory, when subatomic particles bump into one
another, they exchange packets of energy, or quanta. Electrons and neutrinos
interact by exchanging a quantum of the weak force, called the W particle, (b) To
calculate the complete interaction of electrons and neutrinos, we must add up an
infinite series of graphs, called Feynman diagrams, where the quanta are
exchanged in increasingly complicated geometric patterns. This process of adding
up an infinite series of Feynman graphs is called perturbation theory.
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Quantum Heresy
121
explain later), the Yang-Mills field acquires a mass b u t remains a finite
theory. He d e m o n s t r a t e d that the infinities d u e to the loop graphs can
all be canceled or shuffled a r o u n d until they b e c o m e harmless.
Almost 20 years after its being p r o p o s e d by Yang a n d Mills, 't Hooft
finally showed that the Yang-Mills field is a well-defined theory of particle interactions. News of 't Hooft's work spread like a flash fire. Nobel
laureate Sheldon Glashow r e m e m b e r s that when he h e a r d the news, he
exclaimed, " E i t h e r this guy's a total idiot, or h e ' s the biggest genius to
hit physics in years!" Developments came thick a n d fast. An earlier
theory of the weak interactions, proposed in 1967 by Steven Weinberg
a n d Abdus Salam, was rapidly shown to be the correct theory of the weak
interactions. By the mid-1970s, the Yang-Mills field was applied to the
strong interactions. In the 1970s came the stunning realization that the
secret of all nuclear matter could be unlocked by the Yang-Mills field.
3
This was the missing piece in the puzzle. T h e secret of wood that
b o u n d matter together was the Yang-Mills field, n o t the geometry of
Einstein. It a p p e a r e d as t h o u g h this, a n d n o t geometry, was the central
lesson of physics.
The Standard Model
Today, the Yang-Mills field has m a d e possible a comprehensive theory
of all matter. In fact, we are so confident of this theory that we blandly
call it the Standard Model.
T h e Standard Model can explain every piece of experimental data
c o n c e r n i n g subatomic particles, up to a b o u t 1 trillion electron volts in
energy (the energy created by accelerating an electron by 1 trillion
volts). This is a b o u t the limit of the atom smashers currently on line.
Consequently, it is no exaggeration to state that the Standard Model is
the most successful theory in the history of science.
According to the Standard Model, each of the forces binding the
various particles is created by exchanging different kinds of quanta. Let
us now discuss each force separately, a n d t h e n assemble t h e m into the
Standard Model.
The Strong Force
T h e Standard Model states that the protons, n e u t r o n s , a n d o t h e r heavy
particles are n o t fundamental particles at all, b u t consist of some even
tinier particles, called quarks. These quarks, in turn, c o m e in a wide
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variety: t h r e e " c o l o r s " a n d six "flavors." (These n a m e s have n o t h i n g to
do with actual colors a n d flavors.) T h e r e are also t h e antimatter counterparts of t h e quarks, called antiquarks. (Antimatter is identical to matter in all respects, except that the charges are reversed a n d it annihilates
on contact with ordinary matter.) This gives us a total of 3 X 6 X 2 =
36 quarks.
T h e quarks, in t u r n , are h e l d t o g e t h e r by the e x c h a n g e of small packets of energy, called gluons. Mathematically, these gluons are described
by the Yang-Mills field, which " c o n d e n s e s " into a sticky, taffylike substance t h a t " g l u e s " t h e quarks p e r m a n e n t l y together. T h e gluon f i e l d
is so powerful a n d b i n d s the quarks so tightly t o g e t h e r that the quarks
can never be torn away from o n e a n o t h e r . This is called quark confinement,
a n d may explain why free quarks have never b e e n seen experimentally.
For e x a m p l e , t h e p r o t o n a n d n e u t r o n can be c o m p a r e d to t h r e e steel
balls (quarks) h e l d t o g e t h e r by a Y-shaped string (gluon) in the shape
of a bola. O t h e r strongly interacting particles, such as t h e pi meson, can
be c o m p a r e d to a q u a r k a n d an a n t i q u a r k h e l d t o g e t h e r by a single string
(Figure 5.3).
Obviously, by kicking this a r r a n g e m e n t of steel balls, we can set this
c o n t r a p t i o n vibrating. In the q u a n t u m world, only a discrete set of vibrations is allowed. Each vibration of this set of steel balls or quarks corres p o n d s to a different type of subatomic particle. T h u s this simple (but
powerful) picture explains the fact that t h e r e are an infinite n u m b e r of
strongly interacting particles. This p a r t of the S t a n d a r d Model describing t h e strong force is called q u a n t u m c h r o m o d y n a m i c s ( Q C D ) — t h a t
is, t h e q u a n t u m theory of t h e color force.
The Weak Force
In t h e S t a n d a r d Model, t h e weak force governs the p r o p e r t i e s of "lept o n s , " such as the electron, the m u o n , a n d the tau m e s o n , a n d their
n e u t r i n o p a r t n e r s . Like t h e o t h e r forces, the leptons interact by
e x c h a n g i n g q u a n t a , called W a n d Z bosons. T h e s e q u a n t a are also
described mathematically by t h e Yang-Mills field. Unlike the gluon
force, the force g e n e r a t e d by e x c h a n g i n g the W a n d Z bosons is too weak
to b i n d t h e leptons into a r e s o n a n c e , so we do n o t see an infinite n u m b e r
of l e p t o n s e m e r g i n g from o u r a t o m smashers.
The Electromagnetic Force
T h e S t a n d a r d Model includes the theory of Maxwell interacting with the
o t h e r particles. This p a r t of the S t a n d a r d Model governing the interac-
Quantum Heresy
123
quark
Condensed
Yang-Mills
field
Proton,
neutron,
etc.
quark
quark
Condensed
Yang-Mills
field
Meson
anti-quark
Figure 5.3. Strongly interacting particles are actually composites of even smaller
particles, called quarks, which are bound together by a taffylike "glue, " which is
described by the Yang-Mills field. The proton and neutron are each made up of
three quarks, while mesons are made up of a quark and an antiquark.
tion of electrons a n d light is called q u a n t u m electrodynamics ( Q E D ) ,
which has b e e n experimentally verified to be correct to within o n e p a r t
in 10 million, technically m a k i n g it the most accurate theory in history.
In sum, the fruition of 50 years of research, a n d several h u n d r e d million
dollars in g o v e r n m e n t funds, has given us the following picture of subatomic matter: All matter consists of quarks and leptons, which interact by
exchanging different types of quanta, described by the Maxwell and Yang-Mills
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fields. In o n e s e n t e n c e , we have c a p t u r e d t h e essence of t h e past century
of frustrating investigation into the subatomic realm. From this simple
picture o n e can derive, from p u r e mathematics alone, all the myriad
a n d baffling p r o p e r t i e s of matter. (Although it all seems so easy now,
N o b e l laureate Steven W e i n b e r g , o n e of the creators of the Standard
Model, o n c e reflected on how t o r t u o u s the 50-year j o u r n e y to discover
the m o d e l h a d b e e n . He wrote, " T h e r e ' s a long tradition of theoretical
physics, which by no m e a n s affected everyone b u t certainly affected m e ,
that said t h e s t r o n g interactions [were] too complicated for the h u m a n
mind." )
4
Symmetry in Physics
T h e details of the S t a n d a r d Model are actually r a t h e r b o r i n g a n d unimp o r t a n t . T h e most interesting feature of t h e S t a n d a r d Model is that it is
based on symmetry. W h a t has p r o p e l l e d this investigation into matter
(wood) is that we can see the unmistakable sign of symmetry within each
of these interactions. Quarks a n d leptons are n o t r a n d o m , b u t occur in
definite p a t t e r n s in t h e S t a n d a r d Model.
Symmetry, of course, is n o t strictly t h e province of physicists. Artists,
writers, poets, a n d m a t h e m a t i c i a n s have l o n g a d m i r e d the beauty that is
to be f o u n d in symmetry. To the p o e t William Blake, symmetry possessed
mystical, even fearful qualities, as expressed in the p o e m "Tyger! Tyger!
burning bright":
Tyger! Tyger! burning bright
In the forests of the night
What immortal h a n d or eye
C o u l d f r a m e thy f e a r f u l symmetry?"
5
To m a t h e m a t i c i a n Lewis Carroll, symmetry r e p r e s e n t e d a familiar,
almost playful c o n c e p t . In the " T h e H u n t i n g of the S n a r k , " he c a p t u r e d
the essence of symmetry w h e n he wrote:
Y o u boil it in sawdust:
Y o u salt i t i n g l u e :
You c o n d e n s e it with locusts in tape:
Still k e e p i n g o n e p r i n c i p a l o b j e c t i n v i e w —
T o p r e s e r v e its s y m m e t r i c a l s h a p e .
Quantum Heresy
125
In o t h e r words, symmetry is the preservation of t h e s h a p e of an object
even after we d e f o r m or rotate it. Several kinds of symmetries o c c u r
repeatedly in n a t u r e . T h e first is the symmetry of rotations a n d reflections. For e x a m p l e , a snowflake r e m a i n s the same if we rotate it by 60
degrees. T h e symmetry of a kaleidoscope, a flower, or a starfish is of this
type. We call these s p a c e - t i m e symmetries, which a r e created by r o t a t i n g
the object t h r o u g h a d i m e n s i o n of space or time. T h e symmetry of special relativity is of this type, since it describes rotations between space
a n d time.
A n o t h e r type of symmetry is created by reshuffling a series of objects.
T h i n k of a shell g a m e , w h e r e a huckster shuffles t h r e e shells with a p e a
h i d d e n b e n e a t h o n e of t h e m . W h a t makes t h e g a m e difficult is that t h e r e
are many ways in which the shells can be a r r a n g e d . In fact, t h e r e a r e six
different ways in which t h r e e shells can be shuffled. Since the p e a is
h i d d e n , these six configurations a r e identical to t h e observer. Mathematicians like to give n a m e s to these various symmetries. T h e n a m e for
the symmetries of a shell g a m e is called S , which describes the n u m b e r
of ways that t h r e e identical objects may be i n t e r c h a n g e d .
If we replace t h e shells with quarks, t h e n t h e e q u a t i o n s of particle
physics must r e m a i n the same if we shuffle t h e quarks a m o n g themselves.
If we shuffle t h r e e colored quarks a n d t h e e q u a t i o n s r e m a i n t h e same,
t h e n we say that the equations possess s o m e t h i n g called SU(3) symmetry.
T h e 3 represents t h e fact that we have t h r e e types of colors, a n d the SU
stands for a specific m a t h e m a t i c a l p r o p e r t y of the symmetry.* We say
that t h e r e are t h r e e quarks in a multiplet. T h e quarks in a multiplet can
be shuffled a m o n g o n e a n o t h e r without c h a n g i n g the physics of the
theory.
Similarly, the weak force governs t h e p r o p e r t i e s of two particles, t h e
electron a n d the n e u t r i n o . T h e symmetry that i n t e r c h a n g e s these particles, yet leaves the e q u a t i o n t h e same, is called S U ( 2 ) . This m e a n s that
a multiplet of the weak force contains an electron a n d a n e u t r i n o , which
can be r o t a t e d into each o t h e r . Finally, t h e e l e c t r o m a g n e t i c force has
U ( l ) symmetry, which rotates the c o m p o n e n t s of t h e Maxwell field i n t o
itself.
Each of these symmetries is simple a n d elegant. However, t h e most
controversial aspect of t h e S t a n d a r d Model is that it " u n i f i e s " t h e t h r e e
fundamental forces by simply splicing all t h r e e theories i n t o o n e large
symmetry, SU(3) X SU(2) X U ( l ) , which is j u s t t h e p r o d u c t of t h e
3
*SU stands for "special unitary" matrices—that is, matrices that have unit determinant
and are unitary.
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symmetries of the individual forces. (This can be c o m p a r e d to assembling a jigsaw puzzle. If we have t h r e e jigsaw pieces that d o n ' t quite fit,
we can always take Scotch tape a n d splice t h e m t o g e t h e r by h a n d . This
is h o w t h e S t a n d a r d Model is formed, by taping t h r e e distinct multiplets
t o g e t h e r . This may n o t be aesthetically pleasing, b u t at least the three
jigsaw puzzles n o w h a n g t o g e t h e r by tape.)
Ideally, o n e m i g h t have e x p e c t e d that " t h e ultimate t h e o r y " would
have all t h e particles inside j u s t a single multiplet. Unfortunately, the
S t a n d a r d Model has t h r e e distinct multiplets, which c a n n o t be rotated
among one another.
Beyond the Standard Model
P r o m o t e r s of t h e S t a n d a r d Model can say truthfully that it fits all known
e x p e r i m e n t a l data. T h e y can correctly p o i n t o u t that t h e r e are no experimental results that contradict the S t a n d a r d Model. Nonetheless,
n o b o d y , n o t even its most fervent advocates, believes it is the final theory
of matter. T h e r e are several d e e p reasons why it c a n n o t be the final
theory.
First, the S t a n d a r d Model d o e s n o t describe gravity, so it is necessarily
i n c o m p l e t e . W h e n attempts are m a d e to splice Einstein's theory with the
S t a n d a r d Model, t h e resulting theory gives nonsensical answers. W h e n
we calculate, say, t h e probability of an electron b e i n g deflected by a
gravitational field, t h e hybrid theory gives us an infinite probability,
which makes no sense. Physicists say that q u a n t u m gravity is nonrenormalizable, m e a n i n g that it c a n n o t yield sensible, finite n u m b e r s to
describe simple physical processes.
Second, a n d p e r h a p s most i m p o r t a n t , it is very ugly because it crudely
splices t h r e e very different interactions t o g e t h e r . Personally, I think that
the S t a n d a r d Model can be c o m p a r e d to crossing t h r e e entirely dissimilar types of animals, such as a m u l e , an e l e p h a n t , a n d a whale. In fact,
it is so ugly a n d contrived t h a t even its creators are a bit embarrassed.
T h e y are t h e first to apologize for its shortcomings a n d a d m i t that it
c a n n o t b e t h e f i n a l theory.
This ugliness is obvious w h e n we write down t h e details of the quarks
a n d leptons. To describe how ugly t h e theory is, let us list the various
particles a n d forces within t h e S t a n d a r d Model:
1. Thirty-six quarks, c o m i n g in six "flavors" a n d t h r e e " c o l o r s , "
a n d their a n t i m a t t e r c o u n t e r p a r t s to describe the strong interactions
Quantum Heresy
127
2. Eight Yang-Mills fields to describe the gluons, which b i n d t h e
quarks
3. F o u r Yang-Mills fields to describe t h e weak a n d e l e c t r o m a g n e t i c
forces
4. Six types of leptons to describe t h e weak interactions (including
the electron, m u o n , tau l e p t o n , a n d their respective n e u t r i n o
counterparts)
5. A large n u m b e r of mysterious " H i g g s " particles necessary to
fudge the masses a n d t h e constants describing the particles
6. At least 19 arbitrary constants that describe t h e masses of t h e
particles a n d the strengths of the various interactions. T h e s e 19
constants must be p u t in by h a n d ; they are n o t d e t e r m i n e d by
the theory in any way
Worse, this long list of particles can be b r o k e n down into t h r e e "families" of quarks a n d leptons, which are practically indistinguishable from
o n e a n o t h e r . In fact, these t h r e e families of particles a p p e a r to be exact
copies of o n e a n o t h e r , giving a threefold r e d u n d a n c y in t h e n u m b e r of
supposedly " e l e m e n t a r y " particles (Figure 5.4). (It is disturbing to realize that we now have vastly m o r e " e l e m e n t a r y " particles t h a n t h e total
n u m b e r of subatomic particles that were discovered by the 1940s. It
makes o n e w o n d e r how e l e m e n t a r y these e l e m e n t a r y particles really
are.)
T h e ugliness of the S t a n d a r d Model can be contrasted to t h e simplicity of Einstein's equations, in which everything was d e d u c e d from
first principles. To u n d e r s t a n d t h e aesthetic contrast between the Stand a r d Model a n d Einstein's theory of general relativity, we m u s t realize
that when physicists speak of " b e a u t y " in their theories, they really m e a n
that their theory possesses at least two essential features:
1. A unifying symmetry
2. T h e ability to explain vast a m o u n t s of e x p e r i m e n t a l data with
the most economical m a t h e m a t i c a l expressions
T h e S t a n d a r d Model fails on b o t h counts. Its symmetry, as we have
seen, is actually formed by splicing t h r e e smaller symmetries, o n e for
each of the t h r e e forces. Second, the theory is unwieldy a n d awkward in
form. It is certainly n o t e c o n o m i c a l by any m e a n s . For e x a m p l e , Einstein's equations, written o u t in their entirety, are only a b o u t an inch
long a n d w o u l d n ' t even fill up o n e line of this book. F r o m this o n e line
of equations, we can go b e y o n d Newton's laws a n d derive the warping
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Up quark
Electron
Generation
# 1
Down quark
Charmed quark
• Neutrino
Muon
Generation
# 2
'Strange quark
Mu-neutrino
Top q u a r k
Tau
B o t t o m quark
Tau-neutrino
Generation
#3
Figure 5.4. In the Standard Model, the first generation of particles consists of the
"up" and "down" quark (in three colors, with their associated antiparticles) and
the electron and neutrino. The embarrassing feature of the Standard Model is
that there are three generation of such particles, each generation being nearly an
exact copy of the previous generation. It's hard to believe that nature would be so
redundant as to create, at a fundamental level, three identical copies of particles.
of space, t h e Big Bang, a n d o t h e r astronomically i m p o r t a n t p h e n o m e n a .
However, j u s t to write down t h e S t a n d a r d Model in its entirety would
r e q u i r e two-thirds of this p a g e a n d would look like a blizzard of c o m p l e x
symbols.
N a t u r e , scientists like to believe, prefers e c o n o m y in its creations a n d
always seems to avoid unnecessary r e d u n d a n c i e s in creating physical,
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biological, a n d chemical structures. W h e n n a t u r e creates p a n d a bears,
p r o t e i n molecules, or black holes, it is sparing in its design. O r , as N o b e l
laureate C. N. Yang o n c e said, " N a t u r e seems to take advantage of t h e
simple mathematical r e p r e s e n t a t i o n s of the symmetry laws. W h e n o n e
pauses to consider t h e e l e g a n c e a n d the beautiful perfection of t h e
mathematical r e a s o n i n g involved a n d contrast it with the c o m p l e x a n d
far-reaching physical c o n s e q u e n c e s , a d e e p sense of respect for t h e
power of the symmetry laws never fails to d e v e l o p . " However, at t h e
most fundamental level, we now find a gross violation of this rule. T h e
existence of t h r e e identical families, each o n e with an o d d a s s o r t m e n t
of particles, is o n e of t h e most disturbing features of the S t a n d a r d Model,
a n d raises a persistent p r o b l e m for physicists: Should t h e S t a n d a r d
Model, the most spectacularly successful theory in the history of science,
be thrown o u t j u s t because it is ugly?
6
Is Beauty Necessary?
I o n c e a t t e n d e d a c o n c e r t in Boston, where p e o p l e were visibly m o v e d
by the power a n d intensity of Beethoven's N i n t h Symphony. After the
concert, with the rich melodies still fresh in my m i n d , I h a p p e n e d to
walk past the empty orchestra pit, w h e r e I noticed s o m e p e o p l e staring
in w o n d e r at the sheet music left by t h e musicians.
To the u n t r a i n e d eye, I t h o u g h t , the musical score of even t h e most
moving musical piece must a p p e a r to be a raw mass of unintelligible
squiggles, b e a r i n g m o r e r e s e m b l a n c e to a chaotic j u m b l e of scratches
than a beautiful work of art. However, to t h e ear of a trained musician,
this mass of bars, clefs, keys, sharps, flats, a n d notes c o m e s alive a n d
resonates in the m i n d . A musician can " h e a r " beautiful h a r m o n i e s a n d
rich resonances by simply looking at a musical score. A sheet of music,
therefore, is m o r e than j u s t the s u m of its lines.
Similarly, it would be a disservice to define a p o e m as "a short collection of words organized according to some p r i n c i p l e . " N o t only is
the definition sterile, b u t it is ultimately inaccurate because it fails to
take into a c c o u n t the subtle interaction between t h e p o e m a n d the e m o tions that it evokes in the reader. P o e m s , because they crystallize a n d
convey the essence of the feelings a n d images of the a u t h o r , have a
reality m u c h greater t h a n the words p r i n t e d on a sheet of p a p e r . A few
short words of a haiku p o e m , for e x a m p l e , may t r a n s p o r t t h e r e a d e r into
a new realm of sensations a n d feelings.
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Like music or art, m a t h e m a t i c a l equations can have a natural progression a n d logic that can evoke rare passions in a scientist. Although
t h e lay public considers m a t h e m a t i c a l equations to be r a t h e r o p a q u e , to
a scientist an e q u a t i o n is very m u c h like a m o v e m e n t in a larger symphony.
Simplicity. Elegance. T h e s e are the qualities that have inspired some
of the greatest artists to create their masterpieces, a n d they are precisely
the same qualities that motivate scientists to search for t h e laws of n a t u r e .
Like a work of art or a h a u n t i n g p o e m , e q u a t i o n s have a beauty a n d
r h y t h m all their own.
Physicist Richard Feynman expressed this w h e n he said,
Y o u c a n r e c o g n i z e t r u t h b y its b e a u t y a n d s i m p l i c i t y . W h e n y o u g e t i t r i g h t ,
it is o b v i o u s that it is r i g h t — a t least if y o u have any e x p e r i e n c e — b e c a u s e
usually what h a p p e n s is that m o r e c o m e s o u t than g o e s in. . . . T h e inexp e r i e n c e d , t h e crackpots, a n d p e o p l e like that, m a k e g u e s s e s that are simple, b u t y o u c a n i m m e d i a t e l y s e e that they are w r o n g , so that d o e s n o t
c o u n t . O t h e r s , t h e i n e x p e r i e n c e d s t u d e n t s , m a k e g u e s s e s t h a t a r e very
c o m p l i c a t e d , a n d i t s o r t o f l o o k s a s i f i t i s all r i g h t , b u t I k n o w i t i s n o t t r u e
b e c a u s e t h e t r u t h always t u r n s o u t t o b e s i m p l e r t h a n y o u t h o u g h t .
7
T h e F r e n c h m a t h e m a t i c i a n H e n r i Poincare expressed it even m o r e
frankly w h e n he wrote, " T h e scientist d o e s n o t study N a t u r e because it
is useful; he studies it because he delights in it, a n d he delights in it
because it is beautiful. If N a t u r e were n o t beautiful, it would n o t be worth
knowing, a n d if N a t u r e were n o t worth knowing, life would n o t be worth
living." In some sense, the equations of physics are like the p o e m s of
n a t u r e . T h e y are s h o r t a n d are organized a c c o r d i n g to some principle,
a n d the most beautiful of t h e m convey the h i d d e n symmetries of n a t u r e .
For e x a m p l e , Maxwell's equations, we recall, originally consisted of
eight equations. T h e s e equations are n o t " b e a u t i f u l . " They d o n o t possess m u c h symmetry. In their original form, they are ugly, b u t they are
the b r e a d a n d b u t t e r of every physicist or e n g i n e e r w h o has ever e a r n e d
a living working with radar, radio, microwaves, lasers, or plasmas. These
eight e q u a t i o n s are what a tort is to a lawyer or a stethoscope is to a
doctor. However, w h e n rewritten using time as t h e fourth dimension,
this r a t h e r awkward set of eight e q u a t i o n s collapses into a single tensor
e q u a t i o n . This is what a physicist calls " b e a u t y , " because b o t h criteria
are now satisfied. By increasing the n u m b e r of dimensions, we reveal
the true, four-dimensional symmetry of t h e theory a n d can now explain
vast a m o u n t s of experimental data with a single e q u a t i o n .
Quantum Heresy
131
As we have repeatedly seen, the addition of h i g h e r d i m e n s i o n s causes
the laws of n a t u r e to simplify.
O n e of the greatest mysteries confronting science today is t h e explanation of the origin of these symmetries, especially in t h e subatomic
world. W h e n o u r powerful m a c h i n e s blow a p a r t the nuclei of atoms by
slamming t h e m with energies beyond 1 trillion electron volts, we find
that the fragments can be a r r a n g e d a c c o r d i n g to these symmetries.
S o m e t h i n g rare a n d precious is unquestionably h a p p e n i n g w h e n we
p r o b e down to subatomic distances.
T h e p u r p o s e of science, however, is n o t to marvel at the elegance of
natural laws, b u t to explain t h e m . T h e f u n d a m e n t a l p r o b l e m facing subatomic physicists is that, historically, we h a d no idea of why these symmetries were e m e r g i n g in o u r laboratories a n d o u r blackboards.
And h e r e is precisely why the S t a n d a r d Model fails. No m a t t e r how
successful the theory is, physicists universally believe that it must be
replaced by a h i g h e r theory. It fails b o t h " t e s t s " for beauty. It n e i t h e r
has a single symmetry g r o u p n o r describes the subatomic world e c o n o m ically. But m o r e i m p o r t a n t , the Standard Model d o e s n o t explain w h e r e
these symmetries originally came from. They are j u s t spliced t o g e t h e r by
fiat, without any d e e p e r u n d e r s t a n d i n g of their origin.
GUTs
Physicist Ernest Rutherford, w h o discovered t h e n u c l e u s of t h e a t o m ,
o n c e said, "All science is either physics or s t a m p c o l l e c t i n g . "
By this, he m e a n t that science consists of two parts. T h e first is physics, which is based on the foundation of physical laws or principles. T h e
second is taxonomy ( " b u g c o l l e c t i n g " or s t a m p collecting), which is
giving erudite Greek n a m e s for objects you know almost n o t h i n g a b o u t
based on superficial similarities. In this sense, the S t a n d a r d Model is n o t
real physics; it is m o r e like s t a m p collecting, a r r a n g i n g t h e subatomic
particles according to some superficial symmetries, b u t without the vaguest hint of where the symmetries c o m e from.
Similarly, w h e n Charles Darwin n a m e d his b o o k On the Origin of Species, he was going far b e y o n d taxonomy by giving t h e logical e x p l a n a t i o n
for the diversity of animals in n a t u r e . W h a t is n e e d e d in physics is a
c o u n t e r p a r t of this book, to be called On the Origin of Symmetry, which
explains the reasons why certain symmetries are f o u n d in n a t u r e .
Because the Standard Model is so contrived, over t h e years attempts
have b e e n m a d e to go b e y o n d it, with mixed success. O n e p r o m i n e n t
8
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a t t e m p t was called the G r a n d Unified T h e o r y ( G U T ) , p o p u l a r in the
late 1970s, which tried to u n i t e t h e symmetries of the strong, weak, a n d
electromagnetic q u a n t a by a r r a n g i n g t h e m into a m u c h larger symmetry
g r o u p [for e x a m p l e , S U ( 5 ) , O ( 1 0 ) , or E ( 6 ) ] . Instead of naively splicing
the symmetry g r o u p s of the t h r e e forces, GUTs tried to start with a larger
symmetry that r e q u i r e d fewer arbitrary constants a n d fewer assumptions.
GUTs vastly increased t h e n u m b e r of particles beyond the Standard
Model, b u t t h e advantage was that t h e ugly SU(3) X SU(2) X U ( 1 ) was
now replaced by a single symmetry g r o u p . T h e simplest of these GUTs,
called SU (5), used 24 Yang-Mills fields, b u t at least all these Yang-Mills
fields b e l o n g e d to a single symmetry, n o t t h r e e separate ones.
T h e aesthetic advantage of t h e GUTs was that they p u t the strongly
interacting quarks a n d t h e weakly interacting leptons on the same footing. In S U ( 5 ) , for e x a m p l e , a multiplet of particles consisted of t h r e e
c o l o r e d quarks, an electron, a n d a n e u t r i n o . U n d e r an SU(5) rotation,
these five particles could rotate into o n e a n o t h e r without c h a n g i n g the
physics.
At first, GUTs were m e t with intense skepticism, because the energy
at which the t h r e e f u n d a m e n t a l forces were unified was a r o u n d 1 0
billion e l e c t r o n volts, j u s t a bit smaller t h a n the Planck energy. This was
far b e y o n d the energy of any a t o m smasher on the earth, a n d that was
discouraging. However, physicists gradually warmed up to the idea of
GUTs when it was realized that they m a d e a clear, testable prediction:
the decay of the p r o t o n .
We recall t h a t in t h e S t a n d a r d Model, a symmetry like SU(3) rotates
t h r e e quarks into o n e a n o t h e r ; that is, a multiplet consists of three
quarks. This m e a n s that each of the quarks can t u r n into o n e of the
o t h e r quarks u n d e r certain conditions (such as t h e e x c h a n g e of a Y a n g Mills particle). However, quarks c a n n o t t u r n into electrons. T h e multiplets do n o t mix. But in SU(5) G U T , t h e r e are five particles within a
multiplet that can rotate into o n e a n o t h e r : t h r e e quarks, the electron,
a n d t h e n e u t r i n o . This m e a n s that o n e can, u n d e r certain circumstances,
t u r n a p r o t o n ( m a d e of quarks) i n t o an electron or a n e u t r i n o . In o t h e r
words, GUTs say that t h e p r o t o n , which was l o n g held to be a stable
particle with an infinite lifetime, is actually unstable. In principle, it also
m e a n s that all a t o m s in the universe will eventually disintegrate into
radiation.. If correct, it m e a n s that t h e chemical elements, which are
t a u g h t in e l e m e n t a r y chemistry classes to be stable, are actually all unstable.
15
This d o e s n ' t m e a n that we should expect the atoms in o u r body to
disintegrate into a burst of radiation anytime soon. T h e time for the
Quantum Heresy
133
31
p r o t o n to decay i n t o l e p t o n s was calculated to be on t h e o r d e r of 1 0
years, far b e y o n d t h e lifetime of the universe (15 to 20 billion years).
Although this time scale was astronomically long, this d i d n ' t faze t h e
experimentalists. Since an ordinary tank of water contains an a s t r o n o m ical a m o u n t of p r o t o n s , t h e r e is a measurable probability that some proton within the tank will decay, even if the p r o t o n s on the average decay
on a cosmological time scale.
The Search for Proton Decay
Within a few years, this abstract theoretical calculation was p u t to t h e
test: Several expensive, multimillion-dollar e x p e r i m e n t s were c o n d u c t e d
by several groups of physicists a r o u n d the world. T h e construction of
detectors sensitive e n o u g h to d e t e c t p r o t o n decay involved highly e x p e n sive a n d sophisticated t e c h n i q u e s . First, experimentalists n e e d e d to construct e n o r m o u s vats in which to d e t e c t p r o t o n decay. T h e n they h a d to
fill the vats with a hydrogen-rich fluid (such as water or cleaning fluid)
that h a d b e e n filtered with special t e c h n i q u e s in o r d e r to eliminate all
impurities a n d c o n t a m i n a n t s . Most i m p o r t a n t , they t h e n h a d to bury
these gigantic tanks d e e p in t h e e a r t h to eliminate any c o n t a m i n a t i o n
from highly p e n e t r a t i n g cosmic rays. A n d finally, they h a d to construct
thousands of highly sensitive detectors to record t h e faint tracks of subatomic particles emitted from p r o t o n decay.
Remarkably, by the late 1980s six gigantic detectors were in o p e r a t i o n
a r o u n d the world, such as the Kamioka d e t e c t o r in J a p a n a n d t h e 1MB
(Irvine, Michigan, Brookhaven) d e t e c t o r n e a r Cleveland, O h i o . T h e y
c o n t a i n e d vast a m o u n t s of p u r e fluid (such as water) r a n g i n g in weight
from 60 to 3,300 tons. ( T h e 1MB detector, for e x a m p l e , is the world's
largest a n d is c o n t a i n e d in a h u g e 20-meter c u b e hollowed o u t of a salt
m i n e u n d e r n e a t h Lake Erie. Any p r o t o n that spontaneously decayed in
the purified water would p r o d u c e a microscopic burst of light, which in
t u r n would be picked up by some of t h e 2,048 photoelectric tubes.)
T o u n d e r s t a n d how these m o n s t r o u s detectors can m e a s u r e t h e proton lifetime, by analogy think of the A m e r i c a n p o p u l a t i o n . We know that
the average A m e r i c a n can expect to live on the o r d e r of 70 years. However, we d o n ' t have to wait 70 years to find fatalities. Because t h e r e are
so many Americans, in fact m o r e t h a n 250 million, we expect to find
some American dying every few m i n u t e s . Likewise, the simplest SU(5)
G U T p r e d i c t e d that t h e half-life of t h e p r o t o n should be a b o u t 1 0 years;
that is, after 1 0 years, half of t h e p r o t o n s in t h e universe will have
29
29
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decayed.* (By contrast, this is a b o u t 10 billion billion times longer t h a n
the life of t h e universe itself.) A l t h o u g h this seems like an e n o r m o u s
lifetime, these detectors should have b e e n able to see these rare, fleeting
events simply because t h e r e were so many p r o t o n s in the detector. In
fact, each ton of water contains over 1 0 p r o t o n s . With that many protons, a handful of p r o t o n s were e x p e c t e d to decay every year.
However, no m a t t e r how l o n g t h e experimentalists waited, they saw
no clear-cut evidence of any p r o t o n decays. At present, it seems that
p r o t o n s m u s t have a lifetime larger t h a n 10 years, which rules o u t the
simpler GUTs, b u t still leaves o p e n the possibility of m o r e complicated
GUTs.
Initially, a certain a m o u n t of e x c i t e m e n t over the GUTs spilled over
into the media. T h e quest for a unified theory of m a t t e r a n d the search
for t h e decay of t h e p r o t o n c a u g h t t h e a t t e n t i o n of science p r o d u c e r s
a n d writers. Public television's " N o v a " devoted several shows to it, a n d
p o p u l a r books a n d n u m e r o u s articles in science magazines were written
a b o u t it. Nevertheless, the fanfare died o u t by t h e late 1980s. No matter
how l o n g physicists waited for the p r o t o n to decay, t h e p r o t o n simply
d i d n ' t c o o p e r a t e . After tens of millions of dollars were s p e n t by various
n a t i o n s looking for this event, it has n o t yet b e e n found. Public interest
in the GUTs b e g a n to fizzle.
T h e p r o t o n may still decay, a n d GUTs may still prove to be correct,
b u t physicists are now m u c h m o r e cautious a b o u t touting the GUTs as
the "final t h e o r y , " for several reasons. As with the S t a n d a r d Model,
GUTs m a k e no m e n t i o n of gravity. If we naively c o m b i n e GUTs with
gravity, the theory p r o d u c e s n u m b e r s that are infinite a n d h e n c e m a k e
no sense. Like t h e S t a n d a r d Model, GUTs are n o n r e n o r m a l i z a b l e . Moreover, the theory is defined at t r e m e n d o u s energies, where we certainly
e x p e c t e d gravitational effects to a p p e a r . T h u s t h e fact that gravity is
missing in the G U T theory is a serious drawback. F u r t h e r m o r e , it is also
p l a g u e d by the mysterious p r e s e n c e of t h r e e identical c a r b o n copies or
families of particles. A n d finally, the theory could n o t predict the fund a m e n t a l constants, such as t h e q u a r k masses. GUTs lacked a larger
physical principle that would fix t h e q u a r k masses a n d the o t h e r constants from first principles. Ultimately, it a p p e a r e d that GUTs were also
s t a m p collecting.
29
32
T h e f u n d a m e n t a l p r o b l e m was t h a t t h e Yang-Mills field was n o t suf-
*Half-life
i s t h e a m o u n t o f t i m e i t takes f o r h a l f o f a s u b s t a n c e t o d i s i n t e g r a t e . A f t e r t w o
half-lives, o n l y o n e - q u a r t e r o f t h e substance r e m a i n s .
Quantum Heresy
135
ficient to provide t h e " g l u e " to unite all four interactions. T h e world of
wood, as described by t h e Yang-Mills field, was n o t powerful e n o u g h to
explain t h e world of m a r b l e .
After half a century of d o r m a n c y , the time h a d c o m e for " E i n s t e i n ' s
revenge."
6
Einstein's Revenge
S u p e r s y m m e t r y is t h e u l t i m a t e p r o p o s a l for a c o m p l e t e unific a t i o n o f all p a r t i c l e s .
A b d u s Salam
The Resurrection of Kaluza-Klein
I
T ' S b e e n called " t h e greatest scientific p r o b l e m of all t i m e . " T h e
press has d u b b e d it the " H o l y G r a i l " of physics, the quest to unite
the q u a n t u m theory with gravity, thereby creating a T h e o r y of Everything. This is t h e p r o b l e m that has frustrated the finest minds of the
twentieth century. W i t h o u t question, the p e r s o n who solves this p r o b l e m
will win the Nobel Prize.
By the 1980s, physics was r e a c h i n g an impasse. Gravity alone stubbornly stood a p a r t a n d aloof from the o t h e r t h r e e forces. Ironically,
a l t h o u g h the classical theory of gravity was t h e first to be u n d e r s t o o d
t h r o u g h the work of Newton, the q u a n t u m theory of gravity was the last
interaction to be u n d e r s t o o d by physicists.
All the giants of physics have h a d their crack at this p r o b l e m , a n d all
have failed. Einstein devoted the last 30 years of his life to his unified
field theory. Even t h e great W e r n e r H e i s e n b e r g , o n e of the founders of
q u a n t u m theory, s p e n t t h e last years of his life chasing after his version
of a unified theory of fields, even p u b l i s h i n g a b o o k on t h e subject. In
1958, H e i s e n b e r g even broadcast on r a d i o that he a n d his colleague
136
Einstein's Revenge
137
Wolfgang Pauli h a d finally succeeded in finding the unified field theory,
a n d that only t h e technical details were missing. ( W h e n the press got
wind of this s t u n n i n g declaration, Pauli was furious that H e i s e n b e r g h a d
prematurely m a d e that a n n o u n c e m e n t . Pauli s e n d a letter to his collaborator, consisting of a blank sheet of p a p e r with t h e caption, " T h i s is to
show the world that I can p a i n t like Titian. Only technical details are
missing." )
Later that year, w h e n Wolfgang Pauli finally gave a lecture on the
H e i s e n b e r g - P a u l i unified field theory, many eager physicists were in the
a u d i e n c e , anxious to h e a r the missing details. W h e n he was finished,
however, the talk received a mixed response. Niels B o h r finally stood up
a n d said, " W e are all a g r e e d that your theory is crazy. T h e question
which divides us is w h e t h e r it is crazy e n o u g h . " In fact, so many attempts
have b e e n m a d e at the "final synthesis" that it has created a backlash
of skepticism. Nobel laureate J u l i a n Schwinger has said, " I t ' s n o t h i n g
m o r e t h a n a n o t h e r symptom of the u r g e that afflicts every g e n e r a t i o n
of physicist—the itch to have all t h e f u n d a m e n t a l questions answered in
their own lifetimes."
However, by the 1980s, the " q u a n t u m theory of w o o d , " after a halfcentury of almost u n i n t e r r u p t e d success, was b e g i n n i n g to r u n o u t of
steam. I can vividly r e m e m b e r t h e sense of frustration a m o n g j a d e d
y o u n g physicists d u r i n g this p e r i o d . Everyone sensed that the S t a n d a r d
Model was b e i n g killed by its own success. It was so successful that every
international physics conference s e e m e d like j u s t a n o t h e r r u b b e r s t a m p
of approval. All the talks c o n c e r n e d finding yet a n o t h e r b o r i n g experim e n t a l success for t h e S t a n d a r d Model. At o n e physics c o n f e r e n c e , I
glanced back at t h e a u d i e n c e a n d f o u n d that half of t h e m were slowly
dozing off to sleep; the speaker was d r o n i n g on with c h a r t after c h a r t
showing how t h e latest data could be fit a c c o r d i n g to t h e S t a n d a r d
Model.
1
2
3
I felt like the physicists at the t u r n of the century. They, too, s e e m e d
to be facing a d e a d e n d . They s p e n t d e c a d e s tediously filling up tables
of figures for the spectral lines of various gases, or calculating t h e solutions to Maxwell's equations for increasingly complicated metal surfaces.
Since the Standard Model h a d 19 free p a r a m e t e r s that could be arbitrarily " t u n e d " to any value, like the dials on a radio, I i m a g i n e d that
physicists would s p e n d decades finding the precise values of all 19
parameters.
T h e time h a d c o m e for a revolution. W h a t b e c k o n e d the n e x t generation of physicists was the world of m a r b l e .
Of course, several p r o f o u n d p r o b l e m s stood in t h e way of a g e n u i n e
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q u a n t u m theory of gravity. O n e p r o b l e m with constructing a theory of
gravity is that the force is so m a d d e n i n g l y weak. For e x a m p l e , it takes
the e n t i r e mass of the earth to k e e p pieces of p a p e r on my desk. However, by b r u s h i n g a c o m b t h r o u g h my hair, I can pick up these pieces of
p a p e r , overwhelming the force of the p l a n e t earth. T h e electrons in my
c o m b are m o r e powerful t h a n t h e gravitational pull of the entire planet.
Similarly, if I were to try to construct an " a t o m " with electrons attracted
to t h e nucleus by the gravitational force, a n d n o t the electrical force,
the a t o m would be the size of the universe.
Classically, we see that the gravitational force is negligible c o m p a r e d
with the electromagnetic force, a n d h e n c e is extraordinarily difficult to
m e a s u r e . But if we a t t e m p t to write down a q u a n t u m theory of gravity,
t h e n t h e tables are t u r n e d . T h e q u a n t u m corrections d u e to gravity are
on the o r d e r of t h e Planck energy, or 10 billion electron volts, far
b e y o n d anything achievable on the p l a n e t earth in this century. This
p e r p l e x i n g situation d e e p e n s w h e n we try to construct a c o m p l e t e theory
of q u a n t u m gravity. We recall that when q u a n t u m physicists try to quantize a force, they break it up into tiny packets of energy, called quanta.
If you blindly try to quantize t h e theory of gravity, you postulate that it
functions by t h e e x c h a n g e of tiny packets of gravity, called gravitons. T h e
r a p i d e x c h a n g e of gravitons between m a t t e r is what binds t h e m together
gravitationally. In this picture, what holds us to the floor, a n d keeps us
from flying into o u t e r space at a t h o u s a n d miles p e r h o u r , is the invisible
e x c h a n g e of trillions of tiny graviton particles. But whenever physicists
tried to p e r f o r m simple calculations to calculate q u a n t u m corrections
to Newton's a n d Einstein's laws of gravity, they f o u n d that the result is
infinite, which is useless.
19
For e x a m p l e , let us e x a m i n e what h a p p e n s when two electrically neutral particles b u m p into each o t h e r . To calculate the Feynman diagrams
for this theory, we have to m a k e an a p p r o x i m a t i o n , so we assume that
the curvature of s p a c e - t i m e is small, a n d h e n c e the R i e m a n n metric
tensor is close to 1. For a first guess, we assume that s p a c e - t i m e is close
to b e i n g flat, n o t curved, so we divide the c o m p o n e n t s of the metric
tensor as g = 1 + h , w h e r e 1 r e p r e s e n t s flat space in o u r equations
a n d h is the graviton field. (Einstein, of course, was horrified that quant u m physicists would mutilate his e q u a t i o n s in this way by b r e a k i n g up
t h e metric tensor. This is like taking a beautiful piece of marble a n d
hitting it with a sledge h a m m e r in o r d e r to break it.) After this mutilation
is p e r f o r m e d , we arrive at a conventional-looking q u a n t u m theory. In
Figure 6.1 (a), we see that the two n e u t r a l particles e x c h a n g e a q u a n t u m
of gravity, labeled by the field h.
11
11
11
Figure 6.1. (a) In quantum theory, a quantum of the gravitational force, labeled
h, is called the graviton, which is formed by breaking up Riemann's metric. In
this theory, objects interact by exchanging this packet of gravity. In this way, we
completely lose the beautiful geometric picture of Einstein, (b) Unfortunately, all
the diagrams with loops in them are infinite, which has prevented a unification
of gravity with the quantum theory for the past half-century. A quantum theory
of gravity that unites it with the other forces is the Holy Grail of physics.
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T h e p r o b l e m arises when we sum over all l o o p diagrams: We find
that they diverge, as in Figure 6.1 ( b ) . For t h e Yang-Mills field, we could
use clever sleight-of-hand tricks to shuffle a r o u n d these infinite quantities until they e i t h e r cancel or are a b s o r b e d into quantities that c a n ' t be
m e a s u r e d . However, it can be shown that the usual renormalization prescriptions fail completely w h e n we apply t h e m to a q u a n t u m theory of
gravity. In fact, t h e efforts of physicists over half a century to eliminate
or absorb these infinities has b e e n in vain. In o t h e r words, the bruteforce a t t e m p t to smash m a r b l e into pieces failed miserably.
T h e n , in t h e early 1980s, a curious p h e n o m e n o n o c c u r r e d . KaluzaKlein theory, we recall, h a d b e e n a d o r m a n t theory for 60 years. But
physicists were so frustrated in their attempts to unify gravity with the
o t h e r q u a n t u m forces that they b e g a n to overcome their prejudice a b o u t
u n s e e n d i m e n s i o n s a n d hyperspace. They were ready for an alternative,
a n d that was Kaluza-Klein theory.
T h e late physicist H e i n z Pagels s u m m a r i z e d this e x c i t e m e n t over the
r e - e m e r g e n c e of Kaluza-Klein theory:
A f t e r t h e 1 9 3 0 s , t h e K a l u z a - K l e i n i d e a fell o u t o f favor, a n d f o r m a n y y e a r s
i t lay d o r m a n t . B u t r e c e n t l y , a s p h y s i c i s t s s e a r c h e d o u t e v e r y p o s s i b l e aven u e f o r t h e u n i f i c a t i o n o f gravity w i t h o t h e r f o r c e s , i t h a s a g a i n s p r u n g t o
p r o m i n e n c e . T o d a y , in contrast with t h e 1920s, physicists are c h a l l e n g e d
t o d o m o r e t h a n u n i f y gravity w i t h j u s t e l e c t r o m a g n e t i s m — t h e y w a n t t o
u n i f y gravity w i t h t h e w e a k a n d s t r o n g i n t e r a c t i o n s a s w e l l . T h i s r e q u i r e s
e v e n m o r e d i m e n s i o n s , b e y o n d the fifth.
4
Even Nobel laureate Steven W e i n b e r g was swept up by the enthusiasm g e n e r a t e d by Kaluza-Klein theory. However, t h e r e were still physicists skeptical of t h e Kaluza-Klein renaissance. Harvard's Howard
Georgi, r e m i n d i n g W e i n b e r g how difficult it is to m e a s u r e experimentally these compactified d i m e n s i o n s that have curled u p , c o m p o s e d the
following p o e m :
Steve W e i n b e r g , returning from T e x a s
brings dimensions galore to perplex us
B u t t h e e x t r a o n e s all
are r o l l e d up in a ball
s o tiny i t n e v e r a f f e c t s u s .
5
A l t h o u g h Kaluza-Klein theory was still n o n r e n o r m a l i z a b l e , what
sparked the intense interest in t h e theory was that it gave the h o p e of a
Einstein's Revenge
141
theory m a d e of m a r b l e . T u r n i n g the ugly, confused j u m b l e of wood i n t o
the p u r e , elegant m a r b l e of g e o m e t r y was, of course, Einstein's d r e a m .
But in the 1930s a n d 1940s, almost n o t h i n g was known a b o u t t h e n a t u r e
of wood. However, by t h e 1970s, t h e S t a n d a r d Model h a d finally
u n l o c k e d the secret of wood: that m a t t e r consists of quarks a n d l e p t o n s
held t o g e t h e r by the Yang-Mills field, obeying the symmetry SU(3) X
SU(2) X U ( l ) . T h e p r o b l e m was how to derive these particles a n d mysterious symmetries from m a r b l e .
At first, that s e e m e d impossible. After all, these symmetries are t h e
result of i n t e r c h a n g i n g p o i n t particles a m o n g o n e a n o t h e r . If N quarks
within a multiplet are shuffled a m o n g o n e a n o t h e r , t h e n t h e symmetry
is SU(N). T h e s e symmetries s e e m e d to be exclusively the symmetries of
wood, n o t marble. W h a t did SU(N) have to do with geometry?
Turning Wood into Marble
T h e first small clue came in t h e 1960s, w h e n physicists f o u n d , m u c h to
their delight, that t h e r e is an alternative way in which to i n t r o d u c e symmetries into physics. W h e n physicists e x t e n d e d the old five-dimensional
theory of Kaluza-Klein to N dimensions, they realized that t h e r e is the
freedom to impose a symmetry on hyperspace. W h e n t h e fifth d i m e n s i o n
was curled u p , they saw that t h e Maxwell field p o p p e d o u t of R i e m a n n ' s
metric. But when N d i m e n s i o n s were curled u p , physicists f o u n d t h e
celebrated Yang-Mills field, t h e key to the S t a n d a r d Model, p o p p i n g o u t
of their equations!
To see how symmetries e m e r g e from space, consider an ordinary
beach ball. It has a symmetry: We can rotate it a r o u n d its center, a n d
the b e a c h ball retains its s h a p e . T h e symmetry of a b e a c h ball, or a
sphere, is called 0 ( 3 ) , or rotations in t h r e e d i m e n s i o n . Similarly, in
h i g h e r dimensions, a h y p e r s p h e r e can also be r o t a t e d a r o u n d its c e n t e r
a n d maintain its s h a p e . T h e h y p e r s p h e r e has a symmetry called O(N).
Now consider vibrating t h e b e a c h ball. Ripples form on the surface
of the ball. If we carefully vibrate the b e a c h ball in a certain way, we can
i n d u c e regular vibrations on it that are called resonances. T h e s e resonances, unlike ordinary ripples, can vibrate at only certain frequencies.
In fact, if we vibrate the b e a c h ball fast e n o u g h , we can create musical
tones of a definite frequency. T h e s e vibrations, in t u r n , can be cataloged
by the symmetry 0 ( 3 ) .
T h e fact that a m e m b r a n e , like a b e a c h ball, can i n d u c e r e s o n a n c e
frequencies is a c o m m o n p h e n o m e n o n . T h e vocal c h o r d s in o u r throat,
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for e x a m p l e , are stretched m e m b r a n e s that vibrate at definite frequencies, or resonances, a n d can thereby p r o d u c e musical tones. A n o t h e r
e x a m p l e is o u r h e a r i n g . S o u n d waves of all types i m p i n g e on o u r eard r u m s , which t h e n resonate at definite frequencies. These vibrations are
t h e n t u r n e d into electrical signals that are sent into o u r brain, which
interprets t h e m as sounds. This is also t h e principle b e h i n d the telep h o n e . T h e metallic d i a p h r a g m c o n t a i n e d in any t e l e p h o n e is set into
m o t i o n by electrical signals in the t e l e p h o n e wire. This creates mechanical vibrations or r e s o n a n c e s in t h e d i a p h r a g m , which in turn create the
s o u n d waves we h e a r on t h e p h o n e . This is also the principle b e h i n d
stereo speakers as well as orchestral d r u m s .
For a h y p e r s p h e r e , the effect is t h e same. Like a m e m b r a n e , it can
resonate at various frequencies, which in t u r n can be d e t e r m i n e d by its
symmetry O(N). Alternatively, m a t h e m a t i c i a n s have d r e a m e d up m o r e
sophisticated surfaces in h i g h e r d i m e n s i o n s that are described by complex n u m b e r s . ( C o m p l e x n u m b e r s use the square r o o t of — 1 , '— 1.)
T h e n it is straightforward to show that t h e symmetry c o r r e s p o n d i n g to
a c o m p l e x " h y p e r s p h e r e " is S U ( N ) .
T h e key p o i n t is now this: If the wave function of a particle vibrates
a l o n g this surface, it will i n h e r i t this SU(N) symmetry. T h u s the mysterious SU(N) symmetries arising in subatomic physics can now be seen
as by-products of vibrating hyperspace! In o t h e r words, we now have an explan a t i o n for the origin of the mysterious symmetries of wood: They are
really t h e h i d d e n symmetries c o m i n g from marble.
If we now take a Kaluza-Klein theory defined in 4 + N dimensions
a n d t h e n curl up N d i m e n s i o n s , we will find that t h e equations split into
two pieces. T h e first piece is Einstein's usual equations, which we retrieve
as e x p e c t e d . But the s e c o n d piece will n o t be the theory of Maxwell. We
find that the r e m a i n d e r is precisely t h e Yang-Mills theory, which forms
the basis of all subatomic physics! This is the key to t u r n i n g the symmetries of wood into the symmetries of m a r b l e .
At first, it seems almost mystical that the symmetries of wood, which
were discovered painfully by trial a n d e r r o r — t h a t is, by painstakingly
e x a m i n i n g t h e debris from a t o m s m a s h e r s — e m e r g e almost automatically from h i g h e r dimensions. It is miraculous that the symmetries found
by shuffling quarks a n d leptons a m o n g themselves should arise from
hyperspace. An analogy may h e l p us u n d e r s t a n d this. Matter may be
likened to clay, which is formless a n d lumpy. Clay lacks any of the beautiful symmetries that are i n h e r e n t in g e o m e t r i c figures. However, clay
may be pressed i n t o a mold, which can have symmetries. For example,
the m o l d may preserve its shape if it is rotated by a certain angle. T h e n
v
Einstein's Revenge
143
the clay will also i n h e r i t t h e symmetry of t h e m o l d . Clay, like m a t t e r ,
inherits its symmetry b e c a u s e t h e m o l d , like s p a c e - t i m e , has a symmetry.
If correct, t h e n this m e a n s t h a t the strange symmetries we see a m o n g
t h e quarks a n d leptons, which were discovered largely by accident over
several decades, can now be seen as by-products of vibrations in hyperspace. For e x a m p l e , if t h e u n s e e n d i m e n s i o n s have t h e symmetry S U ( 5 ) ,
t h e n we can write SU(5) G U T as a Kaluza-Klein theory.
This can also be seen from R i e m a n n ' s metric tensor. We recall that
it resembles Faraday's field e x c e p t that it has many m o r e c o m p o n e n t s .
It can be a r r a n g e d like the squares of a c h e c k e r b o a r d . By separating o u t
the fifth c o l u m n a n d row of the c h e c k e r b o a r d , we can split off Maxwell's
field from Einstein's field. Now p e r f o r m the same trick with KaluzaKlein theory in (4 + N)-dimensional space. If you split off t h e N c o l u m n s
a n d rows from the first four c o l u m n s a n d rows, t h e n you obtain a m e t r i c
tensor that describes b o t h Einstein's theory a n d Yang-Mills theory. In
Figure 6.2, we have carved up the metric tensor of a (4 + N)-dimensional
Figure 6.2. If we go to the Nth dimension, then the metric tensor is a series of
N numbers that can be arranged in an N X N block. By slicing off the fifth
and higher columns and rows, we can extract the Maxwell electromagnetic field
and the Yang-Mills field. Thus, in one stroke, the hyperspace theory allows us to
unify the Einstein field (describing gravity), the Maxwell field (describing the
electromagnetic force), and the Yang-Mills field (describing the weak and strong
force). The fundamental forces fit together exactly like a jigsaw puzzle.
2
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Kaluza-Klein theory, splitting off Einstein's field from the Yang-Mills
field.
Apparently, o n e of the first physicists to p e r f o r m this r e d u c t i o n was
University of Texas physicist Bryce DeWitt, w h o has s p e n t many years
studying q u a n t u m gravity. O n c e this trick of splitting up the metric tensor was discovered, the calculation for extracting t h e Yang-Mills field is
straightforward. DeWitt felt that extracting the Yang-Mills field from Nd i m e n s i o n a l gravity theory was such a simple mathematical exercise that
he assigned it as a h o m e w o r k p r o b l e m at the Les H o u c h e s Physics Summ e r School in France in 1963. [Recently, it was revealed by Peter F r e u n d
that Oskar Klein h a d i n d e p e n d e n t l y discovered t h e Yang-Mills field in
1938, p r e c e d i n g the work of Yang, Mills, a n d o t h e r s by several decades.
In a c o n f e r e n c e h e l d in Warsaw titled " N e w Physical T h e o r i e s , " Klein
a n n o u n c e d that he was able to generalize the work of Maxwell to include
a h i g h e r symmetry, O(3). Unfortunately, because of the chaos u n l e a s h e d
by World War II a n d because Kaluza-Klein theory was buried by the
e x c i t e m e n t g e n e r a t e d by q u a n t u m theory, this i m p o r t a n t work was forg o t t e n . It is ironic that Kaluza-Klein theory was killed by t h e e m e r g e n c e
of q u a n t u m theory, which is now based on the Yang-Mills field, which
was first discovered by analyzing Kaluza-Klein theory. In the e x c i t e m e n t
to develop q u a n t u m theory, physicists h a d i g n o r e d a central discovery
c o m i n g from Kaluza-Klein theory.]
Extracting t h e Yang-Mills field o u t of Kaluza-Klein theory was only
t h e first step. A l t h o u g h t h e symmetries of w o o d could now be seen as
arising from t h e h i d d e n symmetries of u n s e e n dimensions, the n e x t step
was to create wood itself ( m a d e of quarks a n d leptons) entirely o u t of
m a r b l e . This n e x t step would be called supergravity.
Supergravity
T u r n i n g wood into m a r b l e still faced formidable p r o b l e m s because,
a c c o r d i n g to t h e S t a n d a r d Model, all particles are " s p i n n i n g . " Wood,
for e x a m p l e , we now know is m a d e of quarks a n d leptons. They, in t u r n ,
have 1/2 u n i t of q u a n t u m spin (measured in units of Planck's constant h-bar.
Particles with half-integral spin (1/2, 3/2, 5/2, a n d so on) are called fermions
( n a m e d after E n r i c o Fermi, w h o first investigated their strange p r o p e r ties). However, forces are described by q u a n t a with integral spin. For
e x a m p l e , t h e p h o t o n , the q u a n t u m of light, has o n e u n i t of spin. So does
the Yang-Mills field. T h e graviton, the hypothetical packet of gravity,
has two units of spin. They are called bosons (after the Indian physicist
Satyendra Bose).
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145
Traditionally, q u a n t u m theory k e p t fermions a n d bosons strictly
apart. I n d e e d , any a t t e m p t to t u r n wood into m a r b l e would inevitably
c o m e to grips with the fact that fermions a n d b o s o n s are worlds a p a r t
in their properties. For e x a m p l e , SU(N) may shuffle quarks a m o n g o n e
a n o t h e r , b u t fermions a n d bosons were never s u p p o s e d to mix. It c a m e
as a shock, therefore, w h e n a new symmetry, called supersymmetry, was
discovered, that did exactly that. Equations that are supersymmetric
allow the i n t e r c h a n g e of a fermion with a b o s o n a n d still k e e p the equations intact. In o t h e r words, o n e multiplet of supersymmetry consists of
equal n u m b e r s of bosons a n d fermions. By shuffling t h e b o s o n s a n d
fermions within t h e same multiplet, the supersymmetric e q u a t i o n s
remain the same.
This gives us the tantalizing possibility of p u t t i n g all t h e particles in
the universe into o n e multiplet! As Nobel laureate Abdus Salam has
emphasized, " S u p e r s y m m e t r y is the ultimate proposal for a c o m p l e t e
unification of all particles."
Supersymmetry is based on a new kind of n u m b e r system t h a t would
drive any schoolteacher insane. Most of t h e o p e r a t i o n s of multiplication
a n d division that we take for g r a n t e d fail for supersymmetry. For example, if a a n d b are two " s u p e r n u m b e r s , " t h e n a X b = —b X a. This, of
course, is strictly impossible for ordinary n u m b e r s . Normally, any schoolteacher would throw these s u p e r n u m b e r s o u t the window, because you
can show that a X a = — a X a, or, in o t h e r words, a X a = 0. If these
were ordinary n u m b e r s , t h e n this m e a n s that a = 0, a n d t h e n u m b e r
system collapses. However, with super n u m b e r s , the system does n o t collapse; we have the r a t h e r astonishing s t a t e m e n t that a X a = 0 even
when a =/ 0. A l t h o u g h these s u p e r n u m b e r s violate almost everything
we have learned a b o u t n u m b e r s since c h i l d h o o d , they can be shown to
yield a self-consistent a n d highly nontrivial system. Remarkably, an
entirely new system of s u p e r calculus can be based on t h e m .
Soon, t h r e e physicists (Daniel F r e e d m a n , Sergio Ferrara, a n d P e t e r
van Nieuwenhuizen, at the State University of New York at Stony Brook)
wrote down the theory of supergravity in 1976. Supergravity was the first
realistic a t t e m p t to construct a world m a d e entirely of m a r b l e . In a supersymmetric theory, all particles have s u p e r p a r t n e r s , called sparticles. T h e
supergravity theory of the Stony Brook g r o u p contains j u s t two fields:
the spin-two graviton field (which is a b o s o n ) a n d its s p i n - 3 / 2 p a r t n e r ,
called the gravitino (which m e a n s "little gravity"). Since this is n o t
e n o u g h particles to i n c l u d e the S t a n d a r d Model, a t t e m p t s were m a d e to
couple the theory to m o r e complicated particles.
T h e simplest way to i n c l u d e m a t t e r is to write down t h e supergravity
theory in 11-dimensional space. In o r d e r to write down the s u p e r
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Figure 6.3. Supergravity almost fulfills Einstein's dream of giving a purely geometric derivation of all the forces and particles in the universe. To see this, notice
that if we add supersymmetry to the Riemann metric tensor, the metric doubles in
size, giving us the super Riemann metric. The new components of the super Riemann tensor correspond to quarks and leptons. By slicing the super Riemann
tensor into its components, we find that it includes almost all the fundamental
particles and forces in nature: Einstein's theory of gravity, the Yang-Mills and
Maxwell fields, and the quarks and leptons. But the fact that certain particles
are missing in this picture forces us to go a more powerful formalism: superstring
Kaluza-Klein theory in 11 dimensions, o n e has to increase the c o m p o n e n t s within the R i e m a n n tensor vastly, which now b e c o m e s the super
R i e m a n n tensor. To visualize how supergravity converts wood into marble, let us write down the metric tensor a n d show how supergravity manages to fit the Einstein field, t h e Yang-Mills field, a n d the m a t t e r fields
into o n e supergravity field (Figure 6.3). T h e essential feature of this
diagram is that matter, a l o n g with t h e Yang-Mills a n d Einstein equations, is now i n c l u d e d in the same 11-dimensional supergravity field.
Supersymmetry is the symmetry that reshuffles the wood into marble
a n d vice versa within the supergravity field. T h u s they are all manifestations of the same force, the superforce. W o o d no l o n g e r exists as a
single, isolated entity. It is now m e r g e d with marble, to form supermarble (Figure 6.4)!
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147
Physicist Peter van N i e u w e n h u i z e n , o n e of supergravity's creators,
was deeply impressed by t h e implication of this superunification. He
wrote that supergravity " m a y unify g r a n d unified theories . . . with gravity, leading to a m o d e l with almost no free p a r a m e t e r s . It is the u n i q u e
theory with a local gauge symmetry between fermions a n d bosons. It is
the most beautiful gauge theory known, so beautiful, in fact, that N a t u r e
should be aware of i t ! "
I fondly r e m e m b e r a t t e n d i n g a n d giving lectures at many of these
supergravity conferences. T h e r e was an intense, exhilarating feeling that
we were on the verge of s o m e t h i n g i m p o r t a n t . At o n e m e e t i n g in Moscow, I r e m e m b e r well, a series of lively toasts were m a d e to t h e c o n t i n u e d
success of the supergravity theory. It s e e m e d that we were finally on the
verge of carrying o u t Einstein's d r e a m of a universe of m a r b l e after 60
years of neglect. Some of us jokingly called it "Einstein's r e v e n g e . "
On April 29, 1980, when cosmologist S t e p h e n Hawking assumed the
Lucasian Professorship (previously h e l d by s o m e of t h e i m m o r t a l s of
physics, including Isaac Newton a n d P. A. M. Dirac), he gave a lecture
with the auspicious title "Is the E n d in Sight for The ore tic a l Physics?"
6
Figure 6.4. In supergravity, we almost get a unification of all the known forces
(marble) with matter (wood). Like a jigsaw puzzle, they fit inside Riemann's metric
tensor. This almost fulfills Einstein's dream.
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A s t u d e n t r e a d for h i m : " [ W ] e have m a d e a lot of progress in r e c e n t
years a n d , as I shall describe, t h e r e are s o m e g r o u n d s for cautious optimism that we may see a c o m p l e t e theory within the lifetime of some of
those p r e s e n t h e r e . "
Supergravity's fame gradually spread into the general public a n d
even b e g a n to have a following a m o n g religious groups. For example,
t h e c o n c e p t of " u n i f i c a t i o n " is a central belief within t h e transcendental
m e d i t a t i o n m o v e m e n t . Its followers therefore published a large poster
c o n t a i n i n g the c o m p l e t e e q u a t i o n s describing 11-dimensional supergravity. Each t e r m in t h e e q u a t i o n , they claimed, r e p r e s e n t e d s o m e t h i n g
special, such as " h a r m o n y , " " l o v e , " " b r o t h e r h o o d , " a n d so o n . (This
poster h a n g s on t h e wall of the theoretical institute at Stony Brook. This
is t h e first time that I am aware of that an abstract e q u a t i o n from theoretical physics has inspired a following a m o n g a religious group!)
Super Metric Tensors
P e t e r van N i e u w e n h u i z e n cuts a r a t h e r d a s h i n g figure in physics circles.
Tall, t a n n e d , athletic looking, a n d well dressed, he looks m o r e like an
actor p r o m o t i n g s u n t a n lotion on television t h a n o n e of the original
creators of supergravity. He is a Dutch physicist w h o is now a professor
at Stony Brook; he was a s t u d e n t of Veltman, as was 't Hooft, a n d was
therefore long interested in t h e question of unification. He is o n e of the
few physicists I have ever m e t with a truly inexhaustible capacity for
m a t h e m a t i c a l p u n i s h m e n t . W o r k i n g with supergravity requires an
e x t r a o r d i n a r y a m o u n t of p a t i e n c e . We recall that the simple metric tensor i n t r o d u c e d by R i e m a n n in t h e n i n e t e e n t h century h a d only ten comp o n e n t s . R i e m a n n ' s metric tensor has now b e e n replaced by the super
metric tensor of supergravity, which has literally h u n d r e d s of c o m p o n e n t s . This is n o t surprising, since any theory that has h i g h e r dimensions
a n d makes the claim of unifying all m a t t e r has to have e n o u g h c o m p o n e n t s to describe it, b u t this vastly increases t h e mathematical complexity
of t h e equations. (Sometimes I w o n d e r what R i e m a n n would think,
knowing that after a century his metric tensor would blossom into a
s u p e r metric m a n y times larger t h a n a n y t h i n g a nineteenth-century
m a t h e m a t i c i a n could conceive.)
T h e c o m i n g of supergravity a n d s u p e r metric tensors has m e a n t that
t h e a m o u n t of m a t h e m a t i c s a g r a d u a t e s t u d e n t must master has
e x p l o d e d within t h e past d e c a d e . As Steven W e i n b e r g observes, " L o o k
what's h a p p e n e d with supergravity. T h e p e o p l e who've b e e n working o n
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Einstein's Revenge
it for the past ten years are enormously bright. Some of t h e m a r e
brighter t h a n a n y o n e I knew in my early y e a r s . "
Peter is n o t only a s u p e r b calculator, b u t also a trendsetter. Because
calculations for a single supergravity e q u a t i o n can easily e x c e e d a s h e e t
of paper, he eventually started using large, oversize artist's sketch b o a r d s .
I went to his h o u s e o n e day, a n d saw how he o p e r a t e d . He would start
at the upper4eft-hand c o r n e r of the p a d , a n d start writing his e q u a t i o n s
in his microscopic handwriting. He would t h e n p r o c e e d to work across
a n d down the sketch p a d until it was completely filled, a n d t h e n t u r n
the page a n d start again. This process would t h e n go on for h o u r s , until
the calculation was c o m p l e t e d . T h e only time he would ever be interr u p t e d was when he inserted his pencil into a nearby electric pencil
sharpener, a n d t h e n within seconds he would r e s u m e his calculation
without missing a symbol. Eventually, he would store these artist's n o t e pads on his shelf, as t h o u g h they were volumes of s o m e scientific j o u r n a l .
Peter's sketch pads gradually b e c a m e n o t o r i o u s a r o u n d c a m p u s . Soon,
a fad started; all the g r a d u a t e students in physics b e g a n to buy these
bulky artist's sketch pads a n d could be seen on c a m p u s h a u l i n g t h e m
awkwardly b u t proudly u n d e r their arms.
7
O n e time, Peter, his friend Paul T o w n s e n d (now at C a m b r i d g e University), a n d I were collaborating on an exceptionally difficult supergravity p r o b l e m . T h e calculation was so difficult that it c o n s u m e d several
h u n d r e d pages. Since n o n e of us totally trusted o u r calculations, we
decided to m e e t in my d i n i n g r o o m a n d collectively check o u r work. We
faced a d a u n t i n g challenge: Several t h o u s a n d terms h a d to s u m up to
exactly zero. (Usually, we theoretical physicists can "visualize" blocks of
equations in o u r h e a d s a n d m a n i p u l a t e t h e m without having to use
paper. However, because of the s h e e r length a n d delicacy of this p r o b lem, we h a d to check every single m i n u s sign in t h e calculation.)
We t h e n divided the p r o b l e m into several large c h u n k s . Sitting
a r o u n d the dining-room table, each of us would busily calculate the same
c h u n k . After an h o u r or so, we would t h e n cross-check o u r results. Usually two o u t of t h r e e would get it right, a n d t h e third would be asked to
find his mistake. T h e n we would go to the n e x t c h u n k , a n d r e p e a t t h e
same process until all t h r e e of us a g r e e d on the same answer. This repetitive cross-checking went on late i n t o t h e night. We knew that even o n e
mistake in several h u n d r e d pages would give us a totally worthless calculation. Finally, well past m i d n i g h t we c h e c k e d the last a n d final t e r m .
It was zero, as we h a d h o p e d . We t h e n toasted o u r result. ( T h e a r d u o u s
calculation must have e x h a u s t e d even an indefatigable w o r k h o r s e like
Peter. After leaving my a p a r t m e n t , he p r o m p t l y forgot w h e r e his wife's
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new a p a r t m e n t was in M a n h a t t a n . He k n o c k e d on several d o o r s of an
a p a r t m e n t h o u s e , b u t got only angry responses; he h a d chosen the wrong
building. After a futile search, Peter a n d Paul reluctantly h e a d e d back
to Stony Brook. But because Peter h a d forgotten to replace a clutch
cable, t h e cable s n a p p e d , a n d they h a d to p u s h his car. They eventually
straggled into Stony Brook in their b r o k e n car at 5:00 in the morning!)
The Decline of Supergravity
T h e critics, however, gradually b e g a n to see p r o b l e m s with supergravity.
After an intensive search, sparticles were n o t seen in any e x p e r i m e n t .
For e x a m p l e , t h e s p i n - 1 / 2 electron does n o t have any spin-0 p a r t n e r . In
fact, t h e r e is, at the present, n o t o n e shred of e x p e r i m e n t a l evidence for
sparticles in o u r low-energy world. However, the firm belief of physicists
working in this area is that, at the e n o r m o u s energies found at the instant
of Creation, all particles were a c c o m p a n i e d by their super p a r t n e r s . Only
at this incredible energy do we see a perfectly supersymmetric world.
But after a few years of fervent interest a n d scores of international
conferences, it b e c a m e clear t h a t this theory could n o t be quantized
correctly, thus temporarily derailing the d r e a m of creating a theory
purely o u t of m a r b l e . Like every o t h e r a t t e m p t to construct a theory of
m a t t e r entirely from m a r b l e , supergravity failed for a very simple reason:
W h e n e v e r we tried to calculate n u m b e r s from these theories, we would
arrive at meaningless infinities. T h e theory, a l t h o u g h it h a d fewer infinities t h a n t h e original Kaluza-Klein theory, was still n o n r e n o r m a l i z a b l e .
T h e r e were o t h e r p r o b l e m s . T h e highest symmetry that supergravity
could include was called 0 ( 8 ) , which was too small to a c c o m m o d a t e the
symmetry of t h e S t a n d a r d Model. Supergravity, it a p p e a r e d , was j u s t
a n o t h e r step in the l o n g j o u r n e y toward a unified theory of the universe.
It c u r e d o n e p r o b l e m ( t u r n i n g wood into m a r b l e ) , only to fall victim to
several o t h e r diseases. However, j u s t as interest in supergravity began to
wane, a new theory c a m e a l o n g t h a t was p e r h a p s the strangest b u t most
powerful physical theory ever p r o p o s e d : t h e ten-dimensional superstring
theory.
7
Superstrings
S t r i n g t h e o r y i s twenty-first c e n t u r y p h y s i c s t h a t fell a c c i d e n tally i n t o t h e t w e n t i e t h c e n t u r y .
Edward Witten
E
DWARD Witten, of t h e Institute for Advanced Study in P r i n c e t o n ,
New Jersey, d o m i n a t e s the world of theoretical physics. Witten is
currently the " l e a d e r of t h e p a c k , " the most brilliant high-energy physicist, who sets t r e n d s in the physics c o m m u n i t y the way Picasso would set
trends in the art world. H u n d r e d s of physicists follow his work religiously
to get a glimmer of his path-breaking ideas. A colleague at P r i n c e t o n ,
Samuel T r e i m a n , says, " H e ' s h e a d a n d shoulders above t h e rest. H e ' s
started whole g r o u p s of p e o p l e on new paths. He p r o d u c e s elegant,
breathtaking proofs which p e o p l e gasp at, which leave t h e m in a w e . "
T r e i m a n t h e n concludes, " W e s h o u l d n ' t toss comparisons with Einstein
a r o u n d too freely, b u t w h e n it comes to Witten . . . " '
W i t t e n c o m e s f r o m a family of physicists. His f a t h e r is Louis
Witten, p r o f e s s o r of physics at t h e University of C i n c i n n a t i a n d a
l e a d i n g a u t h o r i t y o n Einstein's t h e o r y o f g e n e r a l relativity. (His
father, in fact, s o m e t i m e s states t h a t his g r e a t e s t c o n t r i b u t i o n to
physics was p r o d u c i n g his son.) His wife is C h i a r a N a p p i , also a
theoretical physicist at t h e institute.
Witten is n o t like o t h e r physicists. Most of t h e m begin their r o m a n c e
with physics at an early age (such as in j u n i o r high school or even elementary school). Witten has defied most conventions, starting o u t as a
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history major at Brandeis University with an intense interest in linguistics. After g r a d u a t i n g in 1971, he worked on G e o r g e McGovern's presidential c a m p a i g n . McGovern even wrote h i m a letter of r e c o m m e n d a tion for g r a d u a t e school. Witten has published articles in The Nation a n d
the New Republic. (Scientific American, in an interview with Witten, comm e n t e d , "yes, a m a n who is arguably the smartest p e r s o n in the world
is a liberal D e m o c r a t . " )
But o n c e Witten d e c i d e d that physics was his chosen profession, he
l e a r n e d physics with a vengeance. He b e c a m e a g r a d u a t e s t u d e n t at
P r i n c e t o n , t a u g h t at Harvard, a n d t h e n rocketed a full professorship at
P r i n c e t o n at the age of 28. He also received t h e prestigious MacArthur
Fellowship (sometimes d u b b e d the " g e n i u s " award by the press). Spinoffs from his work have also deeply affected t h e world of mathematics.
In 1990, he was awarded the Fields Medal, which is as prestigious as the
Nobel Prize in the world of mathematics.
Most of t h e time, however, Witten sits a n d stares o u t the window,
m a n i p u l a t i n g a n d r e a r r a n g i n g vast arrays of equations in his h e a d . His
wife notes, " H e never d o e s calculations except in his m i n d . I will fill
pages with calculations before I u n d e r s t a n d what I ' m doing. But Edward
will sit down only to calculate a m i n u s sign, or a factor of t w o . " Witten
says, "Most p e o p l e w h o h a v e n ' t b e e n trained in physics probably think
of what physicists do as a question of incredibly complicated calculations,
b u t that's n o t really t h e essence of it. T h e essence of it is that physics is
a b o u t concepts, wanting to u n d e r s t a n d t h e concepts, the principles by
which t h e world w o r k s . "
Witten's n e x t project is t h e most ambitious a n d d a r i n g of his career.
A new theory called superstring theory has created a sensation in the
world of physics, claiming to be t h e theory that can u n i t e Einstein's
theory of gravity with the q u a n t u m theory. Witten is n o t content, however, with the way superstring theory is currently formulated. He has set
for himself the p r o b l e m of finding the origin of superstring theory, which
may prove to be a decisive d e v e l o p m e n t toward explaining the very
instant of Creation. T h e key aspect of this theory, the factor that gives it
its power as well as u n i q u e n e s s , is its u n u s u a l geometry: Strings can
vibrate self-consistently only in ten a n d 26 dimensions.
2
3
4
What Is a Particle?
T h e essence of string theory is that it can explain the n a t u r e of b o t h
m a t t e r a n d s p a c e - t i m e — t h a t is, t h e n a t u r e of wood a n d marble. String
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153
theory answers a series of puzzling questions a b o u t particles, such as why
there are so many of t h e m in n a t u r e . T h e d e e p e r we p r o b e into t h e
n a t u r e of subatomic particles, t h e m o r e particles we find. T h e c u r r e n t
" z o o " o f subatomic particles n u m b e r s several h u n d r e d , a n d their p r o p erties fill entire volumes. Even with the Standard Model, we are left with
a bewildering n u m b e r of " e l e m e n t a r y particles." String theory answers
this question because t h e string, a b o u t 100 billion billion times smaller
than a p r o t o n , is vibrating; each m o d e of vibration r e p r e s e n t s a distinct
r e s o n a n c e or particle. T h e string is so incredibly tiny that, from a distance, a r e s o n a n c e of a string a n d a particle are indistinguishable. Only
when we somehow magnify t h e particle can we see that it is n o t a p o i n t
at all, b u t a m o d e of a vibrating string.
In this picture, each subatomic particle c o r r e s p o n d s to a distinct reso n a n c e that vibrates only at a distinct frequency. T h e idea of a r e s o n a n c e
is a familiar o n e from daily life. T h i n k of the e x a m p l e of singing in t h e
shower. Although o u r natural voice may be frail, tinny, or shaky, we know
that we suddenly blossom into o p e r a stars in the privacy of o u r showers.
This is because o u r s o u n d waves b o u n c e rapidly back a n d forth between
the walls of the shower. Vibrations that can fit easily within t h e shower
walls are magnified m a n y times, p r o d u c i n g that r e s o n a n t s o u n d . T h e
specific vibrations are called resonances, while o t h e r vibrations (whose
waves are of an incorrect size) are canceled out.
Or think of a violin string, which can vibrate at different frequencies,
creating musical notes like A, B, a n d C. T h e only m o d e s that can survive
on the string are those that vanish at the e n d p o i n t of the violin string
(because it is bolted down at t h e ends) a n d u n d u l a t e an integral n u m b e r
of times between the e n d p o i n t s . In principle, t h e string can vibrate at
any of an infinite n u m b e r of different frequencies. We know t h a t the
notes themselves are n o t f u n d a m e n t a l . T h e n o t e A is no m o r e fundamental than t h e n o t e B. However, what is f u n d a m e n t a l is the string itself.
T h e r e is no n e e d to study each n o t e in isolation of the others. By u n d e r standing how a violin string vibrates, we immediately u n d e r s t a n d t h e
properties of an infinite n u m b e r of musical notes.
Likewise, t h e particles of the universe are not, by themselves, fund a m e n t a l . An electron is no m o r e f u n d a m e n t a l t h a n a n e u t r i n o . T h e y
a p p e a r to be f u n d a m e n t a l only because o u r microscopes are n o t powerful e n o u g h to reveal their structure. A c c o r d i n g to string theory, if we
could somehow magnify a p o i n t particle, we would actually see a small
vibrating string. In fact, a c c o r d i n g to this theory, m a t t e r is n o t h i n g b u t
the h a r m o n i e s created by this vibrating string. Since t h e r e are an infinite
n u m b e r of h a r m o n i e s that can be c o m p o s e d for the violin, t h e r e are an
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infinite n u m b e r of forms of m a t t e r that can be constructed o u t of vibrating strings. This explains t h e richness of the particles in n a t u r e . Likewise,
t h e laws of physics can be c o m p a r e d to the laws of h a r m o n y allowed on
t h e string. T h e universe itself, c o m p o s e d of countless vibrating strings,
would t h e n be c o m p a r a b l e to a symphony.
String theory can explain n o t only the n a t u r e of particles, b u t that
of s p a c e - t i m e as well. As a string moves in s p a c e - t i m e , it executes a
complicated set of motions. T h e string can, in turn, break into smaller
strings or collide with o t h e r strings to form longer strings. T h e key p o i n t
is that all these q u a n t u m corrections or l o o p diagrams are finite a n d
calculable. This is the first q u a n t u m theory of gravity in the history of
physics to have finite q u a n t u m corrections. (All known previous theories,
we recall—including Einstein's original theory, Kaluza-Klein theory,
a n d supergravity—failed this key criterion.)
In o r d e r to execute these complicated motions, a string must obey a
large set of self-consistency conditions. T h e s e self-consistency conditions
are so stringent that they place extraordinarily restrictive conditions on
s p a c e - t i m e . In o t h e r words, the string c a n n o t self-consistently travel in
any arbitrary s p a c e - t i m e , like a p o i n t particle.
W h e n the constraints that the string places on s p a c e - t i m e were first
calculated, physicists were shocked to find Einstein's equations emerging from the string. This was r e m a r k a b l e ; without assuming any of Einstein's e q u a t i o n s , physicists f o u n d that they e m e r g e d out of the string
theory, as if by magic. Einstein's equations were no longer found to be
f u n d a m e n t a l ; they could be derived from string theory.
If correct, t h e n string theory solves the long-standing mystery a b o u t
the n a t u r e of wood a n d marble. Einstein conjectured that m a r b l e alone
would o n e day explain all the properties of wood. To Einstein, wood was
j u s t a kink or vibration of s p a c e - t i m e , n o t h i n g m o r e or less. Q u a n t u m
physicists, however, t h o u g h t the opposite. They t h o u g h t that marble
could be t u r n e d into w o o d — t h a t is, that Einstein's metric tensor could
be t u r n e d into a graviton, the discrete packet of energy that carries the
gravitational force. T h e s e a r e two diametrically opposite points of view,
a n d it was l o n g t h o u g h t that a c o m p r o m i s e between t h e m was impossible. T h e string, however, is precisely the "missing l i n k " between wood
and marble.
String theory can derive the particles of m a t t e r as resonances vibrating on t h e string. A n d string theory can also derive Einstein's equations
by d e m a n d i n g that t h e string move self-consistently in s p a c e - t i m e . In
this way, we have a comprehensive theory of b o t h m a t t e r - e n e r g y a n d
space-time.
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These self-consistency constraints are surprisingly rigid. For e x a m p l e ,
they forbid the string to move in t h r e e or four d i m e n s i o n s . We will see
that these self-consistency conditions force the string to move in a specific n u m b e r of d i m e n s i o n s . In fact, the only " m a g i c n u m b e r s " allowed
by string theory are ten a n d 26 dimensions. Fortunately, a string theory
defined in these d i m e n s i o n s has e n o u g h " r o o m " to unify all f u n d a m e n tal forces.
String theory, therefore, is rich e n o u g h to explain all the f u n d a m e n tal laws of n a t u r e . Starting from a simple theory of a vibrating string,
o n e can extract the theory of Einstein, Kaluza-Klein theory, supergravity, the Standard Model, a n d even G U T theory. It seems n o t h i n g less
than a miracle that, starting from some purely g e o m e t r i c a r g u m e n t s
from a string, o n e is able to rederive the entire progress of physics for
the past 2 millennia. All the theories so far discussed in this b o o k are
automatically i n c l u d e d in string theory.
T h e c u r r e n t interest in string theory stems from t h e work of J o h n
Schwarz of the California Institute of T e c h n o l o g y a n d his collaborator
Michael G r e e n of Q u e e n Mary's College in L o n d o n . Previously, it was
t h o u g h t that the string m i g h t possess defects that would p r e v e n t a fully
self-consistent theory. T h e n in 1984, these two physicists proved that all
self-consistency conditions on t h e string can be met. This, in t u r n ,
ignited the c u r r e n t s t a m p e d e a m o n g y o u n g physicists to solve t h e theory
a n d win potential recognition. By the late 1980s, a veritable " g o l d r u s h "
b e g a n a m o n g physicists. ( T h e c o m p e t i t i o n a m o n g h u n d r e d s of the
world's brightest theoretical physicists to solve the theory has b e c o m e
quite fierce. In fact, the cover of Discover recently featured string theorist
D. V. N a n o p o u l o u s of Texas, w h o openly boasted that he was h o t on t h e
trail of winning the Nobel Prize in physics. Rarely has such an abstract
theory aroused such passions.)
Why Strings?
I o n c e h a d l u n c h with a Nobel Prize w i n n e r in physics at a Chinese
restaurant in New York. While we were passing the sweet a n d sour pork,
the subject of superstring theory c a m e u p . W i t h o u t warning, he
l a u n c h e d into a long personal discussion of why superstring theory was
n o t the correct p a t h for y o u n g theoretical physicists. It was a wild-goose
chase, he claimed. T h e r e h a d never b e e n anything like it in the history
of physics, so he found it too bizarre for his tastes. It was too alien, t o o
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o r t h o g o n a l to all t h e previous t r e n d s in science. After a long discussion,
it boiled down to o n e question: Why strings? Why n o t vibrating solids or
blobs?
T h e physical world, he r e m i n d e d m e , uses the same concepts over
a n d over again. N a t u r e is like a work by Bach or Beethoven, often starting with a central t h e m e a n d m a k i n g countless variations on it that are
scattered t h r o u g h o u t the symphony. By this criterion, it appears that
strings a r e n o t f u n d a m e n t a l concepts in n a t u r e .
T h e c o n c e p t of orbits, for e x a m p l e , occurs repeatedly in n a t u r e in
different variations; since t h e work of Copernicus, orbits have provided
an essential t h e m e that is constantly r e p e a t e d t h r o u g h o u t n a t u r e in different variations, from the largest galaxy to t h e atom, to the smallest
subatomic particle. Similarly, Faraday's fields have proved to be o n e of
n a t u r e ' s favorite t h e m e s . Fields can describe the galaxy's magnetism a n d
gravitation, or they can describe the electromagnetic theory of Maxwell,
the metric theory of R i e m a n n a n d Einstein, a n d the Yang-Mills fields
f o u n d in the S t a n d a r d Model. Field theory, in fact, has e m e r g e d as the
universal language of subatomic physics, a n d p e r h a p s the universe as
well. It is the single most powerful w e a p o n in the arsenal of theoretical
physics. All known forms of m a t t e r a n d energy have b e e n expressed in
terms of field theory. Patterns, t h e n , like t h e m e s a n d variations in a
symphony, a r e constantly r e p e a t e d .
But strings? Strings do n o t seem to be a p a t t e r n favored by n a t u r e in
d e s i g n i n g the heavens. We do n o t see strings in o u t e r space. In fact, my
colleague e x p l a i n e d to m e , we do n o t see strings anywhere.
A m o m e n t ' s t h o u g h t , however, will reveal that n a t u r e has reserved
the string for a special role, as a basic building block for o t h e r forms.
For e x a m p l e , t h e essential feature of life on earth is the stringlike DNA
molecule, which contains the c o m p l e x information a n d c o d i n g of life
itself. W h e n building t h e stuff of life, as well as subatomic matter, strings
seem to be t h e perfect answer. In b o t h cases, we want to pack a large
a m o u n t of information into a relatively simple, r e p r o d u c i b l e structure.
T h e distinguishing feature of a string is that it is o n e of the most c o m p a c t
ways of storing vast a m o u n t s of d a t a in a way in which information can
be replicated.
For living things, n a t u r e uses t h e d o u b l e strands of t h e DNA molecule, which u n w i n d a n d form duplicate copies of each other. Also,
o u r bodies c o n t a i n billions u p o n billions of p r o t e i n strings,
formed of a m i n o acid building blocks. O u r bodies, in some sense, can
be viewed as a vast collection of strings—protein molecules d r a p e d
around our bones.
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The String Quartet
Currently, t h e most successful version of string theory is t h e o n e created
by Princeton physicists David Gross, Emil Martinec, Jeffrey Harvey, a n d
Ryan R o h m , who are s o m e t i m e s called the P r i n c e t o n string quartet. T h e
most senior of t h e m is David Gross. At most seminars in P r i n c e t o n , Witten may ask questions in his soft voice, b u t Gross's voice is unmistakable:
loud, b o o m i n g , a n d d e m a n d i n g . Anyone w h o gives a s e m i n a r at Princeton lives in fear of the s h a r p , rapid-fire questions that Gross will s h o o t
at t h e m . What is r e m a r k a b l e is that his questions are usually on the mark.
Gross a n d his collaborators p r o p o s e d what is called t h e heterotic string.
Today, it is precisely t h e heterotic string, of all the various Kaluza-Kleintype theories that have b e e n p r o p o s e d in the past, that has t h e greatest
potential of unifying all the laws of n a t u r e into o n e theory.
Gross believes that string theory solves the p r o b l e m of t u r n i n g wood
into marble: " T o build m a t t e r itself from g e o m e t r y — t h a t in a sense is
what string theory does. It can be t h o u g h t of that way, especially in a
theory like the heterotic string which is inherently a theory of gravity in
which the particles of m a t t e r as well as the o t h e r forces of n a t u r e e m e r g e
in the same way that gravity e m e r g e s from g e o m e t r y . "
T h e most r e m a r k a b l e feature of string theory, as we have e m p h a sized, is that Einstein's theory of gravity is automatically c o n t a i n e d in it.
In fact, the graviton (the q u a n t u m of gravity) e m e r g e s as t h e smallest
vibration of the closed string. While GUTs strenuously avoided any m e n tion of Einstein's theory of gravity, the superstring theories d e m a n d that
Einstein's theory be included. For e x a m p l e , if we simply d r o p Einstein's
theory of gravity as o n e vibration of the string, t h e n t h e theory b e c o m e s
inconsistent a n d useless. This, in fact, is t h e reason why Witten was
attracted to string theory in t h e first place. In 1982, he read a review
article by J o h n Schwarz a n d was s t u n n e d to realize that gravity e m e r g e s
from superstring theory from self-consistency r e q u i r e m e n t s a l o n e . He
recalls that it was " t h e greatest intellectual thrill of my life." Witten says,
"String theory is extremely attractive because gravity is forced u p o n us.
All known consistent string theories i n c l u d e gravity, so while gravity is
impossible in q u a n t u m field theory as we have known it, it's obligatory
in string t h e o r y . "
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6
Gross takes satisfaction in believing that Einstein, if he were alive,
would love superstring theory. He would love the fact that t h e beauty
a n d simplicity of superstring theory ultimately c o m e from a g e o m e t r i c
principle, whose precise n a t u r e is still u n k n o w n . Gross claims, "Einstein
would have b e e n pleased with this, at least with the goal, if n o t t h e real-
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i z a t i o n . . . . He would have liked t h e fact that t h e r e is an underlying
geometrical principle—which, unfortunately, we d o n ' t really u n d e r stand."
Witten even goes so far as to say that "all the really great ideas in
physics" are "spinoffs" of superstring theory. By this, he m e a n s that all
t h e great advances in theoretical physics are i n c l u d e d within superstring
theory. He even claims that Einstein's general relativity theory being
discovered before superstring theory was "a m e r e accident of the develo p m e n t o n p l a n e t E a r t h . " H e claims that, somewhere i n o u t e r space,
" o t h e r civilizations in the u n i v e r s e " m i g h t have discovered superstring
theory first, a n d derived general relativity as a by-product.
7
8
Compactification and Beauty
String theory is such a p r o m i s i n g c a n d i d a t e for physics because it gives
a simple origin of t h e symmetries f o u n d in particle physics as well as
g e n e r a l relativity.
We saw in C h a p t e r 6 that supergravity was b o t h n o n r e n o r m a l i z a b l e
a n d too small to a c c o m m o d a t e the symmetry of the Standard Model.
H e n c e , it was n o t self-consistent a n d did n o t begin to realistically
describe the known particles. However, string theory does b o t h . As we
shall soon see, it banishes t h e infinities f o u n d in q u a n t u m gravity, yielding a finite theory of q u a n t u m gravity. T h a t a l o n e would g u a r a n t e e that
string theory should be taken as a serious c a n d i d a t e for a theory of the
universe. However, t h e r e is an a d d e d b o n u s . W h e n we compactify some
of the d i m e n s i o n s of t h e string, we find that t h e r e is " e n o u g h r o o m " to
a c c o m m o d a t e t h e symmetries of t h e S t a n d a r d Model a n d even the
GUTs.
T h e heterotic string consists of a closed string that has two types of
vibrations, clockwise a n d counterclockwise, which are treated differently. T h e clockwise vibrations live in a ten-dimensional space. T h e counterclockwise live in a 26-dimensional space, of which 16 dimensions have
b e e n compactified. (We recall that in Kaluza's original five-dimensional
theory, t h e fifth d i m e n s i o n was compactified by b e i n g w r a p p e d up into
a circle.) T h e h e t e r o t i c string owes its n a m e to t h e fact that the clockwise
a n d t h e counterclockwise vibrations live in two different dimensions b u t
are c o m b i n e d to p r o d u c e a single superstring theory. T h a t is why it is
n a m e d after the Greek word for heterosis, which m e a n s "hybrid vigor."
T h e 16-dimensional compactified space is by far t h e most interesting.
In Kaluza-Klein theory, we recall that the compactified N-dimensional
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Superstrings
space can have a symmetry associated with it, m u c h like a b e a c h ball.
T h e n all the vibrations (or fields) defined on the N-dimensional space
automatically i n h e r i t these symmetries. If the symmetry is S U ( N ) , t h e n
all the vibrations on t h e space must obey SU(N) symmetry (in the same
way that clay inherits t h e symmetries of the m o l d ) . In this way, K a l u z a Klein theory could a c c o m m o d a t e the symmetries of the S t a n d a r d Model.
However, in this way it could also be d e t e r m i n e d that the supergravity
was " t o o small" to contain all the particles of t h e symmetries f o u n d in
the Standard Model. This was sufficient to kill t h e supergravity theory
as a realistic theory of m a t t e r a n d s p a c e - t i m e .
But when the P r i n c e t o n string q u a r t e t analyzed t h e symmetries of
the 16-dimensional space, they f o u n d that it is a monstrously large symmetry, called E(8) X E ( 8 ) , which is m u c h larger t h a n any G U T symmetry
that has ever b e e n tried. This was an u n e x p e c t e d b o n u s . It m e a n t that
that all the vibrations of the string would inherit t h e symmetry of t h e
16-dimensional space, which was m o r e t h a n e n o u g h to a c c o m m o d a t e
the symmetry of the S t a n d a r d Model.
This, t h e n , is the mathematical expression of the central t h e m e of
the book, that the laws of physics simplify in h i g h e r d i m e n s i o n s . In this
case, the 26-dimensional space of t h e counterclockwise vibrations of t h e
heterotic string has r o o m e n o u g h to explain all the symmetries f o u n d
in b o t h Einstein's theory a n d q u a n t u m theory. So, for the first t i m e , .
p u r e geometry has given a simple e x p l a n a t i o n of why t h e subatomic
world must necessarily exhibit certain symmetries that e m e r g e from the
curling up of higher-dimensional space: The symmetries of the subatomic
realm are but remnants of the symmetry of higher-dimensional space.
This m e a n s that the beauty a n d symmetry f o u n d in n a t u r e can ultimately be traced back to higher-dimensional space. For e x a m p l e , snowflakes create beautiful, hexagonal patterns, n o n e of which are precisely
the same. T h e s e snowflakes a n d crystals, in t u r n , have i n h e r i t e d their
structure from the way in which their molecules have b e e n geometrically
a r r a n g e d . This a r r a n g e m e n t is mainly d e t e r m i n e d by the electron shells
of the molecule, which in turn take us back to the rotational symmetries
of the q u a n t u m theory, given by 0 ( 3 ) . All the symmetries of the lowenergy universe that we observe in chemical e l e m e n t s are d u e to t h e
symmetries cataloged by t h e S t a n d a r d Model, which in t u r n can be
derived by compactifying t h e heterotic string.
In conclusion, the symmetries that we see a r o u n d us, from rainbows
to blossoming flowers to crystals, may ultimately be viewed as manifestations of fragments of the original ten-dimensional t h e o r y . R i e m a n n
a n d Einstein h a d h o p e d to find a g e o m e t r i c u n d e r s t a n d i n g of why forces
9
10
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can d e t e r m i n e the m o t i o n a n d t h e n a t u r e of matter. But they were missing a key i n g r e d i e n t in showing the relationship between wood a n d marble. This missing link is most likely superstring theory. With the tend i m e n s i o n a l string theory, we see that t h e geometry of the string may
ultimately be responsible for b o t h the forces a n d the structure of matter.
A Piece of Twenty-First-Century Physics
Given the e n o r m o u s power of its symmetries, it is n o t surprising that
superstring theory is radically different from any o t h e r kind of physics.
It was, in fact, discovered quite by accident. Many physicists have comm e n t e d that if this fortuitous accident h a d never o c c u r r e d , t h e n the
theory would n o t have b e e n discovered until the twenty-first century.
This is because it is such a s h a r p d e p a r t u r e from all the ideas tried in
this century. It is n o t a c o n t i n u o u s extension of trends a n d theories
p o p u l a r in this century; it stands apart.
By contrast, the theory of general relativity h a d a " n o r m a l " a n d logical evolution. First, Einstein postulated the equivalence principle. T h e n
he reformulated this physical principle in the mathematics of a field
theory of gravitation based on Faraday's fields a n d R i e m a n n ' s metric
tensor. Later came the "classical solutions," such as the black hole a n d
the Big Bang. Finally, the last stage is the c u r r e n t a t t e m p t to formulate
a q u a n t u m theory of gravity. T h u s general relativity went t h r o u g h a logical progression, from a physical principle to a q u a n t u m theory:
Geometry —» field theory —» classical theory —» q u a n t u m theory
By contrast, superstring theory has b e e n evolving backward since its
accidental discovery in 1968. T h a t ' s why superstring theory looks so
strange a n d unfamiliar to most physicists. We are still searching for its
underlying physical principle, the c o u n t e r p a r t to Einstein's equivalence
principle.
T h e theory was b o m quite by accident in 1968 when two y o u n g theoretical physicists, Gabriel V e n e z i a n o a n d Mahiko Suzuki, were indep e n d e n t l y leafing t h r o u g h m a t h books, looking for mathematical functions that would describe the interactions of strongly interacting
particles. While studying at CERN, the E u r o p e a n c e n t e r for theoretical
physics in Geneva, Switzerland, they i n d e p e n d e n t l y s t u m b l e d on the
Euler beta function, a mathematical function written down in the ninet e e n t h century by the m a t h e m a t i c i a n L e o n h a r d Euler. They were aston-
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ished to find that t h e Euler beta function fit almost all t h e p r o p e r t i e s
r e q u i r e d to describe t h e strong interactions of e l e m e n t a r y particles.
Over l u n c h at t h e Lawrence Berkeley Laboratory in California, with
a spectacular view of t h e sun blazing down over San Francisco h a r b o r ,
Suzuki o n c e e x p l a i n e d to me the thrill of discovering, quite by accident,
a potentially i m p o r t a n t result. Physics was n o t supposed to h a p p e n that
way.
After finding t h e Euler beta function in a m a t h book, he excitedly
showed his result to a senior physicist at CERN. T h e senior physicist,
after listening to Suzuki, was n o t impressed. In fact, he told Suzuki that
a n o t h e r y o u n g physicist (Veneziano) h a d discovered the identical function a few weeks earlier. He discouraged Suzuki from publishing his
result. Today, this beta function goes by the n a m e of the V e n e z i a n o
model, which has inspired several t h o u s a n d research papers, spawned a
major school of physics, a n d now makes the claim of unifying all physical
laws. (In retrospect, Suzuki, of course, should have p u b l i s h e d his result.
T h e r e is a lesson to all this, I suspect: Never take too seriously t h e advice
of your superiors.)
In 1970, the mystery s u r r o u n d i n g t h e V e n e z i a n o - S u z u k i m o d e l was
partly explained when Yoichiro N a m b u at the University of Chicago a n d
Tetsuo Goto at N i h o n University discovered that a vibrating string lies
b e h i n d its w o n d r o u s properties.
Because string theory was discovered backward a n d by accident, physicists still do n o t know the physical principle that u n d e r l i e s string theory.
T h e last step in the evolution of the theory ( a n d the first step in the
evolution of general relativity) is still missing.
Witten adds that
h u m a n b e i n g s o n p l a n e t Earth n e v e r h a d t h e c o n c e p t u a l framework that
w o u l d lead t h e m to invent string theory on p u r p o s e . . . . No o n e i n v e n t e d
i t o n p u r p o s e , i t was i n v e n t e d i n a l u c k y a c c i d e n t . B y r i g h t s , t w e n t i e t h c e n t u r y physicists s h o u l d n ' t have h a d t h e privilege o f s t u d y i n g this theory.
By rights, string theory s h o u l d n ' t have b e e n i n v e n t e d until o u r k n o w l e d g e
of s o m e of the ideas that are prerequisite for string theory h a d d e v e l o p e d
t o t h e p o i n t t h a t i t was p o s s i b l e f o r u s t o h a v e t h e r i g h t c o n c e p t o f w h a t i t
was all a b o u t .
11
Loops
T h e formula discovered by V e n e z i a n o a n d Suzuki, which they h o p e d
would describe the p r o p e r t i e s of interacting subatomic particles, was still
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i n c o m p l e t e . It violated o n e of t h e p r o p e r t i e s of physics: unitarity, or the
conservation of probability. By itself, t h e Veneziano-Suzuki formula
would give incorrect answers for particle interactions. So the next step
in the theory's evolution was to a d d small q u a n t u m correction terms
that would restore this property. In 1969, even before the string interp r e t a t i o n of N a m b u a n d Goto, t h r e e physicists (Keiji Kikkawa, Bunji
Sakita, a n d Miguel A. Virasoro, t h e n all at the University of Wisconsin)
p r o p o s e d t h e correct solution: a d d i n g increasingly smaller terms to the
Veneziano-Suzuki formula in o r d e r to restore unitarity.
A l t h o u g h these physicists h a d to guess at how to construct the series
from scratch, today it is most easily u n d e r s t o o d in the framework of the
string picture of N a m b u . For e x a m p l e , w h e n a b u m b l e b e e flies in space,
its p a t h can be described as a wiggly line. W h e n a piece of string drifting
in the air moves in space, its p a t h can be likened to an imaginary twodimensional sheet. W h e n a closed string floats in space, its p a t h resembles a t u b e .
Strings interact by b r e a k i n g into smaller strings a n d by j o i n i n g with
o t h e r strings. W h e n these interacting strings move, they trace o u t the
configurations shown in Figure 7.1. Notice that two tubes c o m e in from
the left, with o n e t u b e fissioning in half, e x c h a n g e the m i d d l e tube, a n d
t h e n veer off to t h e right. This is how tubes interact with each other.
This diagram, of course, is s h o r t h a n d for a very complicated m a t h e m a t ical expression. W h e n we calculate the n u m e r i c a l expression corres p o n d i n g to these diagrams, we get back t h e Euler beta function.
In the string picture, the essential trick p r o p o s e d by Kikkawa-SakitaVirasoro (KSV) a m o u n t e d to a d d i n g all possible diagrams where strings
can collide a n d b r e a k apart. T h e r e are, of course, an infinite n u m b e r of
these diagrams. T h e process of a d d i n g an infinite n u m b e r of " l o o p "
diagrams, with each d i a g r a m c o m i n g closer to t h e final answer, is perturbation theory a n d is o n e of most i m p o r t a n t weapons in the arsenal
of any q u a n t u m physicist. (These string diagrams possess a beautiful
symmetry that has never b e e n seen in physics before, which is known as
conformal symmetry in two dimensions. This conformal symmetry allows
us to treat these tubes a n d sheets as t h o u g h they were m a d e of rubber:
We can pull, stretch, b e n d , a n d shrink these diagrams. T h e n , because
of conformal symmetry, we can prove that all these m a t h e m a t i c a l expressions r e m a i n the same.)
KSV claimed that t h e s u m total of all these l o o p diagrams would yield
the precise m a t h e m a t i c a l formula explaining h o w subatomic particles
interact. However, t h e KSV p r o g r a m consisted of a series of u n p r o v e n
conjectures. S o m e o n e h a d to construct these loops explicitly, or else
these conjectures were useless.
+
. . .
Figure 7.1. In string theory, the gravitational force is represented by the exchange
of closed strings, which sweep out tubes in space-time. Even if we add up an
infinite series of diagrams with a large number of holes, infinities never appear
in the theory, giving us a finite theory of quantum gravity.
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I n t r i g u e d by the p r o g r a m b e i n g initiated by KSV, I decided to try my
luck at solving the p r o b l e m . This was a bit difficult, because I was dodging m a c h i n e - g u n bullets at t h e time.
Boot Camp
I r e m e m b e r clearly when the KSV p a p e r came o u t in 1969. KSV was
p r o p o s i n g a p r o g r a m for future work, r a t h e r t h a n giving precise details.
I d e c i d e d t h e n to calculate all possible loops explicitly a n d complete the
KSV p r o g r a m .
It's h a r d to forget those times. T h e r e was a war raging overseas, a n d
the university campuses from Kent State to t h e University of Paris, were
in a state of turmoil. I h a d g r a d u a t e d from Harvard the year before,
w h e n President L y n d o n J o h n s o n revoked deferments for g r a d u a t e students, s e n d i n g p a n i c t h r o u g h o u t g r a d u a t e schools in t h e country. Chaos
g r i p p e d the campuses. Suddenly, my friends were d r o p p i n g o u t of college, t e a c h i n g high school, packing their bags a n d h e a d i n g to Canada,
or trying to r u i n their health in o r d e r to flunk the army physical.
Promising careers were b e i n g shattered. O n e of my g o o d friends in
physics from MIT vowed that he would go to jail r a t h e r t h a n fight in
Vietnam. He told us to send copies of t h e Physical Review to his jail cell
so he could k e e p up with d e v e l o p m e n t s in the Veneziano m o d e l . O t h e r
friends, w h o quit college to teach in high schools r a t h e r t h a n fight in
the war, t e r m i n a t e d p r o m i s i n g scientific careers. (Many of t h e m still
teach in these high schools.)
T h r e e days after g r a d u a t i o n , I left C a m b r i d g e a n d found myself in
the U n i t e d States Army stationed at Fort B e n n i n g , Georgia (the largest
infantry training c e n t e r in the w o r l d ) , a n d later at Fort Lewis, Washington. T e n s of t h o u s a n d s of raw recruits with no previous military training
were b e i n g h a m m e r e d into a fighting force a n d t h e n shipped to Vietn a m , r e p l a c i n g t h e 500 GIs w h o were dying every week.
O n e day, while throwing live g r e n a d e s u n d e r the grueling Georgia
sun a n d seeing t h e deadly s h r a p n e l scatter in all directions, my thoughts
b e g a n to wander. How many scientists t h r o u g h o u t history h a d to face
t h e p u n i s h i n g ravages of war? How many p r o m i s i n g scientists were
snuffed o u t by a bullet in the p r i m e of their youth?
I r e m e m b e r e d that Karl Schwarzschild h a d died in the kaiser's army
on t h e Russian front d u r i n g World War I j u s t a few m o n t h s after he
f o u n d the basic solution to Einstein's e q u a t i o n s used in every black hole
calculation. ( T h e Schwarzschild radius of a black h o l e is n a m e d in his
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h o n o r . Einstein a d d r e s s e d t h e Prussian Academy in 1916 to c o m m e m orate Schwarzschild's work after his untimely d e a t h at t h e front lines.)
And how many p r o m i s i n g p e o p l e were cut down even before they could
begin their careers?
Infantry training, I discovered, is rigorous; it is designed to t o u g h e n
the spirit a n d dull the intellect. I n d e p e n d e n c e of t h o u g h t is g r o u n d o u t
of you. After all, t h e military d o e s n o t necessarily want s o m e wit w h o will
question the sergeant's o r d e r s in the middle of a firefight. U n d e r s t a n d ing this, I d e c i d e d to b r i n g a l o n g some physics p a p e r s . I n e e d e d something to keep my m i n d active while peeling potatoes in KP or firing
m a c h i n e guns, so I b r o u g h t a l o n g a copy of the KSV p a p e r .
D u r i n g night infantry training, I h a d to go past an obstacle course,
which m e a n t d o d g i n g live m a c h i n e - g u n bullets, froglegging u n d e r
b a r b e d wire, a n d crawling t h r o u g h thick brown m u d . Because t h e automatic fire h a d tracers on t h e m , I could see the beautiful crimson streaks
m a d e by t h o u s a n d s of m a c h i n e - g u n bullets sailing a few feet over my
head. However, my t h o u g h t s k e p t drifting back to the KSV p a p e r a n d
how their p r o g r a m could be carried out.
Fortunately, the essential feature of the calculation was strictly topological. It was clear to me that these loops were i n t r o d u c i n g an entirely
new language to physics, t h e l a n g u a g e of topology. Never before in the
history of physics h a d Mobius strips or Klein bottles b e e n used in a fund a m e n t a l way.
Because I rarely h a d any p a p e r or pencils while practicing with
m a c h i n e guns, I forced myself to visualize in my h e a d how strings could
be twisted into loops a n d t u r n e d inside out. Machine-gun training was
actually a blessing in disguise because it forced me to m a n i p u l a t e large
blocks of equations in my h e a d . By t h e time I finished t h e advanced
machine-gun-training p r o g r a m , I was convinced that I could c o m p l e t e
the p r o g r a m of calculating all loops.
Finally, I m a n a g e d to squeeze time from the army to go to t h e University of California at Berkeley, w h e r e I furiously worked o u t the details
that were racing in my h e a d . I sank several h u n d r e d h o u r s of intense
t h o u g h t into the question. This, in fact, b e c a m e my Ph.D. dissertation.
By 1970, the final calculation took up several h u n d r e d densely filled
n o t e b o o k pages. U n d e r the careful supervision of my adviser, Stanley
Mandelstam, my colleague Loh-ping Yu a n d I successfully calculated an
explicit expression for all possible l o o p diagrams known at t h a t time.
However, I wasn't satisfied with this work. T h e KSV p r o g r a m consisted
of a h o d g e - p o d g e of rules of t h u m b a n d intuition, n o t a rigorous set of
basic principles from which these loops could be derived. String theory,
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we saw, was evolving backward, since its accidental discovery by Veneziano a n d Suzuki. T h e n e x t step in the backward evolution of the string
was to follow in the footsteps of Faraday, R i e m a n n , Maxwell, a n d Einstein
a n d construct a field theory of strings.
Field Theory of Strings
Ever since t h e p i o n e e r i n g work of Faraday, every physical theory h a d
b e e n written in terms of fields. Maxwell's theory of light was based on
field theory. So was Einstein's. In fact, all of particle physics was based
on field theory. T h e only theory not based on field theory was string
theory. T h e KSV p r o g r a m was m o r e a set of convenient rules t h a n a field
theory.
My n e x t goal was to rectify that situation. T h e p r o b l e m with a field
theory of strings, however, was that many of the p i o n e e r i n g figures in
physics a r g u e d against it. T h e i r a r g u m e n t s were simple. These giants of
physics, such as Hideki Yukawa a n d W e r n e r H e i s e n b e r g , h a d labored
for years to create a field theory t h a t was n o t based on p o i n t particles.
Elementary particles, they t h o u g h t , m i g h t be pulsating blobs of matter,
r a t h e r t h a n points. However, no m a t t e r how h a r d they tried, field theories based on blobs always violated causality.
If we were to shake the b l o b at o n e point, the interactions would
spread faster t h a n the speed of light t h r o u g h o u t the blob, violating special relativity a n d creating all sorts of time paradoxes. T h u s " n o n l o c a l
field t h e o r i e s " based on blobs were known to be a monstrously difficult
p r o b l e m . Many physicists, in fact, insisted that only local field theories
based on p o i n t particles could be consistent. Nonlocal field theories
m u s t violate relativity.
T h e s e c o n d a r g u m e n t was even m o r e convincing. T h e Veneziano
m o d e l h a d many magical p r o p e r t i e s (including s o m e t h i n g called duality) that h a d never b e e n seen before in field theory. Years earlier, Richa r d Feynman h a d given " r u l e s " that any field theory should obey. However, these Feynman rules were in direct violation of duality. T h u s many
string theorists were convinced that a field theory of strings was impossible because string theory necessarily violated the properties of the
Veneziano m o d e l . String theory, they said, was u n i q u e in all of physics
because it could n o t be recast as a field theory.
I collaborated with Keiji Kikkawa on this difficult b u t i m p o r t a n t p r o b lem. Step by step we built o u r field theory, in m u c h the same way that
o u r predecessors h a d constructed field theories for o t h e r forces. Follow-
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ing Faraday, we i n t r o d u c e d a field at every p o i n t in s p a c e - t i m e . However,
for a field theory of strings, we h a d to generalize t h e c o n c e p t of Faraday
a n d postulate a field t h a t was defined for all possible configurations of
a string vibrating in s p a c e - t i m e .
T h e second step was to postulate the field e q u a t i o n s that t h e string
obeyed. T h e field e q u a t i o n for a single string moving a l o n e in s p a c e time was easy. As e x p e c t e d , o u r field equations r e p r o d u c e d an infinite
series of string resonances, e a c h c o r r e s p o n d i n g to a subatomic particle.
Next, we f o u n d that t h e objections of Yukawa a n d H e i s e n b e r g were
solved by string field theory. If we jiggled t h e string, t h e vibrations traveled down the string at less t h a n t h e speed of light.
Soon, however, we hit a brick wall. W h e n we tried to i n t r o d u c e interacting strings, we could n o t r e p r o d u c e t h e V e n e z i a n o a m p l i t u d e correctly. Duality a n d t h e c o u n t i n g of g r a p h s given by Feynman for any
field theory were in direct conflict. J u s t as the critics e x p e c t e d , t h e Feynm a n g r a p h s were incorrect. This was d i s h e a r t e n i n g . It a p p e a r e d that
field theory, which h a d f o r m e d t h e f o u n d a t i o n of physics for t h e past
century, was fundamentally i n c o m p a t i b l e with string theory.
Discouraged, I r e m e m b e r mulling over the p r o b l e m late into t h e
night. For h o u r s , I began systematically to check all t h e possible alternatives to this p r o b l e m . But t h e conclusion that duality h a d to be b r o k e n
s e e m e d inescapable. T h e n I r e m e m b e r e d what Sherlock H o l m e s , in
A r t h u r C o n a n Doyle's " T h e Sign of F o u r , " said to Watson: " H o w often
have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the t r u t h . " E n c o u r a g e d by this
idea, I eliminated all the impossible alternatives. T h e only i m p r o b a b l e
alternative r e m a i n i n g was to violate the p r o p e r t i e s of the V e n e z i a n o Suzuki formula. At a b o u t 3:00 A.M., the resolution finally hit m e . I realized that physicists h a d overlooked t h e obvious fact that o n e can split
the V e n e z i a n o - S u z u k i formula into two pieces. Each p a r t t h e n corresponds to o n e of F e y n m a n ' s diagrams, a n d each p a r t violates duality,
b u t the sum obeys all t h e correct p r o p e r t i e s of a field theory.
I quickly took o u t some p a p e r a n d went over the calculation. I s p e n t
the n e x t 5 h o u r s c h e c k i n g a n d r e c h e c k i n g the calculation from all possible directions. T h e conclusion was inescapable: Field theory does violate duality, as everyone e x p e c t e d , b u t this is acceptable because the final
sum r e p r o d u c e s the V e n e z i a n o - S u z u k i formula.
I h a d now solved most of the p r o b l e m . However, o n e m o r e F e y n m a n
diagram, r e p r e s e n t i n g t h e collision of four strings, was still lacking. T h a t
year, I was teaching introductory electricity a n d m a g n e t i s m to u n d e r graduates at the City University of New York, a n d we were studying Far-
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aday's lines of force. I would ask the students to draw the lines of force
e m a n a t i n g from different configurations of charges, r e p e a t i n g the same
steps p i o n e e r e d by Faraday in t h e n i n e t e e n t h century. Suddenly, it
d a w n e d on me that the squiggly lines that I was asking my students to
draw h a d exactly the same topological structure as t h e collision of
strings. T h u s by r e a r r a n g i n g charges in a freshman laboratory, I h a d
f o u n d the correct configuration describing the collision of four strings.
Was it that simple?
I r u s h e d h o m e to check my h u n c h , a n d I was right. By employing
pictorial t e c h n i q u e s that even a freshman can use, I could show that the
four-string interaction must be h i d d e n within the Veneziano formula.
By the winter of 1974, using m e t h o d s dating back to Faraday, Kikkawa
a n d I c o m p l e t e d the field theory of strings, the first successful a t t e m p t
to c o m b i n e string theory with the formalism of field theory.
O u r field theory, a l t h o u g h it correctly e m b o d i e d the entire inform a t i o n c o n t a i n e d within string theory, still n e e d e d improvement.
Because we were constructing the field theory backward, many of the
symmetries were still obscure. For e x a m p l e , t h e symmetries of special
relativity were p r e s e n t b u t n o t in an obvious way. Much m o r e work was
n e e d e d to streamline the field equations we h a d found. But j u s t as we
were b e g i n n i n g to explore t h e p r o p e r t i e s of o u r field theory, the m o d e l
u n e x p e c t e d l y suffered a severe setback.
T h a t year, physicist Claude Lovelace of Rutgers University discovered
that the bosonic string (describing integral spins) is self-consistent only
in 26 dimensions. O t h e r physicists verified this result a n d showed that
the superstring (describing b o t h integral a n d half-integral spin) is selfconsistent only in ten dimensions. It was soon realized that, in dimensions o t h e r t h a n ten or 26 d i m e n s i o n s , t h e theory completely loses all
its beautiful m a t h e m a t i c a l properties. But no o n e believed that a theory
defined in ten or 26 d i m e n s i o n s h a d anything to do with reality.
Research in string theory abruptly g r o u n d to a halt. Like Kaluza-Klein
theory before it, string theory lapsed into a d e e p h i b e r n a t i o n . For 10
l o n g years, the m o d e l was b a n i s h e d to obscurity. (Although most string
physicists, myself i n c l u d e d , a b a n d o n e d t h e m o d e l like a sinking ship, a
few die-hards, like physicists J o h n Schwarz a n d the late J o e l Scherk, tried
to k e e p t h e m o d e l alive by steadily m a k i n g i m p r o v e m e n t s . For e x a m p l e ,
string theory was originally t h o u g h t to be j u s t a theory of the strong
interactions, with e a c h m o d e of vibration c o r r e s p o n d i n g to a r e s o n a n c e
of t h e q u a r k m o d e l . Schwarz a n d Scherk correctly showed that the string
m o d e l was really a unified theory of all forces, n o t j u s t the strong interactions.)
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Research in q u a n t u m gravity went into o t h e r direction. F r o m 1974
to 1984, w h e n string theory was in eclipse, a large n u m b e r of alternative
theories of q u a n t u m gravity were successively studied. D u r i n g this
period, the original Kaluza-Klein theory a n d t h e n the supergravity theory enjoyed great popularity, b u t each time t h e failures of these m o d e l s
also b e c a m e a p p a r e n t . For e x a m p l e , b o t h Kaluza-Klein a n d supergravity
theories were shown to be n o n r e n o r m a l i z a b l e .
T h e n s o m e t h i n g strange h a p p e n e d d u r i n g that d e c a d e . O n the o n e
h a n d , physicists b e c a m e frustrated by t h e growing list of m o d e l s that
were tried a n d t h e n discarded d u r i n g this p e r i o d . Everything failed. T h e
realization came slowly that Kaluza-Klein theory a n d supergravity theory
were probably on the right track, b u t they w e r e n ' t sophisticated e n o u g h
to solve t h e p r o b l e m of nonrenormalizability. But the only theory complex e n o u g h to contain b o t h Kaluza-Klein theory a n d t h e supergravity
theory was superstring theory. On t h e o t h e r h a n d , physicists slowly
b e c a m e accustomed to working in hyperspace. Because of t h e K a l u z a Klein renaissance, t h e idea of hyperspace d i d n ' t seem that farfetched or
forbidding a n y m o r e . Over time, even a theory defined in 26 d i m e n s i o n s
d i d n ' t seem that outlandish. T h e original resistance to 26 d i m e n s i o n s
began to slowly melt away with time.
Finally, in 1984, G r e e n a n d Schwarz proved that superstring theory
was the only self-consistent theory of q u a n t u m gravity, a n d the s t a m p e d e
began. In 1985, Edward Witten m a d e a significant advance in t h e field
theory of strings, which m a n y p e o p l e think is o n e of the most beautiful
achievements of t h e theory. He showed that o u r old field theory could
be derived using powerful m a t h e m a t i c a l a n d g e o m e t r i c t h e o r e m s (coming from s o m e t h i n g called cohomology theory) with a fully relativistic form.
With Witten's new field theory, t h e true m a t h e m a t i c a l elegance of
string field theory, which was c o n c e a l e d in o u r formalism, was revealed.
Soon, almost a h u n d r e d scientific p a p e r s were written to explore t h e
fascinating mathematical p r o p e r t i e s of Witten's field t h e o r y .
12
No One Is Smart Enough
Assuming that string field theory is correct, in principle we should be
able to calculate the mass of the p r o t o n from first principles a n d m a k e
contact with known data, such as the masses of t h e various particles. If
the numerical answers are wrong, t h e n we will have to throw the theory
out the window. However, if the theory is correct, it will r a n k a m o n g t h e
most significant advances in physics in 2,000 years.
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After t h e intense, e u p h o r i c fanfare of the late 1980s (when it
a p p e a r e d that t h e theory would be completely solved within a few years
a n d t h e Nobel Prizes h a n d e d o u t by the d o z e n ) , a certain d e g r e e of cold
realism has set in. A l t h o u g h t h e theory is well defined mathematically,
no o n e has b e e n able to solve t h e theory. No o n e .
T h e p r o b l e m is that no one is smart enough to solve the field theory of
strings or any o t h e r n o n p e r t u r b a t i v e a p p r o a c h to string theory. This is a
well-defined p r o b l e m , b u t the irony is that solving field theory requires
t e c h n i q u e s that are currently b e y o n d the skill of any physicist. This is
frustrating. Sitting before us is a perfectly well-defined theory of strings.
Within it is the possibility of settling all t h e controversy s u r r o u n d i n g
higher-dimensional space. T h e d r e a m of calculating everything from
first principles is staring us in the face. T h e p r o b l e m is how to solve it.
O n e is r e m i n d e d of Julius Caesar's famous r e m a r k in Shakespeare's play:
" T h e fault, d e a r Brutus, is n o t in o u r stars, b u t in ourselves." For a string
theorist, t h e fault is n o t in the theory, b u t in o u r primitive mathematics.
T h e reason for this pessimism is that o u r m a i n calculational tool,
p e r t u r b a t i o n theory, fails. P e r t u r b a t i o n theory begins with a Venezianolike formula a n d t h e n calculates q u a n t u m corrections to it (which have
the s h a p e of loops). It was the h o p e of string theorists that they could
write down a m o r e advanced Veneziano-like formula defined in four
d i m e n s i o n s that would uniquely describe the known s p e c t r u m of particles. In retrospect, they were too successful. T h e p r o b l e m is that millions
u p o n millions of Veneziano-like formulas have now b e e n discovered.
Embarrassingly, string theorists are literally d r o w n i n g in these perturbative solutions.
T h e f u n d a m e n t a l p r o b l e m that has stalled progress in superstring
theory in t h e past few years is that no o n e knows how to select the correct
solution o u t of t h e millions that have b e e n discovered. Some of these
solutions c o m e remarkably close to describing the real world. With a few
m o d e s t assumptions, it is easy to extract the S t a n d a r d Model as o n e
vibration of t h e string. Several g r o u p s have a n n o u n c e d , in fact, that they
can find solutions that agree with the known data a b o u t subatomic particles.
T h e p r o b l e m , we see, is that t h e r e are also millions u p o n millions of
o t h e r solutions describing universes that do n o t a p p e a r anything like
o u r universe. In s o m e of these solutions, t h e universe has no quarks or
too many quarks. In most of t h e m , life as we know it c a n n o t exist. O u r
universe may be lost s o m e w h e r e a m o n g t h e millions of possible universes
that have b e e n f o u n d in string theory. To find t h e correct solution, we
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must use n o n p e r t u r b a t i v e techniques, which are notoriously difficult.
Since 9 9 % of what we know a b o u t high-energy physics is based on perturbation theory, this m e a n s that we are at a total loss to find t h e o n e
true solution to the theory.
T h e r e is some r o o m for optimism, however. N o n p e r t u r b a t i v e solutions that have b e e n f o u n d for m u c h simpler theories show that m a n y
of the solutions are actually unstable. After a time, these incorrect, unstable solutions will m a k e a q u a n t u m leap to t h e correct, stable solution. If
this is true for string theory, t h e n p e r h a p s the millions of solutions that
have b e e n f o u n d are actually unstable a n d will decay over time to the
correct solution.
To u n d e r s t a n d the frustration that we physicists feel, think, for a
m o m e n t , of how n i n e t e e n t h - c e n t u r y physicists m i g h t react if a p o r t a b l e
c o m p u t e r were given to t h e m . T h e y could easily learn to t u r n t h e dials
a n d press the b u t t o n s . They could learn to master video games or watch
educational p r o g r a m s on the m o n i t o r . Being a century b e h i n d in technology, they would marvel at t h e fantastic calculational ability of t h e
c o m p u t e r . Within its m e m o r y could easily be stored all known scientific
knowledge of that century. In a short p e r i o d of time, they could learn
to perform mathematical feats t h a t would amaze any of their colleagues.
However, o n c e they d e c i d e to o p e n up the m o n i t o r to see what is inside,
they would be horrified. T h e transistors a n d microprocessors would be
totally alien to anything they could u n d e r s t a n d . T h e r e would be really
n o t h i n g in their e x p e r i e n c e to c o m p a r e with t h e electronic c o m p u t e r .
It would be beyond their ken. T h e y could only stare blankly at the complicated circuitry, n o t knowing in the slightest how it works or what it
all means.
T h e source of their frustration would be that the c o m p u t e r exists
a n d is sitting t h e r e in front of their noses, b u t they would have no refe r e n c e frame from which to explain it. Analogously, string theory
appears to be twenty-first-century physics that was discovered accidentally
in o u r century. String field theory, too, seems to include all physical
knowledge. With little effort, we are able to t u r n a few dials a n d press a
few b u t t o n s with the theory, a n d o u t p o p s the supergravity theory,
Kaluza-Klein theory, a n d the S t a n d a r d Model. But we are at a total loss
to explain why it works. String field theory exists, b u t it taunts us because
we are n o t smart e n o u g h to solve it.
T h e p r o b l e m is that while twenty-first-century physics fell accidentally
into the twentieth century, twenty-first-century m a t h e m a t i c s h a s n ' t b e e n
invented yet. It seems that we may have to wait for twenty-first-century
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m a t h e m a t i c s before we can m a k e any progress, or the c u r r e n t generation of physicists must invent twenty-first-century mathematics on their
own.
Why Ten Dimensions?
O n e of the d e e p e s t secrets of string theory, which is still n o t well u n d e r stood, is why it is defined in only ten a n d 26 dimensions. If the theory
were t h r e e dimensional, it would n o t be able to unify the known laws of
physics in any sensible m a n n e r . T h u s it is t h e geometry of h i g h e r dimensions that is the central feature of the theory.
If we calculate how strings break a n d re-form in N-dimensional space,
we constantly find meaningless terms c r o p p i n g up that destroy the marvelous p r o p e r t i e s of the theory. Fortunately, these u n w a n t e d terms
a p p e a r multiplied by (N — 10). T h e r e f o r e , to m a k e these anomalies
vanish, we have no choice b u t to fix N to be ten. String theory, in fact,
is t h e only known q u a n t u m theory that specifically d e m a n d s that the
d i m e n s i o n of s p a c e - t i m e be fixed at a u n i q u e n u m b e r .
Unfortunately, string theorists are, at present, at a loss to explain why
ten d i m e n s i o n s are singled out. T h e answer lies d e e p within m a t h e m a t ics, in an area called modular functions. W h e n e v e r we m a n i p u l a t e the KSV
l o o p diagrams created by interacting strings, we e n c o u n t e r these strange
m o d u l a r functions, w h e r e the n u m b e r ten a p p e a r s in the strangest
places. T h e s e m o d u l a r functions are as mysterious as the m a n w h o investigated t h e m , t h e mystic from t h e East. P e r h a p s if we better u n d e r s t o o d
the work of this I n d i a n genius, we would u n d e r s t a n d why we live in o u r
p r e s e n t universe.
The Mystery of Modular Functions
Srinivasa R a m a n u j a n was t h e strangest m a n in all of mathematics, probably in the e n t i r e history of science. He has b e e n c o m p a r e d to a bursting
supernova, illuminating the darkest, most p r o f o u n d corners of mathematics, before b e i n g tragically struck down by tuberculosis at the age of
33, like R i e m a n n before h i m . W o r k i n g in total isolation from the main
c u r r e n t s of his field, he was able to rederive 100 years' worth of Western
m a t h e m a t i c s on his own. T h e tragedy of his life is that m u c h of his work
was wasted rediscovering known mathematics. Scattered t h r o u g h o u t the
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obscure equations in his n o t e b o o k s are these m o d u l a r functions, which
are a m o n g the strangest ever f o u n d in mathematics. T h e y r e a p p e a r in
the most distant a n d u n r e l a t e d b r a n c h e s of mathematics. O n e function,
which appears again a n d again in the theory of m o d u l a r functions, is
today called the Ramanujan function in his h o n o r . This bizarre function
contains a t e r m raised to t h e twenty-fourth power.
In the work of Ramanujan, t h e n u m b e r 24 a p p e a r s repeatedly. This
is an e x a m p l e of what m a t h e m a t i c i a n s call magic n u m b e r s , which continually a p p e a r , w h e r e we least expect t h e m , for reasons that no o n e
u n d e r s t a n d s . Miraculously, R a m a n u j a n ' s function also a p p e a r s in string
theory. T h e n u m b e r 24 a p p e a r i n g in R a m a n u j a n ' s function is also the
origin of the miraculous cancellations o c c u r r i n g in string theory. In
string theory, each of the 24 m o d e s in the Ramanujan function corresponds to a physical vibration of the string. W h e n e v e r the string executes
its complex motions in s p a c e - t i m e by splitting a n d r e c o m b i n i n g , a large
n u m b e r of highly sophisticated m a t h e m a t i c a l identities must be satisfied. These are precisely t h e mathematical identities discovered by
Ramanujan. (Since physicists a d d two m o r e d i m e n s i o n s when they c o u n t
the total n u m b e r of vibrations a p p e a r i n g in a relativistic theory,
this m e a n s that s p a c e - t i m e must have 24 + 2 = 26 s p a c e - t i m e d i m e n sions. )
13
W h e n the Ramanujan function is generalized, the n u m b e r 24 is
replaced by the n u m b e r 8. T h u s the critical n u m b e r for the superstring
is 8 + 2, or 10. This is the origin of the tenth d i m e n s i o n . T h e string
vibrates in ten d i m e n s i o n s because it requires these generalized Ramanujan functions in o r d e r to r e m a i n self-consistent. In other words, physicists
have not the slightest understanding of why ten and 26 dimensions are singled
out as the dimension of the string. It's as t h o u g h t h e r e is some kind of d e e p
n u m e r o l o g y b e i n g manifested in these functions that no o n e u n d e r stands. It is precisely these magic n u m b e r s a p p e a r i n g in the elliptic modular function that d e t e r m i n e s the d i m e n s i o n of s p a c e - t i m e to be
ten.
In the final analysis, the origin of t h e ten-dimensional theory is as
mysterious as Ramanujan himself. W h e n asked by a u d i e n c e s why n a t u r e
might exist in ten dimensions, physicists are forced to answer, " W e d o n ' t
know." We know, in vague terms, why some d i m e n s i o n of s p a c e - t i m e
must be selected (or else t h e string c a n n o t vibrate in a self-consistent
q u a n t u m fashion), b u t we d o n ' t know why these particular n u m b e r s are
selected. P e r h a p s the answer lies waiting to be discovered in R a m a n u j a n ' s lost notebooks.
174
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DIMENSIONS
Reinventing 100 Years of Mathematics
R a m a n u j a n was b o r n in 1887 in E r o d e , India, n e a r Madras. Although
his family was B r a h m i n , t h e highest of the H i n d u castes, they were destitute, living off the m e a g e r wages of R a m a n u j a n ' s father's j o b as a clerk
in a clothing m e r c h a n t ' s office.
By the age of 10, it was clear that Ramanujan was n o t like the o t h e r
children. Like R i e m a n n before h i m , he b e c a m e well known in his village
for his awesome calculational powers. As a child, he h a d already rederived Euler's identity between t r i g o n o m e t r i c functions a n d exponentials.
In every y o u n g scientist's life, t h e r e is a t u r n i n g point, a singular
event that helps to c h a n g e the course of his or h e r life. For Einstein, it
was t h e fascination of observing a compass n e e d l e . For R i e m a n n , it was
r e a d i n g L e g e n d r e ' s b o o k on n u m b e r theory. For Ramanujan, it was
w h e n he s t u m b l e d on an obscure, forgotten b o o k on mathematics by
G e o r g e Carr. This b o o k has since b e e n immortalized by the fact that it
m a r k e d R a m a n u j a n ' s only known e x p o s u r e to m o d e r n Western mathematics. According to his sister, " I t was this b o o k which awakened his
genius. He set himself to establish the formulae given t h e r e i n . As he was
without the aid of o t h e r books, each solution was a piece of research so
far as he was c o n c e r n e d . . . . R a m a n u j a n used to say that the goddess of
Namakkal inspired h i m with the formulae in d r e a m s . "
Because of his brilliance, he was able to win a scholarship to high
school. But because he was b o r e d with the t e d i u m of classwork a n d
intensely p r e o c c u p i e d with the e q u a t i o n s that were constantly d a n c i n g
in his h e a d , he failed to e n t e r his senior class, a n d his scholarship was
canceled. Frustrated, he ran away from h o m e . He did finally r e t u r n , b u t
only to fall ill a n d fail his e x a m i n a t i o n s again.
With t h e h e l p of friends, Ramanujan m a n a g e d to b e c o m e a low-level
clerk in t h e P o r t Trust of Madras. It was a menial j o b , paying a paltry
£20 a year, b u t it freed R a m a n u j a n , like Einstein before h i m at the Swiss
p a t e n t office, to follow his d r e a m s in his spare time. Ramanujan t h e n
mailed some of the results of his " d r e a m s " to t h r e e well-known British
m a t h e m a t i c i a n s , h o p i n g for contact with o t h e r mathematical minds.
Two of t h e m a t h e m a t i c i a n s , receiving this letter written by an u n k n o w n
I n d i a n clerk with no formal e d u c a t i o n , p r o m p t l y threw it away. T h e third
o n e was the brilliant C a m b r i d g e m a t h e m a t i c i a n Godfrey H. Hardy.
Because of his stature in England, H a r d y was a c c u s t o m e d to receiving
crank mail a n d t h o u g h t dimly of the letter. Amid t h e d e n s e scribbling
he noticed many t h e o r e m s of m a t h e m a t i c s that were already well known.
1 4
175
Superstrings
T h i n k i n g it t h e obvious work of a plagiarist, he also threw it away. But
s o m e t h i n g wasn't q u i t e right. S o m e t h i n g n a g g e d a t Hardy; h e c o u l d n ' t
h e l p w o n d e r i n g a b o u t this strange letter.
A t d i n n e r that night, J a n u a r y 1 6 , 1 9 1 3 , Hardy a n d his colleague J o h n
Littlewood discussed this o d d letter a n d d e c i d e d to take a s e c o n d look
at its contents. It b e g a n , innocently e n o u g h , with "I b e g to i n t r o d u c e
myself to you as a clerk in t h e Accounts D e p a r t m e n t of the Port T r u s t
Office of Madras on a salary of only 20 p o u n d s p e r a n n u m . " But the
letter from the p o o r Madras clerk c o n t a i n e d t h e o r e m s that were totally
u n k n o w n to Western m a t h e m a t i c i a n s . In all, it c o n t a i n e d 120 t h e o r e m s .
Hardy was s t u n n e d . He recalled that proving s o m e of these t h e o r e m s
"defeated me completely." He recalled, "I h a d never seen anything in
the least like t h e m before. A single look at t h e m is e n o u g h to show t h a t
they could only be written down by a m a t h e m a t i c i a n of t h e highest
class."
1 5
16
Littlewood a n d H a r d y r e a c h e d t h e identical a s t o u n d i n g conclusion:
This was obviously the work of a genius e n g a g e d in rederiving 100 years
o f E u r o p e a n mathematics. " H e h a d b e e n carrying a n impossible h a n d icap, a p o o r a n d solitary H i n d u pitting his brains against t h e a c c u m u lated wisdom of E u r o p e , " recalled H a r d y .
Hardy sent for Ramanujan a n d , after m u c h difficulty, a r r a n g e d for
his stay in C a m b r i d g e in 1914. For t h e first time, Ramanujan could comm u n i c a t e regularly with his peers, the c o m m u n i t y of E u r o p e a n m a t h e maticians. T h e n b e g a n a burst of activity: 3 short, intense years of collaboration with Hardy at Trinity College in C a m b r i d g e .
Hardy later tried to estimate t h e m a t h e m a t i c a l skill that R a m a n u j a n
possessed. He rated David Hilbert, universally recognized as o n e of the
greatest Western m a t h e m a t i c i a n s of the n i n e t e e n t h century, an 80. To
Ramanujan, he assigned a 100. (Hardy r a t e d himself a 25.)
Unfortunately, n e i t h e r Hardy n o r R a m a n u j a n s e e m e d interested in
the psychology or t h i n k i n g process by which R a m a n u j a n discovered
these incredible t h e o r e m s , especially w h e n this flood of material c a m e
p o u r i n g o u t of his " d r e a m s " with such frequency. H a r d y n o t e d , " I t
seemed ridiculous to worry h i m a b o u t how he h a d f o u n d this or that
known t h e o r e m , w h e n he was showing me half a d o z e n new o n e s almost
every d a y . "
Hardy vividly recalled,
17
18
I remember going to see him once when he was lying ill in Putney. I had
ridden in taxi-cab No. 1729, and remarked that the number seemed to be
176
IN
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D I M E N S I O N S
rather a dull o n e , a n d that I h o p e d that it was n o t an unfavorable o m e n .
" N o , " he r e p l i e d , "it is a very i n t e r e s t i n g n u m b e r ; it is t h e smallest n u m b e r
expressible as a s u m of two c u b e s in two different w a y s . "
1 9
(It is t h e s u m of 1 X 1 X 1 a n d 12 X 12 X 12, a n d also the sum of 9 X
9 X 9 a n d 10 X 10 X 10.) On t h e spot, he could recite c o m p l e x t h e o r e m s
in arithmetic that would r e q u i r e a m o d e r n c o m p u t e r to prove.
Always in p o o r health, the austerity of t h e war-torn British e c o n o m y
p r e v e n t e d R a m a n u j a n from m a i n t a i n i n g his strict vegetarian diet, a n d
he was constantly in a n d o u t of sanitariums. After collaborating with
Hardy for 3 years, R a m a n u j a n fell ill a n d never recovered. World War I
i n t e r r u p t e d travel between E n g l a n d a n d India, a n d in 1919 he finally
m a n a g e d to r e t u r n h o m e , where he d i e d a year later.
Modular Functions
R a m a n u j a n ' s legacy is his work, which consists of 4,000 formulas on 400
pages filling t h r e e volumes of notes, all densely packed with t h e o r e m s
of incredible power b u t without any c o m m e n t a r y or, which is m o r e frustrating, any proof. In 1976, however, a new discovery was m a d e . O n e
h u n d r e d a n d thirty pages of scrap p a p e r , c o n t a i n i n g t h e o u t p u t of the
last year of his life, was discovered by accident in a b o x at Trinity College.
This is now called R a m a n u j a n ' s " L o s t N o t e b o o k . " C o m m e n t i n g on t h e
Lost N o t e b o o k , m a t h e m a t i c i a n Richard Askey says, " T h e work of that
o n e year, while he was dying, was the equivalent of a lifetime of work for
a very great m a t h e m a t i c i a n . W h a t he accomplished was unbelievable. If
it were a novel, n o b o d y would believe it." To u n d e r s c o r e the difficulty
of their a r d u o u s task of d e c i p h e r i n g t h e " n o t e b o o k s , " mathematicians
J o n a t h a n Borwein a n d Peter Borwein have c o m m e n t e d , " T o o u r knowle d g e no m a t h e m a t i c a l redaction of this scope or difficulty has ever b e e n
attempted."
20
Looking at t h e progression of R a m a n u j a n ' s e q u a t i o n s , it's as t h o u g h
we have b e e n t r a i n e d for years to listen to t h e Western music of Beethoven, a n d t h e n suddenly we are e x p o s e d to a n o t h e r type of music, an
eerily beautiful Eastern music b l e n d i n g h a r m o n i e s a n d rhythms never
h e a r d before in Western music. J o n a t h a n Borwein says, " H e seems to
have functioned in a way unlike anybody else we know of. He h a d such
a feel for things that they j u s t flowed o u t of his brain. P e r h a p s he d i d n ' t
see t h e m in any way that's translatable. It's like watching somebody at a
feast you h a v e n ' t b e e n invited t o . "
Superstrings
[77
As physicists know, " a c c i d e n t s " do n o t a p p e a r without a reason.
W h e n p e r f o r m i n g a long a n d difficult calculation, a n d t h e n suddenly
having t h o u s a n d s of u n w a n t e d terms miraculously a d d up to zero, physicists know that this does n o t h a p p e n without a d e e p e r , u n d e r l y i n g reason. Today, physicists know that these " a c c i d e n t s " are an indication that
a* symmetry is at work. For strings, the symmetry is called conformal
symmetry, the symmetry of stretching a n d d e f o r m i n g the string's world
sheet.
This is precisely w h e r e R a m a n u j a n ' s work comes in. In o r d e r to p r o tect the original conformal symmetry from b e i n g destroyed by q u a n t u m
theory, a n u m b e r of mathematical identities must be miraculously satisfied. These identities are precisely the identities of R a m a n u j a n ' s m o d ular function.
In summary, we have said that o u r f u n d a m e n t a l premise is that the
laws of n a t u r e simplify when expressed in h i g h e r d i m e n s i o n s . However,
in light of q u a n t u m theory, we must how a m e n d this basic t h e m e . T h e
correct statement should now read: T h e laws of n a t u r e simplify when
self-consistently expressed in h i g h e r d i m e n s i o n s . T h e addition of t h e word
self-consistently is crucial. This constraint forces us to use R a m a n u j a n ' s
m o d u l a r functions, which fixes the d i m e n s i o n of s p a c e - t i m e to be ten.
This, in turn, may give us t h e decisive clue to explain the origin of the
universe.
Einstein often asked himself w h e t h e r God h a d any choice in creating
the universe. According to superstring theorists, o n c e we d e m a n d a unification of q u a n t u m theory a n d general relativity, G o d h a d no choice.
Self-consistency alone, they claim, must have forced God to create t h e
universe as he did.
Although the mathematical sophistication i n t r o d u c e d by superstring
theory has r e a c h e d dizzying heights a n d has startled the m a t h e m a t i c i a n s ,
the critics of the theory still p o u n d it at its weakest point. Any theory,
they claim, must be testable. Since any theory defined at the Planck
energy of 10 billion electron volts is n o t testable, superstring theory is
n o t really a theory at all!
T h e m a i n p r o b l e m , as we have p o i n t e d out, is theoretical r a t h e r t h a n
experimental. If we were smart e n o u g h , we could solve the theory exactly
a n d find the true n o n p e r t u r b a t i v e solution of the theory. However, this
does n o t excuse us from finding s o m e m e a n s by which to verify the
theory experimentally. To test the theory, we must wait for signals from
the t e n t h d i m e n s i o n .
19
8
Signals from
the Tenth Dimension
H o w s t r a n g e i t w o u l d b e i f t h e final t h e o r y w e r e t o b e d i s c o v e r e d i n o u r l i f e t i m e s ! T h e d i s c o v e r y o f t h e final laws o f n a t u r e
will m a r k a d i s c o n t i n u i t y i n h u m a n i n t e l l e c t u a l history, t h e
s h a r p e s t t h a t h a s o c c u r r e d s i n c e t h e b e g i n n i n g o f m o d e r n scie n c e in the seventeenth century. Can we now imagine what
t h a t w o u l d b e like?
Steven W e i n b e r g
Is Beauty a Physical Principle?
A
L T H O U G H superstring theory gives us a compelling formulation
of the theory of the universe, the f u n d a m e n t a l p r o b l e m is that an
e x p e r i m e n t a l test of t h e theory seems beyond o u r present-day technology. In fact, the theory predicts that the unification of all forces occurs
at t h e Planck energy, or 1 0 billion e l e c t r o n volts, which is a b o u t 1
quadrillion times larger than energies currently available in o u r accelerators.
Physicist David Gross, c o m m e n t i n g on the cost of g e n e r a t i n g this
fantastic energy, says, " T h e r e is n o t e n o u g h m o n e y in the treasuries of
all t h e c o u n t r i e s in t h e world p u t together. It's truly a s t r o n o m i c a l . " '
This is disappointing, because it m e a n s that e x p e r i m e n t a l verification, t h e e n g i n e t h a t drives progress in physics, is no l o n g e r possible with
o u r c u r r e n t g e n e r a t i o n of m a c h i n e s or with any g e n e r a t i o n of m a c h i n e s
19
178
179
Signals from the Tenth Dimension
in the conceivable future. This, in turn, m e a n s that the ten-dimensional
theory is n o t a theory in t h e usual sense, because it is untestable given
the p r e s e n t technological state of o u r planet. We are t h e n left with the
question: Is beauty, by itself, a physical principle that can be substituted
for the lack of e x p e r i m e n t a l verification?
To some, t h e answer is a r e s o u n d i n g n o . T h e y derisively call these
theories "theatrical physics" o r "recreational m a t h e m a t i c s . " T h e most
caustic of the critics is Nobel Prize winner S h e l d o n Glashow of Harvard
University. He has a s s u m e d t h e role of gadfly in this d e b a t e , leading the
charge against t h e claims of o t h e r physicists that h i g h e r d i m e n s i o n s may
exist. Glashow rails against these physicists, c o m p a r i n g t h e c u r r e n t epidemic to the AIDS virus; that is, it's incurable. He also c o m p a r e s t h e
c u r r e n t b a n d w a g o n effect with former President R e a g a n ' s Star Wars program:
H e r e ' s a riddle: N a m e two g r a n d d e s i g n s that are incredibly c o m p l e x ,
r e q u i r e d e c a d e s o f r e s e a r c h t o d e v e l o p , a n d m a y n e v e r w o r k i n t h e real
w o r l d ? Stars W a r s a n d s t r i n g t h e o r y . . . . N e i t h e r a m b i t i o n c a n b e a c c o m p l i s h e d w i t h e x i s t i n g t e c h n o l o g y , a n d n e i t h e r m a y a c h i e v e its s t a t e d o b j e c tives. B o t h a d v e n t u r e s a r e c o s t l y i n t e r m s o f s c a r c e h u m a n r e s o u r c e s . A n d ,
in b o t h cases, the Russians are trying desperately to catch u p .
2
To stir up m o r e controversy, Glashow even p e n n e d a p o e m , which
ends:
T h e T h e o r y of Everything, if you dare to be bold,
Might be s o m e t h i n g m o r e than a string orbifold.
While s o m e of your leaders have got old a n d sclerotic,
N o t to be trusted a l o n e with things heterotic,
Please h e e d o u r advice that y o u are n o t s m i t t e n —
T h e B o o k i s n o t f i n i s h e d , t h e last w o r d i s n o t W i t t e n .
3
Glashow has vowed (unsuccessfully) to k e e p these theories o u t of Harvard, where he teaches. But he does a d m i t that he is often o u t n u m b e r e d
on this question. He regrets, "I find myself a d i n o s a u r in a world of
upstart m a m m a l s . " (Glashow's views are certainly n o t s h a r e d by o t h e r
Nobel laureates, such as Murray Gell-Mann a n d Steven W e i n b e r g . Physicist Weinberg, in fact, says, " S t r i n g theory provides o u r only p r e s e n t
source of candidates for a final theory—how could a n y o n e expect that
many of the brightest y o u n g theorists would not work on i t ? " )
To u n d e r s t a n d the implications of this d e b a t e c o n c e r n i n g the uni4
5
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IN
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D I M E N S I O N S
fication of all forces, a n d also the p r o b l e m s with its e x p e r i m e n t a l verification, it is instructive to consider t h e following analogy, the " p a r a b l e
of t h e g e m s t o n e . "
In the b e g i n n i n g , let us say, was a g e m s t o n e of great beauty, which
was perfectly symmetrical in t h r e e d i m e n s i o n s . However, this g e m s t o n e
was unstable. O n e day, it burst a p a r t a n d sent fragments in all directions;
they eventually r a i n e d down on t h e two-dimensional world of Flatland.
Curious, the residents of Flatland e m b a r k e d on a quest to reassemble
the pieces. They called t h e original explosion the Big Bang, b u t did n o t
u n d e r s t a n d why these fragments were scattered t h r o u g h o u t their world.
Eventually, two kinds of fragments were identified. Some fragments were
polished a n d s m o o t h o n o n e side, a n d Flatlanders c o m p a r e d t h e m t o
" m a r b l e . " O t h e r fragments were entirely j a g g e d a n d ugly, with no regularity whatsoever, a n d Flatlanders c o m p a r e d these pieces to " w o o d . "
Over t h e years, t h e Flatlanders divided into two camps. T h e first
c a m p began to piece t o g e t h e r t h e polished fragments. Slowly, s o m e of
t h e polished pieces begin to fit together. Marveling at how these polished fragments were b e i n g assembled, these Flatlanders were convinced
that s o m e h o w a powerful new geometry must be o p e r a t i n g . These Flatlanders called their partially assembled piece "relativity."
T h e second g r o u p devoted their efforts to assembling the j a g g e d ,
irregular fragments. They, too, h a d limited success in finding patterns
a m o n g these fragments. However, t h e j a g g e d pieces p r o d u c e d only a
larger b u t even m o r e irregular c l u m p , which they called the Standard
Model. No o n e was inspired by the ugly mass called t h e Standard Model.
After years of painstaking work trying to fit these various pieces
together, however, it a p p e a r e d as t h o u g h t h e r e was no way to p u t the
polished pieces t o g e t h e r with the j a g g e d pieces.
T h e n o n e day an ingenious Flatlander hit u p o n a marvelous idea.
He d e c l a r e d that t h e two sets of pieces could be reassembled into o n e
piece if they were moved " u p " — t h a t is, in s o m e t h i n g he called the third
d i m e n s i o n . Most Flatlanders were bewildered by this new a p p r o a c h ,
because n o o n e could u n d e r s t a n d what " u p " m e a n t . However, h e was
able to show by c o m p u t e r that the " m a r b l e " fragments could be viewed
as o u t e r fragments of s o m e object, a n d were h e n c e polished, while the
" w o o d " fragments were the i n n e r fragments. W h e n b o t h sets of fragm e n t s were assembled in the third d i m e n s i o n , t h e Flatlanders gasped at
what was revealed in the c o m p u t e r : a dazzling g e m s t o n e with perfect
three-dimensional symmetry. In o n e stroke, t h e artificial distinction
between the two sets of fragments was resolved by p u r e geometry.
This solution, however, left several questions u n a n s w e r e d . Some Flat-
181
landers still w a n t e d e x p e r i m e n t a l proof, n o t j u s t theoretical calculations,
that the pieces could really be assembled into this g e m s t o n e . This theory
gave a c o n c r e t e n u m b e r for the energy it would take to build powerful
m a c h i n e s that could hoist these fragments " u p " off Flatland a n d assemble the pieces in three-dimensional space. But t h e energy r e q u i r e d was
a b o u t a quadrillion times the largest energy source available to t h e Flatlanders.
For some, t h e theoretical calculation was sufficient. Even lacking
e x p e r i m e n t a l verification, they felt that " b e a u t y " was m o r e t h a n sufficient to settle the question of unification. History h a d always shown, they
p o i n t e d out, that t h e solution to t h e most difficult p r o b l e m s in n a t u r e
h a d b e e n the ones with the most beauty. They also correctly p o i n t e d o u t
that the three-dimensional theory h a d no rival.
O t h e r Fladanders, however, raised a howl. A theory that c a n n o t be
tested is n o t a theory, they fumed. Testing this theory would drain t h e
best m i n d s a n d waste valuable resources on a wild-goose chase, they
claimed.
T h e d e b a t e in Flatland, as well as in the real world, will persist for
some t i m e , which is a g o o d thing. As t h e eighteenth-century p h i l o s o p h e r
J o s e p h J o u b e r t o n c e said, " I t is b e t t e r to d e b a t e a question without settling it t h a n to settle a question without d e b a t i n g it."
The Superconducting Supercollider: Window on Creation
T h e eighteenth-century English p h i l o s o p h e r David H u m e , w h o was
famous for advancing the thesis that every theory m u s t be g r o u n d e d on
the foundation of e x p e r i m e n t , was at a loss to explain how o n e can
experimentally verify a theory of Creation. T h e essence of e x p e r i m e n t ,
he claimed, is reproducibility. Unless an e x p e r i m e n t can be d u p l i c a t e d
over a n d over, in different locations a n d at different times with t h e same
results, the theory is unreliable. But how can o n e p e r f o r m an e x p e r i m e n t
with Creation itself? Since C r e a d o n , by definition, is n o t a r e p r o d u c i b l e
event, H u m e h a d to c o n c l u d e that it is impossible to verify any theory
of Creation. Science, he claimed, can answer almost all questions conc e r n i n g the universe except for o n e , Creation, the only e x p e r i m e n t that
cannot be reproduced.
In some sense, we are e n c o u n t e r i n g a m o d e r n version of t h e p r o b l e m
identified by H u m e in t h e e i g h t e e n t h century. T h e p r o b l e m r e m a i n s
the same: T h e energy necessary to re-create Creation exceeds a n y t h i n g
available on the p l a n e t earth. However, a l t h o u g h direct e x p e r i m e n t a l
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verification of t h e ten-dimensional theory in o u r laboratories is n o t possible, t h e r e are several ways to a p p r o a c h this question indirectly. T h e
most logical a p p r o a c h was to h o p e that t h e s u p e r c o n d u c t i n g supercollider (SSC) would find subatomic particles that show the distinctive sign a t u r e of t h e superstring, such as supersymmetry. A l t h o u g h the SSC
could n o t have p r o b e d t h e Planck energy, it m i g h t have given us strong,
indirect evidence of t h e correctness of superstring theory.
T h e SSC (killed off by formidable political opposition) would have
b e e n a truly m o n s t r o u s m a c h i n e , t h e last of its type. W h e n c o m p l e t e d
outside Dallas, Texas, a r o u n d t h e year 2000, it would have consisted of
a gigantic t u b e 50 miles in circumference s u r r o u n d e d by h u g e magnets.
(If it were c e n t e r e d in M a n h a t t a n , it would have e x t e n d e d well into
C o n n e c t i c u t a n d New Jersey.) Over 3,000 full-time a n d visiting scientists
a n d staff would have c o n d u c t e d e x p e r i m e n t s a n d analyzed the data from
the machine.
T h e p u r p o s e of the SSC was to whip two b e a m s of p r o t o n s a r o u n d
inside this t u b e until they r e a c h e d a velocity very close to the speed of
light. Because these b e a m s would be traveling clockwise a n d counterclockwise, it would have b e e n a simple m a t t e r to make t h e m collide
within t h e t u b e w h e n they r e a c h e d their m a x i m u m energy. T h e p r o t o n s
would have s m a s h e d into o n e a n o t h e r at an energy of 40 trillion electron
volts (TeV), thereby g e n e r a t i n g an intense burst of subatomic debris
analyzed by detectors. This kind of collision has n o t o c c u r r e d since the
Big Bang itself ( h e n c e t h e n i c k n a m e for the SSC: "window on creat i o n " ) . A m o n g t h e debris, physicists h o p e d to find exotic subatomic
particles that would have s h e d light on the ultimate form of matter.
N o t surprisingly, the SSC was an extraordinary e n g i n e e r i n g a n d physics project, stretching t h e limits of known technology. Because the magnetic fields necessary to b e n d t h e p r o t o n s a n d a n t i p r o t o n s within the
t u b e are so exceptionally large (on the o r d e r of 100,000 times the e a r t h ' s
m a g n e t i c field), extraordinary p r o c e d u r e s would have b e e n necessary to
g e n e r a t e a n d m a i n t a i n t h e m . For e x a m p l e , t o r e d u c e the h e a t i n g a n d
electrical resistance within t h e wires, the m a g n e t s would have b e e n
cooled down nearly to absolute zero. T h e n they would have b e e n specially reinforced because the magnetic fields are so intense that otherwise they would have warped the metal of t h e m a g n e t itself.
Projected to cost $11 billion, the SSC b e c a m e a prized p l u m a n d a
m a t t e r of intense political jockeying. In t h e past, the sites for a t o m
smashers were d e c i d e d by u n a b a s h e d political h o r s e trading. For example, t h e state of Illinois was able to land the Fermilab accelerator in
Batavia, j u s t outside Chicago, because ( a c c o r d i n g to Physics Today) Pres-
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ident Lyndon J o h n s o n n e e d e d Illinois senator Everett Dirkson's crucial
vote on the V i e t n a m War. T h e SSC was probably no different. A l t h o u g h
many states vigorously c o m p e t e d for the project, it probably c a m e as no
surprise that in 1988 t h e great state of Texas l a n d e d t h e SSC, especially
when b o t h the president-elect of the United States a n d the D e m o c r a t i c
vice-presidential c a n d i d a t e c a m e from Texas.
A l t h o u g h billions of dollars have b e e n s p e n t on t h e SSC, it will never
be completed. To the h o r r o r of t h e physics community, the H o u s e of
Representatives voted in 1993 to cancel the project completely. I n t e n s e
lobbying failed to restore funding for the project. To Congress, an
expensive a t o m smasher can be seen in two ways. It can be a juicy p l u m ,
g e n e r a t i n g t h o u s a n d s of j o b s a n d billions of dollars in federal subsidies
for t h e state that has it. Or it can be viewed as an incredible b o o n d o g g l e ,
a waste of m o n e y that g e n e r a t e s no direct c o n s u m e r benefits. In lean
times, they a r g u e , an expensive toy for high-energy physicists is a luxury
the country c a n n o t afford. (In all fairness, t h o u g h , f u n d i n g for the SSC
project must be p u t into p r o p e r perspective. Star Wars f u n d i n g for j u s t
1 year costs $4 billion. It costs a b o u t $1 billion to refurbish an aircraft
carrier. A single space-shuttle mission costs $1 billion. A n d a single B-2
stealth b o m b e r costs almost $1 billion.)
Although the SSC is d e a d , what m i g h t we have discovered with it? At
the very least, scientists h o p e d to find exotic particles, such as t h e mysterious Higgs particle p r e d i c t e d by the S t a n d a r d Model. It is t h e Higgs
particle that generates symmetry b r e a k i n g a n d is therefore the origin of
the mass of the quarks. T h u s we h o p e d that t h e SSC would have f o u n d
the "origin of mass." All objects s u r r o u n d i n g us that have weight owe
their mass to the Higgs particle.
T h e betting a m o n g physicists, however, was that t h e r e was an even
c h a n c e that the SSC would find exotic particles beyond the S t a n d a r d
Model. (Possibilities i n c l u d e d " T e c h n i c o l o r " particles, which lie j u s t
beyond the Standard Model, or " a x i o n s , " which may h e l p to explain
the dark m a t t e r p r o b l e m . ) But p e r h a p s t h e most exciting possibility was
the sparticles, which are t h e supersymmetric p a r t n e r s of ordinary particles. T h e gravitino, for e x a m p l e , is t h e supersymmetric p a r t n e r of t h e
graviton. T h e supersymmetric p a r t n e r s of the q u a r k a n d l e p t o n , respectively, are the squark a n d t h e slepton.
If supersymmetric particles are eventually discovered, t h e n t h e r e is a
fighting c h a n c e that we will be seeing t h e r e m n a n t s of the superstring
itself. (Supersymmetry, as a symmetry of a field theory, was first discovered in superstring theory in 1971, even before t h e discovery of supergravity. In fact, the superstring is probably t h e only theory in which
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supersymmetry a n d gravity can be c o m b i n e d in a totally self-consistent
way.) A n d even t h o u g h t h e potential discovery of sparticles will n o t prove
t h e correctness of superstring theory, it will h e l p to quiet the skeptics
w h o have said that t h e r e is n o t o n e shred of physical evidence for
superstring theory.
Signals from Outer Space
Since t h e SSC will never be built, a n d h e n c e will never detect particles
that are low-energy r e s o n a n c e s of the superstring, t h e n a n o t h e r possibility is to m e a s u r e t h e energy of cosmic rays, which are highly energetic
subatomic particles whose origin is still u n k n o w n , b u t must lie d e e p in
o u t e r space b e y o n d o u r galaxy. For e x a m p l e , a l t h o u g h no o n e knows
where they c o m e from, cosmic rays have energies m u c h larger t h a n anyt h i n g f o u n d in o u r laboratories.
Cosmic rays, unlike the controlled rays p r o d u c e d in a t o m smashers,
have u n p r e d i c t a b l e energies a n d c a n n o t p r o d u c e precise energies o n
d e m a n d . In some sense, it's like trying to p u t o u t a fire by either using
h o s e water or waiting for a rainstorm. T h e hose water is m u c h m o r e
convenient: We can t u r n it on any time we please, we can adjust the
intensity of t h e water at will, a n d all t h e water travels at t h e same uniform
velocity. Water from a fire h y d r a n t therefore c o r r e s p o n d s to p r o d u c i n g
controlled b e a m s in a t o m smashers. However, water from a rainstorm
may be m u c h m o r e intense a n d effective t h a n water from a fire hydrant.
T h e p r o b l e m , of course, is that rainstorms, like cosmic rays, are u n p r e dictable. You c a n n o t regulate the rainwater, n o r can you predict its velocity, which may fluctuate wildly.
Cosmic rays were first discovered 80 years ago in e x p e r i m e n t s performed by t h e J e s u i t priest T h e o d o r Wulf a t o p t h e Eiffel Tower in Paris.
F r o m the 1900s to t h e 1930s, c o u r a g e o u s physicists sailed in balloons or
scaled m o u n t a i n s to obtain the best m e a s u r e m e n t s of cosmic rays. But
cosmic-ray research b e g a n to fade d u r i n g t h e 1930s, w h e n Ernest Lawr e n c e invented t h e cyclotron a n d p r o d u c e d controlled b e a m s i n the
laboratory m o r e energetic t h a n most cosmic rays. For e x a m p l e , cosmic
rays, which are as energetic as 100 million electron volts, are as c o m m o n
as rain d r o p s ; they hit the a t m o s p h e r e of the earth at the rate of a few
p e r s q u a r e inch p e r second. However, Lawrence's invention spawned
giant m a c h i n e s that could e x c e e d that energy by a factor of 10 to 100.
Cosmic-ray e x p e r i m e n t s , fortunately, have c h a n g e d dramatically
since F a t h e r Wulf first placed electrified j a r s on t h e Eiffel Tower. Rockets
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a n d even satellites can now s e n d radiation c o u n t e r s high above the
e a r t h ' s surface, so that a t m o s p h e r i c effects are minimized. W h e n a
highly energetic cosmic ray strikes t h e a t m o s p h e r e , it shatters t h e a t o m s
in its wake. These fragments, in t u r n , create a shower of b r o k e n atoms,
or ions, which can t h e n be d e t e c t e d on the g r o u n d by this series of
detectors. A collaboration between the University of Chicago a n d the
University of Michigan has i n a u g u r a t e d t h e most ambitious cosmic-ray
project yet, a vast array of 1,089 detectors scattered over a b o u t a s q u a r e
mile of desert, waiting for t h e cosmic-ray showers to trigger t h e m . T h e s e
detectors are located in an ideal, isolated area: the Dugway Proving
G r o u n d s , 80 miles southwest of Salt Lake City, U t a h .
T h e U t a h d e t e c t o r is sensitive e n o u g h to identify t h e p o i n t of origin
of some of the most energetic cosmic rays. So far, Cygnus X-3 a n d Hercules X-l have b e e n identified as powerful cosmic-ray emitters. T h e y a r e
probably large, s p i n n i n g n e u t r o n stars, or even black holes, that a r e
slowly eating up a c o m p a n i o n star, creating a large vortex of energy a n d
spewing gigantic quantities of radiation (for e x a m p l e , p r o t o n s ) into
o u t e r space.
So far, the most energetic cosmic ray ever d e t e c t e d h a d an energy of
1 0 electron volts. This figure is an incredible 10 million t i m e s the
energy that would have b e e n p r o d u c e d in the SSC. We do n o t expect to
g e n e r a t e energies a p p r o a c h i n g this cosmic energy with o u r m a c h i n e s
within the century. A l t h o u g h this fantastic energy is still 100 million
times smaller t h a n the energy necessary to p r o b e t h e t e n t h d i m e n s i o n ,
we h o p e that energies p r o d u c e d d e e p within black holes in o u r galaxy
will a p p r o a c h the Planck energy. With large, orbiting spacecraft, we
should be able to p r o b e d e e p e r into t h e structure of these energy sources
a n d detect energies even larger t h a n this.
According to o n e favored theory, t h e largest energy s o u r c e within
o u r Milky Way galaxy—far b e y o n d a n y t h i n g p r o d u c e d by Cygnus X-3 or
Hercules X-1—lies at the center, which may consist of millions of black
holes. So, because the SSC was canceled by Congress, we may find t h a t
the ultimate p r o b e for e x p l o r i n g t h e t e n t h d i m e n s i o n may lie in o u t e r
space.
20
Testing the Untestable
Historically speaking, t h e r e have b e e n m a n y times w h e n physicists have
solemnly declared certain p h e n o m e n a t o b e " u n t e s t a b l e " o r " u n p r o v a b l e . " But t h e r e is a n o t h e r attitude t h a t scientists can take c o n c e r n i n g
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t h e inaccessibility of t h e Planck e n e r g y — u n f o r e s e e n b r e a k t h r o u g h s will
m a k e indirect e x p e r i m e n t s possible n e a r the Planck energy.
In t h e n i n e t e e n t h century, s o m e scientists declared that the composition of t h e stars would forever be b e y o n d the reach of e x p e r i m e n t .
In 1825, t h e F r e n c h p h i l o s o p h e r a n d social critic Auguste C o m t e , writing
in Cours de philosophie, declared that we would never know the stars o t h e r
t h a n as u n r e a c h a b l e points of light in the sky because of their e n o r m o u s
distance from us. T h e m a c h i n e s of t h e n i n e t e e n t h century, or any century, he a r g u e d , were n o t powerful e n o u g h to escape from t h e earth a n d
reach t h e stars.
A l t h o u g h d e t e r m i n i n g what t h e stars were m a d e of s e e m e d beyond
the capabilities of any science, ironically at almost t h e same time, the
G e r m a n physicist J o s e p h von F r a u n h o f e r was d o i n g j u s t that. Using a
prism a n d spectroscope, he could separate the white light emitted from
the distant stars a n d d e t e r m i n e t h e chemical composition of those stars.
Since each chemical within t h e stars emits a characteristic " f i n g e r p r i n t , "
or s p e c t r u m of light, it was easy for F r a u n h o f e r to p e r f o r m the " i m p o s s i b l e " a n d to d e t e r m i n e t h a t h y d r o g e n is the most a b u n d a n t e l e m e n t in
the stars.
This, in t u r n , inspired p o e t Ian D. Bush to write:
T w i n k l e , t w i n k l e little star
I d o n ' t w o n d e r what y o u are,
For by spectroscopic ken,
I k n o w that y o u are h y d r o g e n .
6
T h u s a l t h o u g h t h e energy necessary to reach the stars via rockets was far
b e y o n d a n y t h i n g available to C o m t e (or, for that matter, anything available to m o d e r n science), t h e crucial step did n o t involve energy. T h e
key observation was that signals from the stars, r a t h e r t h a n direct meas u r e m e n t , were sufficient to solve the p r o b l e m . Similarly, we can h o p e
that signals from the Planck energy ( p e r h a p s from cosmic rays or perh a p s an as yet u n k n o w n source), r a t h e r t h a n a direct m e a s u r e m e n t from
large a t o m smashers, may be sufficient to p r o b e the t e n t h
dimension.
A n o t h e r e x a m p l e of an " u n t e s t a b l e " idea was t h e existence of atoms.
In t h e n i n e t e e n t h century, the atomic hypothesis proved to be the decisive step in u n d e r s t a n d i n g the laws of chemistry a n d t h e r m o d y n a m i c s .
However, m a n y physicists refused to believe t h a t a t o m s actually exist.
P e r h a p s they were j u s t a mathematical device that, by accident, gave the
c o r r e c t description of the world. For e x a m p l e , t h e p h i l o s o p h e r Ernst
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Mach did n o t believe in the existence of atoms, o t h e r t h a n as a calculational tool. (Even today, we are still u n a b l e to take direct pictures of
the a t o m because of t h e H e i s e n b e r g Uncertainty Principle, a l t h o u g h
indirect m e t h o d s now exist.) In 1905, however, Einstein gave the most
convincing, a l t h o u g h indirect, evidence of the existence of a t o m s w h e n
he showed that Brownian m o t i o n (that is, the r a n d o m m o t i o n of dust
particles s u s p e n d e d in a liquid) can be explained as r a n d o m collisions
between the particles a n d a t o m s in the liquid.
By analogy, we m i g h t h o p e for e x p e r i m e n t a l confirmation of t h e
physics of the t e n t h d i m e n s i o n using indirect m e t h o d s that have n o t yet
b e e n discovered. Instead of p h o t o g r a p h i n g the object we desire, p e r h a p s
we should be satisfied with a p h o t o g r a p h of its " s h a d o w . " T h e indirect
a p p r o a c h would be to e x a m i n e carefully low-energy data from an a t o m
smasher, a n d try to see if ten-dimensional physics affects the data in s o m e
way.
T h e third " u n t e s t a b l e " idea in physics was the existence of t h e elusive n e u t r i n o .
In 1930, physicist Wolfgang Pauli hypothesized a new, u n s e e n particle called the neutrino in o r d e r to a c c o u n t for t h e missing c o m p o n e n t
of energy in certain e x p e r i m e n t s on radioactivity that s e e m e d to violate
the conservation of m a t t e r a n d energy. Pauli realized, t h o u g h , that neutrinos would be almost impossible to observe experimentally, because
they would interact so weakly, a n d h e n c e so rarely, with matter. For
example, if we could construct a solid block of lead that stretched several
light-years from o u r solar system to Alpha C e n t a u r i a n d placed it in t h e
p a t h of a b e a m of n e u t r i n o s , some would still c o m e o u t t h e o t h e r e n d .
They can p e n e t r a t e t h e earth as t h o u g h it d o e s n ' t even exist, a n d , in
fact, trillions of n e u t r i n o s emitted from the sun are always p e n e t r a t i n g
your body, even at night. Pauli a d m i t t e d , "I have c o m m i t t e d t h e ultimate
sin, I have p r e d i c t e d t h e existence of a particle that can never be
observed."
7
So elusive a n d u n d e t e c t a b l e was t h e n e u t r i n o that it even inspired a
p o e m by J o h n U p d i k e , called " C o s m i c Gall":
N e u t r i n o s , they are very small.
T h e y have no charge a n d have no mass
A n d d o n o t i n t e r a c t a t all.
T h e e a r t h is j u s t a silly b a l l
T o t h e m , t h r o u g h w h i c h they simply pass,
L i k e d u s t m a i d s d o w n a drafty h a l l
Or p h o t o n s t h o u g h a s h e e t of glass.
T h e y s n u b the m o s t exquisite gas,
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I g n o r e t h e m o s t s u b s t a n t i a l wall,
C o l d - s h o u l d e r s t e e l a n d s o u n d i n g brass,
I n s u l t t h e s t a l l i o n i n h i s stall,
A n d s c o r n i n g barriers o f class,
I n f i l t r a t e y o u a n d m e ! L i k e tall
A n d p a i n l e s s g u i l l o t i n e s , t h e y fall
D o w n t h r o u g h o u r h e a d s i n t o t h e grass.
At night, they enter at Nepal
A n d p i e r c e t h e l o v e r a n d h i s lass
F r o m u n d e r n e a t h t h e b e d — y o u call
It w o n d e r f u l ; I call it crass.
8
A l t h o u g h the n e u t r i n o , because it barely interacts with o t h e r materials, was o n c e c o n s i d e r e d t h e ultimate " u n t e s t a b l e " idea, today we regularly p r o d u c e b e a m s of n e u t r i n o s in a t o m smashers, perform experim e n t s with t h e n e u t r i n o s emitted from a n u c l e a r reactor, a n d detect
their p r e s e n c e within m i n e s far below t h e e a r t h ' s surface. (In fact, when
a spectacular supernova lit up the sky in the s o u t h e r n h e m i s p h e r e in
1987, physicists noticed a burst of n e u t r i n o s streaming t h r o u g h their
d e t e c t o r s d e e p in these mines. This was the first time that n e u t r i n o detectors were used to m a k e crucial astronomical m e a s u r e m e n t s . ) Neutrinos,
in 3 s h o r t decades, have b e e n transformed from an " u n t e s t a b l e " idea
into o n e of the workhorses of m o d e r n physics.
The Problem Is Theoretical, Not Experimental
T a k i n g the l o n g view on the history of science, p e r h a p s t h e r e is some
cause for optimism. Witten is convinced that science will some day be
able to p r o b e down to Planck energies. He says,
It's n o t always s o e a s y t o tell w h i c h a r e t h e e a s y q u e s t i o n s a n d w h i c h a r e
the hard o n e s . In the 19th century, the q u e s t i o n of why water boils at 100
d e g r e e s was h o p e l e s s l y inaccessible. If y o u t o l d a 19th-century physicist that
by t h e 2 0 t h c e n t u r y y o u w o u l d be able to c a l c u l a t e this, it w o u l d have
s e e m e d l i k e a fairy tale. . . . Q u a n t u m f i e l d t h e o r y is so d i f f i c u l t t h a t n o b o d y
fully b e l i e v e d i t f o r 2 5 y e a r s .
9
In his view, " g o o d ideas always get t e s t e d . "
T h e a s t r o n o m e r A r t h u r E d d i n g t o n even q u e s t i o n e d w h e t h e r scientists were n o t overstating the case w h e n they insisted that everything
s h o u l d be tested. He wrote: "A scientist c o m m o n l y professes to base his
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beliefs on observations, n o t theories. . . . I have never c o m e across anyo n e who carries this profession into practice. . . . Observation is n o t sufficient . . . theory has an i m p o r t a n t share in d e t e r m i n i n g belief." N o b e l
laureate Paul Dirac said it even m o r e bluntly, " I t is m o r e i m p o r t a n t to
have beauty in o n e ' s e q u a t i o n s t h a n to have t h e m fit e x p e r i m e n t . " " O r ,
in the words of CERN physicist J o h n Ellis, " i n the words of a candy
w r a p p e r I o p e n e d a few years ago: 'It is only the optimists w h o achieve
anything in this world.' " Nonetheless, despite a r g u m e n t s that u p h o l d a
certain d e g r e e of optimism, the e x p e r i m e n t a l situation looks bleak. I
share, along with the skeptics, t h e idea that t h e best we can h o p e for is
indirect tests of ten-dimensional theory into t h e twenty-first century. This
is because, in t h e final analysis, this theory is a theory of Creation, a n d
h e n c e testing it necessarily involves re-creating a piece of the Big Bang
in o u r laboratories.
10
Personally, I d o n ' t think that we have to wait a century until o u r
accelerators, space p r o b e s , a n d cosmic-ray c o u n t e r s will be powerful
e n o u g h to p r o b e the t e n t h d i m e n s i o n indirectly. Within a span of years,
a n d certainly within the lifetime of today's physicists, s o m e o n e will be
clever e n o u g h to e i t h e r verify or disprove t h e ten-dimensional theory by
solving the field theory of strings or some o t h e r n o n p e r t u r b a t i v e formulation. T h e p r o b l e m is thus theoretical, n o t e x p e r i m e n t a l .
Assuming that some b r i g h t physicist solves the field theory of strings
a n d derives the known p r o p e r t i e s of o u r universe, t h e r e is still t h e practical p r o b l e m of when we m i g h t be able to h a r n e s s the power of t h e
hyperspace theory. T h e r e are two possibilities:
1. Wait until o u r civilization attains t h e ability to master energies
trillions of times larger t h a n a n y t h i n g we can p r o d u c e today
2. E n c o u n t e r extraterrestrial civilizations that have m a s t e r e d t h e
art of m a n i p u l a t i n g hyperspace
We recall that it took a b o u t 70 years, between t h e work of Faraday
a n d Maxwell to t h e work of Edison a n d his co-workers, to exploit t h e
electromagnetic force for practical p u r p o s e s . Yet m o d e r n civilization
d e p e n d s crucially on t h e h a r n e s s i n g of this force. T h e n u c l e a r force was
discovered n e a r the t u r n of t h e century, a n d 80 years later we still do
n o t have the m e a n s to harness it successfully with fusion reactors. T h e
next leap, to harness the power of t h e unified field theory, requires a
m u c h greater j u m p in o u r technology, b u t o n e that will probably have
vastly m o r e i m p o r t a n t implications.
T h e f u n d a m e n t a l p r o b l e m is that we are forcing superstring theory
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to answer questions a b o u t everyday energies, w h e n its " n a t u r a l h o m e "
lies at t h e Planck energy. This fabulous energy was released only at the
instant of Creation itself. In o t h e r words, superstring theory is naturally
a theory of Creation. Like the caged c h e e t a h , we are d e m a n d i n g that
this s u p e r b animal d a n c e a n d sing for o u r e n t e r t a i n m e n t . T h e real h o m e
of t h e c h e e t a h is t h e vast plains of Africa. T h e real " h o m e " of superstring theory is t h e instant of Creation. Nevertheless, given the sophistication of o u r artificial satellites, t h e r e is p e r h a p s o n e last " l a b o r a t o r y "
in which we may experimentally p r o b e the natural h o m e of superstring
theory, a n d this is the e c h o of Creation!
9
Before Creation
In t h e b e g i n n i n g , was t h e great c o s m i c e g g . Inside t h e e g g was
c h a o s , a n d f l o a t i n g i n c h a o s was P ' a n K u , t h e d i v i n e E m b r y o .
P'an K u m y t h ( C h i n a , t h i r d century)
If G o d created t h e world, w h e r e was He b e f o r e Creation? . . .
Know
that
t h e w o r l d is u n c r e a t e d , as t i m e i t s e l f is, w i t h o u t
beginning and end.
Mahapurana
(India,
ninth
century)
ID God have a m o t h e r ? "
Children, w h e n told that G o d m a d e t h e heavens a n d t h e
earth, innocently ask w h e t h e r G o d h a d a m o t h e r . This deceptively simple question has s t u m p e d the elders of the c h u r c h a n d e m b a r r a s s e d t h e
finest theologians, precipitating s o m e of t h e thorniest theological
debates over t h e centuries. All the great religions have elaborate mythologies s u r r o u n d i n g the divine act of Creation, b u t n o n e of t h e m adequately confronts t h e logical p a r a d o x e s i n h e r e n t in t h e questions that
even children ask.
God may have created the heavens a n d the e a r t h in 7 days, b u t what
h a p p e n e d before t h e first day? If o n e c o n c e d e s that G o d h a d a m o t h e r ,
t h e n o n e naturally asks w h e t h e r she, too, h a d a m o t h e r , a n d so o n ,
forever. However, if G o d did n o t have a m o t h e r , t h e n this answer raises
even m o r e questions: W h e r e did G o d c o m e from? Was G o d always in
existence since eternity, or is G o d b e y o n d time itself?
" D
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Over t h e centuries, even great painters commissioned by the c h u r c h
g r a p p l e d with these ticklish theological debates in their works of art:
W h e n d e p i c t i n g G o d or A d a m a n d Eve, do you give t h e m belly buttons?
Since the navel marks t h e p o i n t of a t t a c h m e n t of the umbilical cord,
t h e n n e i t h e r G o d n o r A d a m a n d Eve could be p a i n t e d with belly buttons.
For e x a m p l e , Michelangelo faced this d i l e m m a in his celebrated depiction of Creation a n d the expulsion of A d a m a n d Eve from the G a r d e n
of E d e n w h e n he p a i n t e d the ceiling of t h e Sistine Chapel. T h e answer
to this theological question is to be f o u n d h a n g i n g in any large m u s e u m :
G o d a n d A d a m a n d Eve simply have no belly b u t t o n s , because they were
the first.
Proofs of the Existence of God
T r o u b l e d by the inconsistencies in c h u r c h ideology, St. T h o m a s Aquinas,
writing in the t h i r t e e n t h century, d e c i d e d to raise t h e level of theological
d e b a t e from the vagueness of mythology to t h e rigor of logic. He proposed to solve these a n c i e n t questions in his celebrated "proofs of the
existence of G o d . "
Aquinas s u m m a r i z e d his proofs in the following p o e m :
T h i n g s a r e i n m o t i o n , h e n c e t h e r e i s a first m o v e r
T h i n g s a r e c a u s e d , h e n c e t h e r e i s a first c a u s e
T h i n g s exist, h e n c e there is a creator
Perfect g o o d n e s s exists, h e n c e it has a s o u r c e
T h i n g s are d e s i g n e d , h e n c e they serve a p u r p o s e .
1
( T h e first t h r e e lines are variations of what is called the cosmological proof;
the fourth argues on moral g r o u n d s ; a n d t h e fifth is called the teleological
proof. T h e m o r a l p r o o f is by far t h e weakest, because morality can be
viewed in terms of evolving social customs.)
Aquinas's " c o s m o l o g i c a l " a n d " t e l e o l o g i c a l " proofs of the existence
of G o d have b e e n used by the c h u r c h for the past 700 years to answer
this sticky theological question. A l t h o u g h these proofs have since b e e n
shown to be flawed in light of the scientific discoveries m a d e over the
past 7 centuries, they were quite i n g e n i o u s for their time a n d show the
influence of t h e Greeks, w h o were t h e first to i n t r o d u c e rigor into their
speculations a b o u t n a t u r e .
Aquinas b e g a n the cosmological p r o o f by postulating that God was
t h e First Mover a n d First Maker. He artfully d o d g e d t h e question of
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" w h o m a d e G o d " by simply asserting that the question m a d e no sense.
God h a d no m a k e r because he was the First. Period. T h e cosmological
proof states that everything that moves must have h a d s o m e t h i n g p u s h
it, which in t u r n must have h a d s o m e t h i n g p u s h it, a n d so o n . But what
started the first push?
Imagine, for the m o m e n t , idly sitting in t h e park a n d seeing a wagon
moving in front of you. Obviously, you think, t h e r e is a y o u n g child
p u s h i n g the wagon. You wait a m o m e n t , only to find a n o t h e r wagon
p u s h i n g the first wagon. Curious, you wait a bit l o n g e r for t h e child, b u t
t h e r e is a third wagon p u s h i n g the first two wagons. As time goes by, you
witness h u n d r e d s of wagons, each o n e p u s h i n g the others, with no child
in sight. Puzzled, you look o u t into the distance. You are surprised to
see an infinite s e q u e n c e of wagons stretching into the h o r i z o n , each
wagon p u s h i n g the others, with no child at all. If it takes a child to p u s h
a wagon, t h e n can an infinite s e q u e n c e of wagons be p u s h e d without
the First Pusher? Can an infinite s e q u e n c e of wagons p u s h itself? N o .
Therefore, God must exist.
T h e teleological p r o o f is even m o r e persuasive. It states that t h e r e
has to be a First Designer. For e x a m p l e , imagine walking on the sands
of Mars, where the winds a n d dust storms have worn even t h e m o u n t a i n s
a n d giant craters. Over tens of millions of years, n o t h i n g has escaped
the corrosive, g r i n d i n g effect of t h e sand storms. T h e n , to your surprise,
you find a beautiful c a m e r a lying in t h e sand d u n e s . T h e lens is smoothly
polished a n d the shutter m e c h a n i s m delicately crafted. Surely, you
think, the sands of Mars could n o t have created such a beautiful piece
of craftsmanship. You c o n c l u d e that s o m e o n e intelligent obviously m a d e
this camera. T h e n , after w a n d e r i n g on t h e surface of Mars s o m e m o r e ,
you c o m e across a rabbit. Obviously, t h e eye of t h e rabbit is infinitely
m o r e intricate t h a n the eye of the camera. T h e muscles of the rabbit's
eye are infinitely m o r e elaborate t h a n the shutter of the c a m e r a .
Therefore, the m a k e r of this rabbit must be infinitely m o r e advanced
than the m a k e r of t h e c a m e r a . This m a k e r must therefore be
God.
Now imagine the m a c h i n e s on t h e e a r t h . T h e r e is no question that
these m a c h i n e s were m a d e by s o m e t h i n g even greater, such as h u m a n s .
T h e r e is no question that a h u m a n is infinitely m o r e complicated t h a n
a m a c h i n e . T h e r e f o r e , the p e r s o n who created us must be infinitely m o r e
complicated t h a n we are. So therefore G o d m u s t exist.
In 1078, St. Anselm, the a r c h b i s h o p of Canterbury, c o o k e d up perhaps the most sophisticated p r o o f of the existence of God, the ontological
proof, which does n o t d e p e n d on First Movers or First Designers at all.
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St. Anselm claimed that he could prove the existence of God from p u r e
logic a l o n e . He defined God as t h e most perfect, most powerful b e i n g
imaginable. It is, however, possible to conceive of two types of God. T h e
first God, we imagine, does n o t exist. T h e second God, we imagine,
actually does exist a n d can p e r f o r m miracles, such as parting the
rivers a n d raising t h e d e a d . Obviously, t h e second God (who exists) is
m o r e powerful a n d m o r e perfect t h a n the first G o d (who does n o t
exist).
However, we defined G o d to be the most perfect a n d powerful b e i n g
imaginable. By t h e definition of God, the second G o d (who exists) is the
m o r e powerful a n d m o r e perfect o n e . T h e r e f o r e , the second God is the
o n e w h o fits the definition. T h e first God (who does n o t exist) is weaker
a n d less perfect t h a n the second God, a n d therefore does n o t fit the
definition of God. H e n c e G o d must exist. In o t h e r words, if we define
G o d as " t h a t b e i n g n o t h i n g g r e a t e r t h a n which can be conceived," t h e n
G o d m u s t exist because if he d i d n ' t , it's possible to conceive of a m u c h
g r e a t e r G o d w h o does exist. This r a t h e r ingenious proof, unlike those
of St. T h o m a s Aquinas, is totally i n d e p e n d e n t of the act of Creation a n d
rests solely on the definition of the perfect being.
Remarkably, these " p r o o f s " of t h e existence of God lasted for over
700 years, defying t h e r e p e a t e d challenges of scientists a n d logicians.
T h e reason for this is that n o t e n o u g h was known a b o u t the fundamental
laws of physics a n d biology. In fact, only within the past century have
new laws of n a t u r e b e e n discovered that can isolate t h e potential flaws
in these proofs.
T h e flaw in t h e cosmological proof, for e x a m p l e , is that the conservation of mass a n d energy is sufficient to explain m o t i o n without appealing to a First Mover. For e x a m p l e , gas molecules may b o u n c e against
t h e walls of a c o n t a i n e r without r e q u i r i n g a n y o n e or anything to get
t h e m moving. In principle, these molecules can move forever, r e q u i r i n g
no b e g i n n i n g or e n d . T h u s t h e r e is no necessity for a First or a Last
Mover as l o n g as mass a n d energy are conserved.
For t h e teleological proof, the theory of evolution shows that it is
possible to create h i g h e r a n d m o r e c o m p l e x life forms from m o r e primitive o n e s t h r o u g h natural selection a n d c h a n c e . Ultimately, we can trace
t h e origin of life itself back to the s p o n t a n e o u s formation of protein
molecules in t h e early e a r t h ' s o c e a n s without a p p e a l i n g to a h i g h e r intelligence. Studies p e r f o r m e d by Stanley L. Miller in 1955 have shown that
sparks sent t h r o u g h a flask c o n t a i n i n g m e t h a n e , a m m o n i a , a n d o t h e r
gases f o u n d in the early e a r t h ' s a t m o s p h e r e can spontaneously create
c o m p l e x h y d r o c a r b o n molecules a n d eventually a m i n o acids (precursors
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to p r o t e i n molecules) a n d o t h e r complex organic molecules. T h u s a
First Designer is n o t necessary to create the essentials for life, which can
apparently e m e r g e naturally o u t of inorganic chemicals if they are given
e n o u g h time.
And, finally, I m m a n u e l Kant was the first to isolate t h e e r r o r in the
ontological p r o o f after centuries of confusion. Kant p o i n t e d o u t that
stating that an object exists does n o t make it m o r e perfect. For e x a m p l e ,
this p r o o f can be used to prove the existence of the u n i c o r n . If we define
the u n i c o r n to be t h e most perfect horse imaginable, a n d if u n i c o r n s
d o n ' t exist, t h e n it's possible to imagine a u n i c o r n that d o e s exist. But
saying that it exists d o e s n o t m e a n that it is m o r e perfect t h a n a u n i c o r n
that does n o t exist. T h e r e f o r e , u n i c o r n s do n o t necessarily have to exist.
And n e i t h e r d o e s God.
Have we m a d e any progress since the time of St. T h o m a s Aquinas
a n d St. Anselm?
Yes a n d n o . We can say that present-day theories of Creation are built
on two pillars: q u a n t u m theory a n d Einstein's theory of gravity. We can
say that, for the first time in a t h o u s a n d years, religious " p r o o f s " of t h e
existence of God are b e i n g replaced by o u r u n d e r s t a n d i n g of t h e r m o dynamics a n d particle physics. However, by replacing G o d ' s act of Creation with the Big Bang, we have s u p p l a n t e d o n e p r o b l e m with a n o t h e r .
Aquinas t h o u g h t he solved t h e p r o b l e m of what came before God by
defining h i m as the First Mover. Today, we are still struggling with t h e
question of what h a p p e n e d before the Big Bang.
Unfortunately, Einstein's equations break down at the e n o r m o u s l y
small distances a n d large energies found at the origin of the universe.
At distances on t h e o r d e r of 1 0
c e n t i m e t e r , q u a n t u m effects take over
from Einstein's theory. T h u s to resolve the philosophical questions
involving the b e g i n n i n g of time, we m u s t necessarily invoke the tendimensional theory.
T h r o u g h o u t this book, we have emphasized the fact that the laws of
physics unify when we a d d h i g h e r dimensions. W h e n studying the Big
Bang, we see the precise reverse of this statement. T h e Big Bang, as we
shall see, p e r h a p s originated in the b r e a k d o w n of t h e original tendimensional universe into a four- a n d a six-dimensional universe. T h u s
we can view the history of the Big Bang as the history of the b r e a k u p of
ten-dimensional space a n d h e n c e the b r e a k u p of previously unified symmetries. This, in t u r n , is the t h e m e of this b o o k in reverse.
It is no w o n d e r , therefore, that piecing t o g e t h e r the dynamics of t h e
Big Bang has b e e n so difficult. In effect, by going backward in time, we
are reassembling the pieces of t h e ten-dimensional universe.
- 3 3
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Experimental Evidence for the Big Bang
Every year, we find m o r e e x p e r i m e n t a l evidence that the Big Bang
o c c u r r e d roughly 15 to 20 billion years ago. Let us review some of these
e x p e r i m e n t a l results.
First, t h e fact that the stars are r e c e d i n g from us at fantastic velocities
has b e e n repeatedly verified by m e a s u r i n g the distortion of their starlight (called the r e d shift). ( T h e starlight of a r e c e d i n g star is shifted to
l o n g e r wavelengths—that is, toward t h e r e d e n d of the s p e c t r u m — i n the
same way that t h e whistle of a r e c e d i n g train s o u n d s h i g h e r t h a n n o r m a l
w h e n a p p r o a c h i n g a n d lower when r e c e d i n g . This is called the D o p p l e r
effect. Also, H u b b l e ' s Law states that the farther from us the star or
galaxy, t h e faster it is r e c e d i n g from us. This fact, first a n n o u n c e d by the
a s t r o n o m e r Edwin H u b b l e in 1929, has b e e n experimentally verified
over t h e past 50 years.) We do n o t see any blue shift of the distant galaxies, which would m e a n a collapsing universe.
Second, we know that t h e distribution of t h e chemical elements in
o u r galaxy are in almost exact a g r e e m e n t with the prediction of heavye l e m e n t p r o d u c t i o n in t h e Big Bang a n d in t h e stars. In the original Big
Bang, because of the e n o r m o u s heat, e l e m e n t a l h y d r o g e n nuclei b a n g e d
into o n e a n o t h e r at large e n o u g h velocities to fuse t h e m , forming a new
e l e m e n t : h e l i u m . T h e Big Bang theory predicts that the ratio of helium
to h y d r o g e n in t h e universe should be approximately 2 5 % helium to
7 5 % h y d r o g e n . This agrees with the observational result for the abund a n c e of h e l i u m in t h e universe.
T h i r d , t h e earliest objects in t h e universe d a t e back 10 to 15 billion
years, in a g r e e m e n t with t h e r o u g h estimate for t h e Big Bang. We do
n o t see any evidence for objects o l d e r t h a n the Big Bang. Since radioactive materials decay (for e x a m p l e , via t h e weak interactions) at a precisely known rate, it is possible to tell t h e age of an object by calculating
t h e relative a b u n d a n c e of certain radioactive materials. For example,
half of a radioactive substance called carbon-14 decays every 5,730 years,
which allows us to d e t e r m i n e the age of archeological artifacts that contain c a r b o n . O t h e r radioactive e l e m e n t s (like u r a n i u m - 2 3 8 , with a halflife of over 4 billion years) allow us to d e t e r m i n e t h e age of m o o n rocks
(from t h e Apollo mission). T h e oldest rocks a n d m e t e o r s found on earth
d a t e to a b o u t 4 to 5 billion years, which is t h e a p p r o x i m a t e age of the
solar system. By calculating the mass of certain stars whose evolution is
known, we can show t h a t the oldest stars in o u r galaxy date back a b o u t
10 billion years.
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F o u r t h , a n d m o s t i m p o r t a n t , the Big Bang p r o d u c e d a cosmic
" e c h o " reverberating t h r o u g h o u t the universe that s h o u l d b e measurable by o u r i n s t r u m e n t s . In fact, A r n o Penzias a n d R o b e r t Wilson of t h e
Bell T e l e p h o n e Laboratories won the Nobel Prize in 1978 for d e t e c t i n g
this e c h o of t h e Big Bang, a microwave radiation that p e r m e a t e s t h e
known universe. T h e fact that the e c h o of the Big B a n g s h o u l d be circulating a r o u n d t h e universe billions of years after t h e event was first
predicted by G e o r g e Gamow a n d his students Ralph A l p h e r a n d R o b e r t
H e r m a n , b u t n o o n e took t h e m seriously. T h e very idea o f m e a s u r i n g
the e c h o of Creation s e e m e d outlandish w h e n they first p r o p o s e d this
idea soon after World War II.
T h e i r logic, however, was very compelling. Any object, w h e n h e a t e d ,
gradually emits radiation. This is the reason why iron gets r e d h o t when
placed in a furnace. T h e h o t t e r t h e iron, the h i g h e r t h e frequency of
radiation it emits. A precise m a t h e m a t i c a l formula, the Stefan-Boltzm a n n law, relates the frequency of light (or the color, in this case) to
the t e m p e r a t u r e . (In fact, this is how scientists d e t e r m i n e t h e surface
t e m p e r a t u r e of a distant star, by e x a m i n i n g its color.) This radiation is
called blackbody radiation.
W h e n the iron cools, the frequency of the e m i t t e d radiation also
decreases, until the iron no l o n g e r emits in the visible r a n g e . T h e iron
r e t u r n s to its n o r m a l color, b u t it c o n t i n u e s to emit invisible infrared
radiation. This is how the army's n i g h t glasses o p e r a t e in the dark. At
night, relatively warm objects such as e n e m y soldiers a n d tank e n g i n e s
may be concealed in the darkness, b u t they c o n t i n u e to e m i t invisible
blackbody radiation in t h e form of infrared radiation, which can be
picked up by special infrared goggles. This is also why your sealed car
gets h o t d u r i n g t h e s u m m e r . Sunlight p e n e t r a t e s t h e glass of your car
a n d heats the interior. As it gets hot, it begins to e m i t blackbody radiation in the form of infrared radiation. However, infrared radiation does
n o t p e n e t r a t e glass very well, a n d h e n c e is t r a p p e d inside your car, dramatically raising its t e m p e r a t u r e . (Similarly, blackbody radiation drives
the g r e e n h o u s e effect. Like glass, rising levels of c a r b o n dioxide in t h e
a t m o s p h e r e , caused by the b u r n i n g of fossil fuels, can t r a p the infrared
blackbody radiation of t h e e a r t h a n d thereby gradually h e a t t h e
planet.)
Gamow r e a s o n e d that the Big B a n g was initially quite h o t , a n d h e n c e
would be an ideal blackbody e m i t t e r of radiation. A l t h o u g h t h e technology of the 1940s was too primitive to pick up this faint signal from
Creation, he could calculate the t e m p e r a t u r e of this radiation a n d con-
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fidently p r e d i c t that o n e day o u r i n s t r u m e n t s would be sensitive e n o u g h
to d e t e c t this "fossil" radiation. T h e logic b e h i n d his thinking was as
follows: A b o u t 300,000 years after the Big Bang, the universe cooled to
t h e p o i n t w h e r e a t o m s could begin to c o n d e n s e ; electrons could begin
to circle p r o t o n s a n d form stable a t o m s that would no longer be b r o k e n
up by t h e i n t e n s e radiation p e r m e a t i n g the universe. Before this time,
t h e universe was so h o t t h a t a t o m s were continually r i p p e d a p a r t by
radiation as soon as they were formed. This m e a n t that t h e universe was
o p a q u e , like a thick, absorbing, a n d i m p e n e t r a b l e fog. After 300,000
years, however, t h e radiation was no l o n g e r sufficiently strong to break
up t h e atoms, a n d h e n c e light could travel long distances without b e i n g
scattered. In o t h e r words, t h e universe suddenly b e c a m e black a n d transp a r e n t after 300,000 years. (We are so used to h e a r i n g a b o u t the "blackness of o u t e r s p a c e " that we forget that t h e early universe was n o t transp a r e n t at all, b u t filled with t u r b u l e n t , o p a q u e radiation.)
After 300,000 years, electromagnetic radiation no l o n g e r interacted
so strongly with matter, a n d h e n c e b e c a m e blackbody radiation. Gradually, as t h e universe cooled, the frequency of this radiation decreased.
Gamow a n d his students calculated that the radiation would be far below
the infrared r a n g e , into t h e microwave region. Gamow r e a s o n e d that by
s c a n n i n g t h e heavens for a uniform, isotropic source of microwave radiation, o n e should be able to d e t e c t this microwave radiation a n d discover
the e c h o of t h e Big Bang.
Gamow's p r e d i c t i o n was forgotten for many decades, until the microwave b a c k g r o u n d radiation was discovered quite by accident in 1965.
Penzias a n d Wilson f o u n d a mysterious b a c k g r o u n d radiation permeating all space w h e n they t u r n e d on their new h o r n reflector a n t e n n a in
H o l m d e l , New Jersey. At first, they t h o u g h t this u n w a n t e d radiation was
d u e to electrical static caused by c o n t a m i n a n t s , such as bird d r o p p i n g s
on their a n t e n n a . But w h e n they disassembled a n d cleaned large portions of t h e a n t e n n a , they f o u n d that t h e " s t a t i c " persisted. At the same
time, physicists R o b e r t Dicke a n d J a m e s Peebles at P r i n c e t o n University
were r e t h i n k i n g Gamow's old calculation. W h e n Penzias a n d Wilson
were finally i n f o r m e d of t h e P r i n c e t o n physicists' work, it was clear that
t h e r e was a direct relationship between their results. W h e n they realized
that this b a c k g r o u n d radiation m i g h t be the e c h o of the original Big
Bang, they are said to have exclaimed, " E i t h e r we've seen a pile of bird
s
t,
or the creation of t h e universe!" T h e y discovered that this uniform b a c k g r o u n d radiation was almost exactly what h a d b e e n p r e d i c t e d
years earlier by G e o r g e Gamow a n d his collaborators if t h e Big Bang h a d
left a residual b l a n k e t of radiation that h a d cooled down to 3°K.
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COBE and the Big Bang
P e r h a p s the most spectacular scientific confirmation of t h e Big B a n g
theory c a m e in 1992 with t h e results of t h e COBE (Cosmic Background
Explorer) satellite. On April 23, newspaper h e a d l i n e s across the country
h e r a l d e d the findings of a t e a m of scientists at t h e University of California at Berkeley, led by G e o r g e Smoot, w h o a n n o u n c e d t h e most dramatic, convincing a r g u m e n t for the Big Bang theory. Journalists a n d
columnists, with no b a c k g r o u n d in physics or theology, were suddenly
waxing e l o q u e n t a b o u t t h e "face of G o d " in their dispatches.
T h e COBE satellite was able to improve vastly t h e earlier work of
Penzias, Wilson, Peebles, a n d Dicke by many o r d e r s of m a g n i t u d e , sufficient to rule o u t all d o u b t that t h e fossil radiation e m i t t e d by t h e Big
Bang h a d b e e n conclusively found. P r i n c e t o n cosmologist J e r e m i a h P.
Ostriker declared, " W h e n fossils were f o u n d in t h e rocks, it m a d e t h e
origin of species absolutely clear-cut. Well, COBE f o u n d its fossils."
L a u n c h e d in late 1989, the COBE satellite was specifically designed to
analyze the microscopic details in t h e structure of the microwave backg r o u n d radiation first postulated by G e o r g e Gamow a n d his colleagues.
T h e mission of COBE also h a d a new task: to resolve an earlier puzzle
arising from the b a c k g r o u n d radiation.
2
T h e original work of Penzias a n d Wilson was c r u d e ; they could show
only that the b a c k g r o u n d radiation was s m o o t h to 10%. W h e n scientists
analyzed the b a c k g r o u n d radiation in m o r e detail, they f o u n d that it was
exceptionally s m o o t h , with no a p p a r e n t ripples, kinks, or blotches. In
fact, it was too s m o o t h . T h e b a c k g r o u n d radiation was like a s m o o t h ,
invisible fog filling up t h e universe, so u n i f o r m that scientists h a d difficulty reconciling it with known astronomical data.
In the 1970s, a s t r o n o m e r s t u r n e d their great telescopes to systematically m a p e n o r m o u s collections of galaxies across large p o r t i o n s of t h e
sky. To their surprise, they f o u n d that, 1 billion years after the Big Bang,
the universe h a d already exhibited a p a t t e r n of c o n d e n s i n g into galaxies
a n d even large clusters of galaxies a n d h u g e , empty spaces called voids.
T h e clusters were e n o r m o u s , c o n t a i n i n g billions of galaxies at a time,
a n d the voids stretched across millions of light-years.
But h e r e lay a cosmic mystery: If t h e Big Bang was exceptionally
s m o o t h a n d uniform, t h e n 1 billion years was n o t e n o u g h time to
develop the c l u m p i n e s s that we see a m o n g t h e galactic clusters. T h e
gross mismatch between t h e original s m o o t h Big B a n g a n d t h e lumpiness of the universe 1 billion years later was a n a g g i n g p r o b l e m that
gnawed at every cosmologist. T h e Big Bang theory itself was never in any
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d o u b t ; what was in trouble was o u r u n d e r s t a n d i n g of t h e post-Big Bang
evolution 1 billion years after Creation. But without sensitive satellites
that could m e a s u r e the cosmic b a c k g r o u n d radiation, the p r o b l e m fest e r e d over the years. In fact, by 1990, journalists without a rigorous scie n c e b a c k g r o u n d b e g a n to write sensational articles saying erroneously
that scientists h a d f o u n d a fatal flaw in the Big Bang theory itself. Many
j o u r n a l i s t s wrote that the Big Bang theory was a b o u t to be overthrown.
Long-discredited alternatives to the Big Bang theory began to resurface
in t h e press. Even t h e New York Times p u b l i s h e d a major article saying
t h a t the Big Bang theory was in serious trouble (which was scientifically
incorrect).
This pseudocontroversy s u r r o u n d i n g t h e Big Bang theory m a d e t h e
a n n o u n c e m e n t of t h e COBE data all t h e m o r e interesting. With u n p r e c e d e n t e d accuracy, capable of d e t e c t i n g variations as small as o n e part in
100,000, t h e COBE satellite was able to scan t h e heavens a n d radio back
the most accurate m a p of t h e cosmic b a c k g r o u n d radiation ever constructed. T h e COBE results r e c o n f i r m e d the Big Bang theory, a n d m o r e .
COBEs data, however, were n o t easy to analyze. T h e team led by
S m o o t h a d to face e n o r m o u s p r o b l e m s . For e x a m p l e , they h a d to subtract carefully t h e effect of the e a r t h ' s m o t i o n in the b a c k g r o u n d radiation. T h e solar system drifts at a velocity of 370 kilometers p e r second
relative to t h e b a c k g r o u n d radiation. T h e r e is also t h e relative motion
of t h e solar system with respect to t h e galaxy, a n d the galaxy's complex
m o t i o n s with respect to galactic clusters. Nevertheless, after painstaking
c o m p u t e r e n h a n c e m e n t , several s t u n n i n g results c a m e o u t of the analysis. First, the microwave b a c k g r o u n d fit the earlier prediction of George
Gamow (adjusted with m o r e accurate e x p e r i m e n t a l n u m b e r s ) to within
0 . 1 % (Figure 9.1). T h e solid line represents t h e prediction; the x's mark
t h e data p o i n t s m e a s u r e d by t h e COBE satellite. W h e n this g r a p h was
flashed on the screen for the first time to a m e e t i n g of a b o u t a t h o u s a n d
a s t r o n o m e r s , everyone in t h e r o o m e r u p t e d in a s t a n d i n g ovation. This
was p e r h a p s t h e first time in the history of science that a simple g r a p h
received such a t h u n d e r o u s applause from so many distinguished scientists.
Second, S m o o t ' s team was able to show that tiny, almost microscopic
blotches did, in fact, a p p e a r in the microwave b a c k g r o u n d . These tiny
blotches were precisely what was n e e d e d to explain the clumpiness a n d
voids f o u n d 1 billion years after the Big Bang. (If these blotches h a d n o t
b e e n f o u n d by COBE, t h e n a major revision in the post-Big Bang analysis
would have h a d to be m a d e . )
T h i r d , the results were consistent with, b u t did n o t prove, the so-
Before Creation
201
Intensity
F r e q u e n c y
o f c o s m i c
b a c k g r o u n d
r a d i a t i o n
Figure 9.1. The solid line represents the prediction made by the Big Bang theory,
which predicts that the background cosmic radiation should resemble blackbody
radiation in the microwave region. The x's represent the actual data collected by
the COBE satellite, giving us one of the most convincing proofs of the Big Bang
theory.
called inflation theory. (This theory, p r o p o s e d by Alan G u t h of MIT, states
that t h e r e was a m u c h m o r e explosive e x p a n s i o n of t h e universe at the
initial instant of Creation t h a n the usual Big B a n g scenario; it h o l d s that
the visible universe we see with o u r telescopes is only the tiniest p a r t of
a m u c h bigger universe whose b o u n d a r i e s lie beyond o u r visible horizon.)
Before Creation: Orbifolds?
T h e COBE results have given physicists confidence that we u n d e r s t a n d
the origin of the universe to within a fraction of a second after the Big
Bang. However, we are still left with the embarrassing questions of what
p r e c e d e d the Big Bang a n d why it o c c u r r e d . G e n e r a l relativity, if taken
to its limits, ultimately yields nonsensical answers. Einstein, realizing that
general relatively simply breaks down at those e n o r m o u s l y small dis-
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tances, tried to e x t e n d general relativity into a m o r e comprehensive theory that could explain these p h e n o m e n a .
At t h e instant of the Big Bang, we expect q u a n t u m effects to be the
d o m i n a n t force, overwhelming gravity. T h e key to the origin of the Big
Bang, therefore, is a q u a n t u m theory of gravity. So far, the only theory
that can claim to solve the mystery of what h a p p e n e d before the Big
B a n g is t h e ten-dimensional superstring theory. Scientists are j u s t now
conjecturing how the ten-dimensional universe split into a four- a n d a
six-dimensional universe. W h a t does o u r twin universe look like?
O n e physicist w h o is struggling with these cosmic questions is Cumr u m Vafa, a Harvard professor w h o has s p e n t several years studying how
o u r ten-dimensional universe may have b e e n torn into two smaller universes. He is, ironically, also a physicist torn between two worlds. Living
in C a m b r i d g e , Massachusetts, Vafa is originally from Iran, which has
b e e n racked by political convulsions for the past d e c a d e . On the o n e
h a n d , he wishes eventually to r e t u r n to his native Iran, p e r h a p s when
t h e social t u m u l t has calmed down. On the o t h e r h a n d , his research
takes h i m far from that t r o u b l e d region of the world, all the way to the
far reaches of six-dimensional space, long before the t u m u l t in the early
universe h a d a c h a n c e to stabilize.
" I m a g i n e a simple video g a m e , " he says. A rocket ship can travel in
t h e video screen, he points out, until it veers too far to the right. Any
video-game player knows that the rocket ship t h e n suddenly appears
from t h e left side of t h e screen, at exactly t h e same height. Similarly, if
the rocket ship wanders too far a n d falls off the b o t t o m of the screen, it
rematerializes at the t o p of the screen. T h u s , Vafa explains, t h e r e is an
entirely self-contained universe in t h a t video screen. You can never leave
the universe defined by that screen. Even so, most teenagers have never
asked themselves what that universe is actually s h a p e d like. Vafa points
o u t , surprisingly e n o u g h , that the topology of the video screen is that of
a n i n n e r tube!
T h i n k of the video screen as a sheet of p a p e r . Since points at the top
of the screen are identical to the points at the b o t t o m , we can seal the
top a n d b o t t o m sides t o g e t h e r with glue. We now have rolled the sheet
of p a p e r into a tube. But the points on the left side of the tube are
identical to the points on t h e right side of the tube. O n e way to glue
these two e n d s is to b e n d the t u b e carefully into a circle, a n d seal the
two o p e n e n d s t o g e t h e r with glue (Figure 9.2).
W h a t we have d o n e is to t u r n a sheet of p a p e r into a d o u g h n u t . A
rocket ship w a n d e r i n g on the video screen can be described as moving
on the surface of an i n n e r tube. Every time the rocket vanishes off the
Figure 9.2. If a rocket disappears off the right side of a video-game screen, it reemerges on the left. If it disappears at the top, it re-emerges at the bottom. Let us
now wrap the screen so that identical points match. We first match the top and
bottom points by wrapping up the screen. Then we match the points on the leftand right-hand sides by rolling up the screen like a tube. In this way, we can
show that a video-game screen has the topology of a doughnut.
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video screen a n d r e a p p e a r s on t h e o t h e r side of the screen, this corres p o n d s to the rocket ship moving across the glued j o i n t of the i n n e r
tube.
Vafa conjectures that o u r sister universe has the shape of some sort
of twisted six-dimensional torus. Vafa a n d his colleagues have p i o n e e r e d
t h e c o n c e p t that o u r sister universe can be described by what mathematicians call an orbifold. In fact, his proposal that o u r sister universe has
t h e topology of an orbifold seems to fit the observed data r a t h e r well.
To visualize an orbifold, think of moving 360 degrees in a circle.
Everyone knows that we c o m e back to t h e same point. In o t h e r words,
if I d a n c e 360 d e g r e e s a r o u n d a May pole, I know that I will c o m e back
to t h e same spot. In an orbifold, however, if we move less t h a n 360
d e g r e e s a r o u n d the May pole, we will still c o m e back to the same point.
A l t h o u g h this may s o u n d p r e p o s t e r o u s , it is easy to construct orbifolds.
T h i n k of Flatlanders living on a c o n e . If they move less t h a n 360 degrees
a r o u n d the a p e x of t h e c o n e , they arrive at t h e same spot. T h u s an
orbifold is a higher-dimensional generalization of a c o n e (Figure 9.3).
To get a feel for orbifolds, imagine that s o m e Flatlanders live on what
is called a Z-orbifold, which is equivalent to the surface of a square b e a n
bag (like those f o u n d at carnivals a n d country fairs). At first, n o t h i n g
seems different from living in Flatland itself. As they explore the surface,
however, they begin to find strange h a p p e n i n g s . For example, if a Flatl a n d e r walks in any direction long e n o u g h , he r e t u r n s to his original
position as t h o u g h he walked in a circle. However, Flatlanders also notice
that t h e r e is s o m e t h i n g strange a b o u t certain points in their universe
(the four points of the b e a n b a g ) . W h e n walking a r o u n d any of these
four points by 180 d e g r e e s (not 360 d e g r e e s ) , they r e t u r n to the same
place from which they started.
T h e r e m a r k a b l e t h i n g a b o u t Vafa's orbifolds is that, with j u s t a few
assumptions, we can derive many of the features of quarks a n d o t h e r
subatomic particles. (This is because, as we saw earlier, the geometry of
space in Kaluza-Klein theory forces the quarks to assume the symmetry
of that space.) This gives us confidence that we are on the right track.
If these orbifolds gave us totally meaningless results, t h e n o u r intuition
would tell us that t h e r e is s o m e t h i n g fundamentally w r o n g with this construction.
If n o n e of t h e solutions of string theory contains t h e S t a n d a r d Model,
t h e n we must throw away superstring theory as a n o t h e r promising b u t
ultimately incorrect theory. However, physicists are excited by the fact
that it is possible to obtain solutions that are tantalizingly close to the
S t a n d a r d Model.
3
Figure 9.3. If we join points A and B, then we form a cone, which is the simplest
example of an orbifold. In string theory, our four-dimensional universe may have
a six-dimensional twin, which has the topology of an orbifold. However, the sixdimensional universe is so small that it is unobservable.
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Mathematicians for the past 80 years have b e e n working o u t the p r o p erties of these weird surfaces in h i g h e r dimensions, ever since the French
m a t h e m a t i c i a n H e n r i Poincare p i o n e e r e d t h e subject of topology in the
early twentieth century. T h u s the ten-dimensional theory is able to incorp o r a t e a large body of m o d e r n m a t h e m a t i c s that previously s e e m e d quite
useless.
Why Are There Three Generations?
In particular, t h e rich s t o r e h o u s e of mathematical t h e o r e m s compiled
by m a t h e m a t i c i a n s over the past century are now b e i n g used to explain
why t h e r e are t h r e e families of particles. As we saw earlier, o n e disastrous
feature of the GUTs is that t h e r e are t h r e e identical families of quarks
a n d leptons. However, orbifolds may explain this disconcerting feature
of t h e G U T s .
Vafa a n d his co-workers have discovered many promising solutions
to the string e q u a t i o n s that a p p e a r to resemble the physical world. With
a remarkably small set of assumptions, in fact, they can rederive the
S t a n d a r d Model, which is an i m p o r t a n t step for the theory. This is, in
fact, b o t h the strength a n d t h e weakness of superstring theory. Vafa a n d
his co-workers have b e e n , in a way, too successful: They have found millions of o t h e r possible solutions to the string equations.
T h e f u n d a m e n t a l p r o b l e m facing superstring theory is this: Of the
millions of possible universes that can be mathematically generated by superstring
theory, which is the correct one? As David Gross has said,
4
[ T ] h e r e are millions a n d millions of solutions that have
t h r e e spatial
d i m e n s i o n s . T h e r e i s a n e n o r m o u s a b u n d a n c e o f p o s s i b l e classical s o l u t i o n s . . . . T h i s a b u n d a n c e o f r i c h e s w a s o r i g i n a l l y very p l e a s i n g b e c a u s e i t
p r o v i d e d e v i d e n c e t h a t a t h e o r y l i k e t h e h e t e r o t i c s t r i n g c o u l d l o o k very
m u c h l i k e t h e real w o r l d . T h e s e s o l u t i o n s , i n a d d i t i o n t o h a v i n g f o u r s p a c e t i m e d i m e n s i o n s , h a d m a n y o t h e r properties that r e s e m b l e o u r w o r l d —
t h e right kinds of particles s u c h as quarks a n d l e p t o n s , a n d the right kinds
of i n t e r a c t i o n s . . . . T h a t was a s o u r c e of e x c i t e m e n t two years a g o .
5
Gross cautions that a l t h o u g h some of these solutions are very close
to t h e S t a n d a r d Model, o t h e r solutions p r o d u c e undesirable physical
p r o p e r t i e s : " I t is, however, slightly embarrassing that we have so many
solutions b u t no g o o d way of c h o o s i n g a m o n g t h e m . It seems even m o r e
embarrassing that these solutions have, in addition to many desired
p r o p e r t i e s , a few potentially disastrous p r o p e r t i e s . " A layperson, hear6
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ing this for t h e first time, may be puzzled a n d ask: Why d o n ' t you j u s t
calculate which solution the string prefers? Since string theory is a welldefined theory, it seems puzzling that physicists c a n n o t calculate t h e
answer.
T h e p r o b l e m is that t h e p e r t u r b a t i o n theory, o n e of t h e m a i n tools
in physics, is of no use. P e r t u r b a t i o n theory (which a d d s up increasingly
small q u a n t u m corrections) fails to b r e a k t h e ten-dimensional theory
down to four a n d six d i m e n s i o n s . T h u s we are forced to use n o n p e r t u r bative m e t h o d s , which are notoriously difficult to use. This, t h e n , is t h e
reason why we c a n n o t solve string theory. As we said earlier, string field
theory, developed by Kikkawa a n d me a n d further i m p r o v e d by Witten,
c a n n o t at p r e s e n t be solved nonperturbatively. No o n e is smart e n o u g h .
I o n c e h a d a r o o m m a t e w h o was a g r a d u a t e s t u d e n t in history. I
r e m e m b e r o n e day h e w a r n e d m e a b o u t the c o m p u t e r revolution, which
eventually m i g h t p u t physicists o u t of a j o b . "After all," he said, " c o m puters can calculate everything, c a n ' t t h e y ? " To h i m , it was only a m a t t e r
of time before m a t h e m a t i c i a n s p u t all physics questions in the c o m p u t e r
a n d physicists got on t h e u n e m p l o y m e n t line.
I was taken aback by t h e c o m m e n t , because to a physicist a c o m p u t e r
is n o t h i n g m o r e t h a n a sophisticated a d d i n g m a c h i n e , an impeccable
idiot. It makes up in speed what it lacks in intelligence. You have to
i n p u t the theory into the c o m p u t e r before it can m a k e a calculation.
T h e c o m p u t e r c a n n o t g e n e r a t e new theories by itself.
F u r t h e r m o r e , even if a theory is known, t h e c o m p u t e r may take an
infinite a m o u n t of time to solve a p r o b l e m . In fact, c o m p u t i n g all t h e
really interesting questions in physics would take an infinite a m o u n t of
c o m p u t e r time. This is the p r o b l e m with string theory. A l t h o u g h Vafa
a n d his colleagues have p r o d u c e d millions of possible solutions, it would
take an infinite a m o u n t of time to d e c i d e which of t h e millions of possibilities was t h e correct o n e , or to calculate solutions to q u a n t u m p r o b lems involving the bizarre process of t u n n e l i n g , o n e of the most difficult
of q u a n t u m p h e n o m e n a to solve.
Tunneling Through Space and Time
In the final analysis, we are asking t h e same question posed by Kaluza
in 1919—Where did the fifth d i m e n s i o n go?—except on a m u c h h i g h e r
level. As Klein p o i n t e d o u t in 1926, t h e answer to this question has to
do with q u a n t u m theory. P e r h a p s t h e most startling ( a n d c o m p l e x ) p h e n o m e n o n in q u a n t u m theory is t u n n e l i n g .
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For e x a m p l e , I am now sitting in a chair. T h e t h o u g h t of my body
suddenly z a p p i n g t h r o u g h the molecules of t h e wall n e x t to me a n d
reassembling, uninvited, in s o m e o n e else's living r o o m is an u n p l e a s a n t
o n e . Also an unlikely o n e . However, q u a n t u m m e c h a n i c s postulates that
t h e r e is a finite ( a l t h o u g h small) probability that even the most unlikely,
bizarre events—such a s waking u p o n e m o r n i n g a n d f i n d i n g o u r b e d i n
t h e m i d d l e of t h e A m a z o n jungle—will actually h a p p e n . All events, no
m a t t e r how strange, are r e d u c e d by q u a n t u m theory to probabilities.
This t u n n e l i n g process s o u n d s m o r e like science fiction than real
science. However, t u n n e l i n g can be m e a s u r e d in t h e laboratory a n d , in
fact, solves t h e riddle of radioactive decay. Normally, t h e nucleus of an
a t o m is stable. T h e p r o t o n s a n d n e u t r o n s within the nucleus are b o u n d
t o g e t h e r by t h e n u c l e a r force. However, t h e r e is a small probability that
t h e n u c l e u s m i g h t fall apart, that t h e p r o t o n s a n d n e u t r o n s m i g h t escape
by t u n n e l i n g past t h e large energy barrier, the nuclear force, that binds
t h e n u c l e u s together. Ordinarily, we would say that all nuclei must therefore be stable. But it is an u n d e n i a b l e fact that u r a n i u m nuclei d o , in
fact, decay w h e n they s h o u l d n ' t ; in fact, t h e conservation of energy law
is briefly violated as t h e n e u t r o n s in t h e n u c l e u s t u n n e l their way t h r o u g h
the barrier.
T h e catch, however, is that these probabilities are vanishingly small
for large objects, such as h u m a n s . T h e probability of o u r t u n n e l i n g
t h r o u g h a wall within the lifetime of t h e known universe is infinitesimally
small. T h u s I can safely assume that I will n o t be ungraciously transp o r t e d t h r o u g h the wall, at least within my own lifetime. Similarly, o u r
universe, which originally m i g h t have b e g u n as a ten-dimensional universe, was n o t stable; it t u n n e l e d a n d e x p l o d e d into a four- a n d a sixd i m e n s i o n a l universe.
To u n d e r s t a n d this form of t u n n e l i n g , t h i n k of an imaginary Charlie
Chaplin film, in which Chaplin is trying to stretch a b e d sheet a r o u n d
an oversize b e d . T h e s h e e t is t h e kind with elastic b a n d s on the corners.
But it is too small, so he has to strain to wrap the elastic b a n d s a r o u n d
each c o r n e r of t h e mattress, o n e at a time. He grins with satisfaction
o n c e he has stretched t h e b e d s h e e t smoothly a r o u n d all four corners
of t h e b e d . But t h e strain is too great; o n e elastic b a n d p o p s off o n e
c o r n e r , a n d the b e d sheet curls u p . Frustrated, he pulls this elastic
a r o u n d t h e c o r n e r , only to have a n o t h e r elastic p o p off a n o t h e r corner.
Every time he yanks an elastic b a n d a r o u n d o n e c o r n e r , a n o t h e r elastic
p o p s off a n o t h e r c o r n e r .
This process is called symmetry breaking. T h e smoothly stretched b e d
sheet possesses a high d e g r e e of symmetry. You can rotate the b e d 180
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degrees along any axis, a n d the b e d sheet r e m a i n s t h e same. This highly
symmetrical state is called the false vacuum. A l t h o u g h the false v a c u u m
appears quite symmetrical, it is n o t stable. T h e s h e e t d o e s n o t want to
be in this stretched c o n d i t i o n . T h e r e is t o o m u c h tension. T h e energy
is too high. T h u s o n e elastic p o p s off, a n d t h e b e d sheet curls u p . T h e
symmetry is b r o k e n , a n d the b e d s h e e t has g o n e to a lower-energy state
with less symmetry. By r o t a t i n g t h e curled-up b e d s h e e t 180 d e g r e e s
a r o u n d an axis, we no l o n g e r r e t u r n to t h e same sheet.
Now replace the b e d s h e e t with ten-dimensional s p a c e - t i m e , t h e
s p a c e - t i m e of ultimate symmetry. At t h e b e g i n n i n g of time, the universe
was perfectly symmetrical. If a n y o n e was a r o u n d at that time, he could
freely pass t h r o u g h any of t h e ten d i m e n s i o n s without p r o b l e m . At that
time, gravity a n d the weak, t h e strong, a n d t h e e l e c t r o m a g n e t i c forces
were all unified by the superstring. All m a t t e r a n d forces were p a r t of
the same string multiplet. However, this symmetry c o u l d n ' t last. T h e tendimensional universe, a l t h o u g h perfectly symmetrical, was unstable, j u s t
like the b e d sheet, a n d in a false vacuum. T h u s t u n n e l i n g to a lowerenergy state was inevitable. W h e n t u n n e l i n g finally o c c u r r e d , a p h a s e
transition took place, a n d symmetry was lost.
Because t h e universe b e g a n to split up into a four- a n d a six-dimensional universe, the universe was no l o n g e r symmetrical. Six d i m e n s i o n s
have curled u p , in t h e same way that the b e d s h e e t curls up w h e n o n e
elastic p o p s off a c o r n e r of a mattress. But notice that t h e r e a r e four
ways in which the b e d sheet can curl u p , d e p e n d i n g on which c o r n e r
pops off first. For the ten-dimensional universe, however, t h e r e are
apparently millions of ways in which to curl u p . To calculate which state
the ten-dimensional universe prefers, we n e e d to solve t h e field theory
of strings using the theory of p h a s e transitions, the most difficult p r o b lem in q u a n t u m theory.
Symmetry Breaking
Phase transitions are n o t h i n g new. T h i n k of o u r own lives. In h e r b o o k
Passages, Gail Sheehy stresses that life is n o t a c o n t i n u o u s stream of experiences, as it often a p p e a r s , b u t actually passes t h r o u g h several stages,
characterized by specific conflicts that must be resolved a n d goals that
must be achieved.
T h e psychologist Erik Erikson even p r o p o s e d a theory of t h e psychological stages of d e v e l o p m e n t . A f u n d a m e n t a l conflict characterizes e a c h
phase. W h e n this conflict is correctly resolved, we move on to the n e x t
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p h a s e . If this conflict is n o t resolved, it may fester a n d even cause regression to an earlier p e r i o d . Similarly, the psychologist J e a n Piaget showed
t h a t early c h i l d h o o d m e n t a l d e v e l o p m e n t is also n o t a s m o o t h process
of learning, b u t is actually typified by a b r u p t stages in a child's ability to
conceptualize. O n e m o n t h , a child may give up looking for a ball o n c e
it has rolled o u t of view, n o t u n d e r s t a n d i n g that an object exists even if
you can no l o n g e r see it. T h e n e x t m o n t h , this is obvious to the child.
This is t h e essence of dialectics. A c c o r d i n g to this philosophy, all
objects ( p e o p l e , gases, the universe itself) go t h r o u g h a series of stages.
Each stage is characterized by a conflict between two o p p o s i n g forces.
T h e n a t u r e of this conflict, in fact, d e t e r m i n e s the n a t u r e of the stage.
W h e n the conflict is resolved, t h e object goes to a h i g h e r stage, called
t h e synthesis, w h e r e a new c o n t r a d i c t i o n begins, a n d the process starts
over again at a h i g h e r level.
P h i l o s o p h e r s call this t h e transition from " q u a n t i t y " to "quality."
Small quantitative c h a n g e s eventually build up until t h e r e is a qualitative
r u p t u r e with t h e past. This theory applies to societies as well. Tensions
in a society can rise dramatically, as they did in France in the late eight e e n t h century. T h e peasants faced starvation, s p o n t a n e o u s food riots
took place, a n d t h e aristocracy r e t r e a t e d b e h i n d its fortresses. W h e n the
tensions r e a c h e d t h e b r e a k i n g point, a p h a s e transition o c c u r r e d from
t h e quantitative to the qualitative: T h e peasants took up arms, seized
Paris, a n d s t o r m e d the Bastille.
Phase transitions can also be quite explosive affairs. F o r example,
think of a river that has b e e n d a m m e d u p . A reservoir quickly fills up
b e h i n d the d a m with water u n d e r e n o r m o u s pressure. Because it is
unstable, t h e reservoir is in the false v a c u u m . T h e water would prefer to
be in its t r u e v a c u u m , m e a n i n g it would prefer to burst the d a m a n d
wash d o w n s t r e a m , to a state of lower energy. T h u s a p h a s e transition
would involve a d a m burst, which could have disastrous consequences.
An even m o r e explosive e x a m p l e is an atomic b o m b . T h e false vacu u m c o r r e s p o n d s t o stable u r a n i u m nuclei. A l t h o u g h the u r a n i u m
n u c l e u s a p p e a r s stable, t h e r e a r e e n o r m o u s , explosive energies t r a p p e d
within t h e u r a n i u m n u c l e u s that are a million times m o r e powerful,
p o u n d for p o u n d , t h a n a chemical explosive. O n c e in a while, the
n u c l e u s t u n n e l s to a lower state, which m e a n s that t h e nucleus spontaneously splits a p a r t all by itself. This is called radioactive decay. However,
it is possible, by s h o o t i n g n e u t r o n s at t h e u r a n i u m nucleus, to release
this p e n t - u p energy all at o n c e . This, of course, is an atomic explosion.
T h e new feature discovered by scientists a b o u t p h a s e transitions is
that they are usually a c c o m p a n i e d by a symmetry breaking. Nobel lau-
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reate Abdus Salam likes t h e following illustration: C o n s i d e r a circular
b a n q u e t table, where all t h e guests are seated with a c h a m p a g n e glass
on either side. T h e r e is a symmetry h e r e . L o o k i n g at t h e b a n q u e t table
t h r o u g h a mirror, we see t h e same thing: each guest seated a r o u n d t h e
table, with c h a m p a g n e glasses on either side. Similarly, we can rotate
the circular b a n q u e t table, a n d the a r r a n g e m e n t is still t h e same.
Now break the symmetry. Assume that t h e first d i n e r picks up t h e
glass on his or h e r right. By custom, all t h e o t h e r guests pick up t h e
c h a m p a g n e glass to their right. Notice that t h e image of t h e b a n q u e t
table as seen in the m i r r o r p r o d u c e s t h e opposite situation. Every d i n e r
has picked up the glass to his or h e r left. T h u s left-right symmetry has
been broken.
A n o t h e r e x a m p l e of symmetry b r e a k i n g comes from an a n c i e n t fairy
tale. This fable c o n c e r n s a princess w h o is t r a p p e d on t o p of a polished
crystal s p h e r e . A l t h o u g h t h e r e are no iron bars confining h e r to the
s p h e r e , she is a prisoner because if she makes t h e slightest move, she
will slip off the s p h e r e a n d kill herself. N u m e r o u s princes have tried to
rescue the princess, b u t each has failed to scale the s p h e r e because it is
too s m o o t h a n d slippery. This is an e x a m p l e of symmetry b r e a k i n g .
While the princess is a t o p the s p h e r e , she is in a perfectly symmetrical
state. T h e r e is no p r e f e r r e d direction for the s p h e r e . We can rotate t h e
s p h e r e at any angle, a n d t h e situation r e m a i n s t h e same. Any false move
off the center, however, will cause the princess to fall, t h e r e b y b r e a k i n g
the symmetry. If she falls to the west, for e x a m p l e , t h e symmetry of rotation is b r o k e n . T h e westerly direction is now singled out.
T h u s the state of m a x i m u m symmetry is often also an unstable state,
a n d h e n c e c o r r e s p o n d s to a false vacuum. T h e true v a c u u m state corresponds to the princess falling off the s p h e r e . So a p h a s e transition
(falling off the s p h e r e ) c o r r e s p o n d s to symmetry b r e a k i n g (selecting t h e
westerly d i r e c t i o n ) .
Regarding superstring theory, physicists assume (but c a n n o t yet
prove) that the original ten-dimensional universe was unstable a n d tunn e l e d its way to a four- a n d a six-dimensional universe. T h u s t h e original
universe was in the state of the false vacuum, t h e state of m a x i m u m
symmetry, while today we are in the b r o k e n state of t h e t r u e v a c u u m .
This raises a disturbing question: W h a t would h a p p e n if o u r universe
were actually n o t the true vacuum? W h a t would h a p p e n if the superstring only temporarily chose o u r universe, b u t the t r u e v a c u u m lay a m o n g
the millions of possible orbifolds? This would have disastrous conseq u e n c e s . In many o t h e r orbifolds, we find that t h e S t a n d a r d Model is
n o t present. T h u s if the true v a c u u m were actually a state w h e r e the
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S t a n d a r d Model was n o t present, t h e n all the laws of chemistry a n d physics, as we know t h e m , would c o m e t u m b l i n g down.
If this o c c u r r e d , a tiny b u b b l e m i g h t suddenly a p p e a r in o u r universe.
Within this b u b b l e , t h e S t a n d a r d Model would no longer hold, so a
different set of chemical a n d physical laws would apply. Matter inside
t h e b u b b l e would disintegrate a n d p e r h a p s re-form in different ways.
This b u b b l e would t h e n e x p a n d at t h e speed of light, swallowing up
e n t i r e star systems, galaxies, a n d galactic clusters, until it g o b b l e d up the
e n t i r e universe.
We would never see it c o m i n g . Traveling at the speed of light, it could
never be observed b e f o r e h a n d . We would never know what hit us.
From Ice Cubes to Superstrings
C o n s i d e r an o r d i n a r y ice c u b e sitting in a pressure cooker in o u r kitchen.
We all know what h a p p e n s if we t u r n on t h e stove. But what h a p p e n s to
an ice c u b e if we h e a t it up to trillions upon trillions of degrees?
If we h e a t the ice c u b e on t h e stove, first it melts a n d turns into water;
that is, it u n d e r g o e s a p h a s e transition. Now let us h e a t the water until
it boils. It t h e n u n d e r g o e s a n o t h e r p h a s e transition a n d turns into steam.
Now c o n t i n u e to h e a t t h e steam to e n o r m o u s t e m p e r a t u r e s . Eventually,
t h e water molecules b r e a k u p . T h e energy of the molecules exceeds the
b i n d i n g energy of t h e molecules, which are r i p p e d a p a r t into elemental
h y d r o g e n a n d oxygen gas.
Now we c o n t i n u e to h e a t it past 3,000°K, until the atoms of hydrogen
a n d oxygen are r i p p e d apart. T h e electrons are pulled from the nucleus,
a n d we now have a plasma (an ionized gas), often called the fourth state
of m a t t e r (after gases, liquids, a n d solids). A l t h o u g h a plasma is n o t p a r t
of c o m m o n e x p e r i e n c e , we can see it every time we look at t h e sun. In
fact, plasma is t h e most c o m m o n state of m a t t e r in t h e universe.
Now c o n t i n u e to h e a t the plasma on the stove to 1 billion°K, until
t h e nuclei of h y d r o g e n a n d oxygen are r i p p e d apart, a n d we have a
" g a s " of individual n e u t r o n s a n d p r o t o n s , similar to the interior of a
n e u t r o n star.
If we h e a t t h e " g a s " of n u c l e o n s even further to 10 trillion°K, these
subatomic particles will t u r n into disassociated quarks. We will now have
a gas of quarks a n d l e p t o n s (the electrons a n d n e u t r i n o s ) .
If we h e a t this gas to 1 quadrillion°K, t h e electromagnetic force a n d
t h e weak force will b e c o m e u n i t e d . T h e symmetry SU(2) X U ( l ) will
e m e r g e at this t e m p e r a t u r e . At 1 0 °K, the electroweak a n d strong forces
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Before Creation
b e c o m e united, a n d the G U T symmetries [ S U ( 5 ) , O ( 1 0 ) , o r E ( 6 ) ]
appear.
Finally, at a fabulous 1 0 °K, gravity unites with t h e G U T force, a n d
all the symmetries of the ten-dimensional s u p e r s t r i n g a p p e a r . We n o w
have a gas of superstrings. At that point, so m u c h energy will have g o n e
into the pressure c o o k e r that t h e g e o m e t r y of s p a c e - t i m e may very well
begin to distort, a n d t h e dimensionality of s p a c e - t i m e may c h a n g e . T h e
space a r o u n d o u r kitchen may very well b e c o m e unstable, a rip may form
in the fabric of space, a n d a w o r m h o l e may a p p e a r in the kitchen. At
this p o i n t , it may be advisable to leave t h e kitchen.
32
Cooling the Big Bang
T h u s by h e a t i n g an ordinary ice c u b e to fantastic t e m p e r a t u r e s , we can
retrieve the superstring. T h e lesson h e r e is that m a t t e r goes t h r o u g h
definite stages of d e v e l o p m e n t as we h e a t it u p . Eventually, m o r e a n d
m o r e symmetry b e c o m e s restored as we increase the energy.
By reversing this process, we can a p p r e c i a t e how t h e Big B a n g
o c c u r r e d as a s e q u e n c e of different stages. Instead of h e a t i n g an ice
cube, we now cool the s u p e r h o t m a t t e r in the universe t h r o u g h different
stages. Beginning with t h e instant of Creation, we have t h e following
stages in the evolution of o u r universe.
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10
seconds T h e ten-dimensional universe breaks down to a foura n d a six-dimensional universe. T h e six-dimensional universe collapses
down to 1 0
c e n t i m e t e r in size. T h e four-dimensional universe inflates
rapidly. T h e t e m p e r a t u r e is 1 0 °K.
10
seconds T h e G U T force breaks; t h e strong force is no l o n g e r
united with the electroweak interactions. SU(3) breaks off from t h e G U T
symmetry. A small speck in t h e larger universe b e c o m e s inflated by a
factor of 1 0 , eventually b e c o m i n g o u r visible universe.
10 seconds T h e t e m p e r a t u r e is now 1 0 °K, a n d t h e electroweak
symmetry breaks into SU(2) a n d U ( l ) .
10 seconds Quarks begin to c o n d e n s e into n e u t r o n s a n d p r o t o n s .
T h e t e m p e r a t u r e is roughly 1 0 °K.
3 minutes
T h e p r o t o n s a n d n e u t r o n s are now c o n d e n s i n g into stable
nuclei. T h e energy of r a n d o m collisions is no l o n g e r powerful e n o u g h
to break up the nucleus of t h e e m e r g i n g nuclei. Space is still o p a q u e to
light because ions do n o t transmit light well.
300,000 years Electrons begin to c o n d e n s e a r o u n d nuclei. Atoms
- 3 2
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-9
15
-3
14
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begin to form. Because light is no l o n g e r scattered or absorbed as m u c h ,
the universe b e c o m e s t r a n s p a r e n t to light. O u t e r space b e c o m e s black.
3 billion years T h e first quasars a p p e a r .
5 billion years T h e first galaxies a p p e a r .
10 to 15 billion years T h e solar system is b o r n . A few billion years
after that, t h e first forms of life a p p e a r on earth.
It seems almost i n c o m p r e h e n s i b l e that we, as intelligent apes on the
third p l a n e t of a m i n o r star in a m i n o r galaxy, would be able to reconstruct the history of o u r universe going back almost to the instant of its
birth, w h e r e t e m p e r a t u r e s a n d pressures e x c e e d e d anything ever found
in o u r solar system. Yet t h e q u a n t u m theory of the weak, electromagnetic, a n d s t r o n g interactions reveals this picture to us.
As startling as this picture of Creation is, p e r h a p s stranger still is the
possibility t h a t w o r m h o l e s can act as gateways to a n o t h e r universe a n d
p e r h a p s even as time m a c h i n e s into the past a n d future. A r m e d with a
q u a n t u m theory of gravity, physicists may be able to answer t h e intriguing questions: Are t h e r e parallel universes? Can the past be changed?
PART III
Wormholes:
Gateways to
Another Universe?
10
Black Holes and
Parallel Universes
Listen, there's a hell of a universe next door: let's go!
e. e. c u m m i n g s
Black Holes: Tunnels Through Space and Time
B
LACK holes have recently seized t h e public's imagination. Books
a n d d o c u m e n t a r i e s have b e e n devoted to e x p l o r i n g this strange
prediction of Einstein's equations, the final stage in t h e d e a t h of a collapsed star. Ironically, the public r e m a i n s largely u n a w a r e of p e r h a p s t h e
most peculiar feature of black holes, that they may be gateways to an
alternative universe. F u r t h e r m o r e , t h e r e is also i n t e n s e speculation in the
scientific c o m m u n i t y that a black h o l e may o p e n up a t u n n e l in time.
To u n d e r s t a n d black holes a n d how difficult they are to find, we m u s t
first u n d e r s t a n d what makes t h e stars shine, h o w they grow, a n d how
they eventually die. A star is b o r n w h e n a massive cloud of h y d r o g e n gas
many times the size of o u r solar system is slowly c o m p r e s s e d by the force
of gravity. T h e gravitational force c o m p r e s s i n g t h e gas gradually heats
up the gas, as gravitational e n e r g y is converted into t h e kinetic energy
of the h y d r o g e n atoms. Normally, t h e repulsive c h a r g e of t h e p r o t o n s
within the h y d r o g e n gas is sufficient to k e e p t h e m apart. But at a certain
point, when the t e m p e r a t u r e rises to 10 to 100 million°K, t h e kinetic
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GATEWAYS
TO
A N O T H E R
UNIVERSE?
energy of t h e p r o t o n s (which are h y d r o g e n nuclei) overcomes their electrostatic repulsion, a n d they slam into o n e a n o t h e r . T h e nuclear force
t h e n takes over from the electromagnetic force, a n d the two hydrogen
nuclei " f u s e " into h e l i u m , releasing vast quantities of energy.
In o t h e r words, a star is a n u c l e a r furnace, b u r n i n g h y d r o g e n fuel
a n d creating n u c l e a r " a s h " in t h e form of waste helium. A star is also a
delicate b a l a n c i n g act between t h e force of gravity, which tends to crush
the star into oblivion, a n d the n u c l e a r force, which tends to blow the
star a p a r t with t h e force of trillions of h y d r o g e n b o m b s . A star t h e n
m a t u r e s a n d ages as it exhausts its n u c l e a r fuel.
To see how energy is extracted from the fusion process a n d to u n d e r stand t h e stages in the life of a star leading to a black hole, we must
analyze Figure 10.1, which shows o n e of t h e most i m p o r t a n t curves in
m o d e r n science, sometimes called t h e binding energy curve. On the horizontal scale is t h e atomic weight of the various elements, from hydrogen
to u r a n i u m . On t h e vertical scale, crudely speaking, is the a p p r o x i m a t e
average " w e i g h t " of each p r o t o n in the nucleus. Notice that h y d r o g e n
a n d u r a n i u m have p r o t o n s that weigh, o n average, m o r e t h a n the protons of o t h e r e l e m e n t s in the c e n t e r of the diagram.
O u r sun is an ordinary yellow star, consisting mainly of hydrogen.
Like t h e original Big Bang, it fuses h y d r o g e n a n d forms helium. However, because t h e p r o t o n s in h y d r o g e n weigh m o r e t h a n the p r o t o n s in
helium, t h e r e is an excess of mass, which is converted into energy via
Einstein's E = mc formula. This energy is what binds the nuclei together.
This is also the e n e r g y released w h e n h y d r o g e n is fused into helium.
This is why t h e sun shines.
However, as t h e h y d r o g e n is slowly used up over several billion years,
a yellow star eventually builds up too m u c h waste h e l i u m , a n d its nuclear
furnace shuts off. W h e n that h a p p e n s , gravity eventually takes over a n d
crushes t h e star. As t e m p e r a t u r e s soar, the star soon b e c o m e s h o t
e n o u g h to b u r n waste h e l i u m a n d convert it into t h e o t h e r elements,
like lithium a n d c a r b o n . Notice that energy can still be released as we
d e s c e n d down t h e curve to t h e h i g h e r elements. In o t h e r words, it is still
possible to b u r n waste h e l i u m (in t h e same way that ordinary ash can
still be b u r n e d u n d e r certain c o n d i t i o n s ) . A l t h o u g h t h e star has
d e c r e a s e d e n o r m o u s l y in size, its t e m p e r a t u r e is quite high, a n d its atmos p h e r e e x p a n d s greatly in size. In fact, w h e n o u r own sun exhausts its
h y d r o g e n supply a n d starts to b u r n h e l i u m , its a t m o s p h e r e may e x t e n d
o u t to t h e orbit of Mars. This is what is called a red giant. This m e a n s , of
course, t h a t the e a r t h will be vaporized in the process. T h u s the curve
also predicts t h e ultimate fate of t h e e a r t h . Since o u r sun is a middle2
Average
weight
Atomic weight
Figure 10.1. The average "weight" of each proton of lighter elements, such as
hydrogen and helium, is relatively large. Thus if we fuse hydrogen to form helium
inside a star, we have excess mass, which is converted to energy via Einstein's
equation E = m c . This is the energy that lights up the stars. But as stars fuse
heavier and heavier elements, eventually we reach iron, and we cannot extract
any more energy. The star then collapses, and the tremendous heat of collapse
creates a supernova. This colossal explosion rips the star apart and seeds the
interstellar space, in which new stars are formed. The process then starts all over
again, like a pinball machine.
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aged star a b o u t 5 billion years old, it still has a n o t h e r 5 billion years
before it c o n s u m e s the e a r t h . (Ironically, the earth was originally b o r n
o u t of t h e same swirling gas cloud that created o u r sun. Physics now
predicts that the earth, which was created with the sun, will r e t u r n to
the sun.)
Finally, when the h e l i u m is used u p , the nuclear furnace again shuts
down, a n d gravity takes over to crush t h e star. T h e r e d giant shrinks to
b e c o m e a white dwarf, a m i n i a t u r e star with t h e mass of an entire star
s q u e e z e d d o w n to a b o u t t h e size of the p l a n e t e a r t h . White dwarfs are
n o t very l u m i n o u s because, after d e s c e n d i n g to the b o t t o m of the curve,
t h e r e is only a little excess energy o n e can squeeze from it t h r o u g h E =
mc . T h e white dwarf b u r n s what little t h e r e is left at the b o t t o m of the
curve.
1
2
O u r sun will eventually t u r n into a white dwarf a n d , over billions of
years, slowly die as it exhausts its n u c l e a r fuel. It will eventually b e c o m e
a dark, b u r n e d - o u t dwarf star. However, it is believed that if a star is
sufficiently massive (several times t h e mass of o u r s u n ) , t h e n most of the
e l e m e n t s in t h e white dwarf will c o n t i n u e to be fused into increasingly
heavier e l e m e n t s , eventually r e a c h i n g i r o n . O n c e we r e a c h iron, we are
n e a r t h e very b o t t o m of t h e curve. We can no l o n g e r extract any m o r e
energy from the excess mass, so t h e n u c l e a r furnace shuts off. Gravity
o n c e again takes over, c r u s h i n g the star until t e m p e r a t u r e s rise explosively a thousandfold, r e a c h i n g trillions of degrees. At this point, the
iron core collapses a n d t h e o u t e r layer of the white dwarf blows off,
releasing t h e largest burst of energy known in the galaxy, an e x p l o d i n g
star called a supernova. J u s t o n e supernova can temporarily o u t s h i n e an
e n t i r e galaxy of 100 billion stars.
In t h e aftermath of t h e supernova, we find a totally d e a d star, a neutron star a b o u t t h e size of M a n h a t t a n . T h e densities in a n e u t r o n star are
so great that, crudely speaking, all t h e n e u t r o n s are " t o u c h i n g " o n e
a n o t h e r . A l t h o u g h n e u t r o n stars are almost invisible, we can still detect
t h e m with o u r i n s t r u m e n t s . Because they e m i t some radiation while they
are rotating, they act like a cosmic lighthouse in o u t e r space. We see
t h e m as a blinking star, or pulsar. (Although this scenario sounds like
science fiction, well over 400 pulsars have b e e n observed since their
initial discovery in 1967.)
C o m p u t e r calculations have shown that most of t h e heavier elements
b e y o n d iron can be synthesized in the h e a t a n d pressure of a supernova.
W h e n t h e star explodes, it releases vast a m o u n t s of stellar debris, consisting of t h e h i g h e r elements, into t h e v a c u u m of space. This debris
eventually mixes with o t h e r gases, until e n o u g h h y d r o g e n gas is accu-
Black Holes and Parallel Universes
221
m u l a t e d to begin t h e gravitational contraction process o n c e again. Second-generation stars that a r e b o r n o u t of this stellar gas a n d dust c o n t a i n
an a b u n d a n c e of heavy e l e m e n t s . Some of these stars (like o u r sun) will
have planets s u r r o u n d i n g t h e m that also c o n t a i n these heavy e l e m e n t s .
This solves a long-standing mystery in cosmology. O u r bodies are
m a d e of heavy e l e m e n t s b e y o n d iron, b u t o u r sun is n o t h o t e n o u g h to
forge t h e m . If the e a r t h a n d t h e atoms of o u r bodies were originally
from t h e same gas cloud, t h e n w h e r e did t h e heavy e l e m e n t s of o u r
bodies c o m e from? T h e conclusion is inescapable: T h e heavy e l e m e n t s
in o u r bodies were synthesized in a supernova that blew up before o u r
sun was created. In o t h e r words, a nameless s u p e r n o v a e x p l o d e d billions
of years ago, seeding the original gas cloud that created o u r solar system.
T h e evolution of a star can be roughly p i c t u r e d as a pinball m a c h i n e ,
as in Figure 10.1, with the s h a p e of t h e b i n d i n g energy curve. T h e ball
starts at the top a n d b o u n c e s from h y d r o g e n , to h e l i u m , from t h e lighter
elements to the heavier e l e m e n t s . Each time it b o u n c e s a l o n g t h e curve,
it b e c o m e s a different type of star. Finally, t h e ball b o u n c e s to the b o t t o m
of the curve, where it lands on iron, a n d is ejected explosively in a supernova. T h e n as this stellar material is collected again i n t o a new hydrogenrich star, the process starts all over again on the pinball.
Notice, however, that t h e r e are two ways for the pinball to b o u n c e
down the curve. It can also start at t h e o t h e r side of the curve, at uran i u m , a n d go down the curve in a single b o u n c e by fissioning t h e uran i u m nucleus into fragments. Since t h e average weight of t h e p r o t o n s
in fission p r o d u c t s , like cesium a n d krypton, is smaller t h a n t h e average
weight of the p r o t o n s in u r a n i u m , t h e excess mass has b e e n c o n v e r t e d
into energy via E = mc . This is t h e source of energy b e h i n d the atomic
bomb.
T h u s the curve of b i n d i n g energy n o t only explains t h e birth a n d
d e a t h of stars a n d the creation of the e l e m e n t s , b u t also makes possible
the existence of h y d r o g e n a n d atomic b o m b s ! (Scientists are often asked
w h e t h e r it would be possible to develop n u c l e a r b o m b s o t h e r t h a n
atomic a n d h y d r o g e n b o m b s . F r o m t h e curve of b i n d i n g energy, we can
see that the answer is n o . Notice t h a t the curve excludes the possibility
of b o m b s m a d e of oxygen or iron. T h e s e e l e m e n t s are n e a r t h e b o t t o m
of the curve, so t h e r e is n o t e n o u g h excess mass to create a b o m b . T h e
various b o m b s m e n t i o n e d in t h e press, such as n e u t r o n b o m b s , are only
variations o n u r a n i u m a n d h y d r o g e n bombs.)
W h e n o n e first hears t h e life history of stars, o n e may be a bit skeptical. After all, no o n e has ever lived 10 billion years to witness their
evolution. However, since t h e r e are u n c o u n t a b l e stars in t h e heavens, it
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is a simple m a t t e r to see stars at practically every stage in their evolution.
(For e x a m p l e , the 1987 supernova, which was visible to the n a k e d eye
in t h e s o u t h e r n h e m i s p h e r e , yielded a treasure trove of astronomical
d a t a that m a t c h e d t h e theoretical predictions of a collapsing dwarf with
an iron c o r e . Also, the spectacular supernova observed by ancient Chinese a s t r o n o m e r s on July 4, 1054, left b e h i n d a r e m n a n t , which has now
b e e n identified as a n e u t r o n star.)
In addition, o u r c o m p u t e r p r o g r a m s have b e c o m e so accurate that
we can essentially p r e d i c t t h e s e q u e n c e of stellar evolution numerically.
I o n c e h a d a r o o m m a t e in g r a d u a t e school w h o was an astronomy major.
He would invariably d i s a p p e a r in t h e early m o r n i n g a n d r e t u r n late at
night. J u s t before he would leave, he would say that he was p u t t i n g a star
in t h e oven to watch it grow. At first, I t h o u g h t he said this in jest.
However, w h e n I pressed h i m on this point, he said with all seriousness
t h a t he was p u t t i n g a star into t h e c o m p u t e r a n d watching it evolve
d u r i n g t h e day. Since t h e t h e r m o d y n a m i c equations a n d t h e fusion
e q u a t i o n s were well known, it was j u s t a m a t t e r of telling the c o m p u t e r
to start with a certain mass of h y d r o g e n gas a n d t h e n letting it n u m e r i cally solve for the evolution of this gas. In this way, we can check that
o u r theory of stellar evolution can r e p r o d u c e t h e known stages of star
life t h a t we see in t h e heavens with o u r telescopes.
Black Holes
If a star was t e n to 50 times the size of o u r sun, t h e n gravity will c o n t i n u e
to squeeze it even after it b e c o m e s a n e u t r o n star. W i t h o u t the force of
fusion to r e p e l t h e gravitational pull, t h e r e is n o t h i n g to o p p o s e the final
collapse of t h e star. At this p o i n t , it b e c o m e s t h e famous black hole.
In s o m e sense, black holes must exist. A star, we recall, is the byp r o d u c t of two cosmic forces: gravity, which tries to crush the star, a n d
fusion, which tries to blow t h e star a p a r t like in a h y d r o g e n b o m b . All
the various phases in t h e life history of a star are a c o n s e q u e n c e of this
delicate b a l a n c i n g act between gravity a n d fusion. S o o n e r or later, when
all t h e n u c l e a r fuel in a massive star is finally e x h a u s t e d a n d the star is
a mass of p u r e n e u t r o n s , t h e r e is n o t h i n g known t h a t can t h e n resist the
powerful force of gravity. Eventually, t h e gravitational force will take over
a n d crush t h e n e u t r o n star i n t o n o t h i n g n e s s . T h e star has c o m e full
circle: It was b o r n w h e n gravity first b e g a n to compress h y d r o g e n gas in
t h e heavens into a star, a n d it will die w h e n t h e nuclear fuel is exhausted
a n d gravity collapses it.
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T h e density of a black h o l e is so large that light, like a rocket
l a u n c h e d from t h e earth, will be forced to orbit a r o u n d it. Since no light
can escape from the e n o r m o u s gravitational field, the collapsed star
b e c o m e s black in color. In fact, that is the usual definition of a black
hole, a collapsed star from which no light can escape.
To u n d e r s t a n d this, we n o t e that all heavenly bodies have what is
called an escape velocity. This is the velocity necessary to escape p e r m a nently the gravitational pull of that body. For e x a m p l e , a space p r o b e
must reach an escape velocity of 25,000 miles p e r h o u r in o r d e r to leave
the gravitational pull of the e a r t h a n d go into d e e p space. O u r space
p r o b e s like the Voyager that have v e n t u r e d i n t o d e e p space a n d have
completely left the solar system (carrying good-will messages to any
aliens who m i g h t pick t h e m u p ) have r e a c h e d the escape velocity of o u r
sun. (The fact that we b r e a t h e oxygen is because t h e oxygen a t o m s do
n o t have e n o u g h velocity to escape t h e e a r t h ' s gravitational field. T h e
fact that J u p i t e r a n d the o t h e r gas giants are m a d e mainly of h y d r o g e n
is because their escape velocity is large e n o u g h to c a p t u r e t h e p r i m o r d i a l
hydrogen of t h e early solar system. T h u s escape velocity helps to explain
the planetary evolution of the planets of o u r solar system over t h e past
5 billion years.)
Newton's theory of gravity, in fact, gives t h e precise relationship
between the escape velocity a n d t h e mass of t h e star. T h e heavier t h e
planet or star a n d the smaller its radius, the larger t h e escape velocity
necessary to escape its gravitational pull. As early as 1783, t h e English
a s t r o n o m e r J o h n Michell used this calculation to p r o p o s e that a s u p e r
massive star m i g h t have an escape velocity equal to the speed of light.
T h e light emitted by such a massive star could never escape, b u t would
orbit a r o u n d it. T h u s , to an outside observer, t h e star would a p p e a r
totally black. Using t h e best knowledge available in t h e e i g h t e e n t h century, he actually calculated t h e mass of such a black h o l e . * Unfortunately, his theory was c o n s i d e r e d to be crazy a n d was soon forgotten.
Nevertheless, today we t e n d to believe that black holes exist because o u r
telescopes a n d i n s t r u m e n t s have seen white dwarfs a n d n e u t r o n stars in
the heavens.
T h e r e are two ways to explain why black holes are black. F r o m t h e
*In the Philosophical Transactions of the Royal Society, he wrote, "If the semi-diameter of
a sphere of the same density with the Sun were to e x c e e d that of the Sun in the proportion
of 500 to 1, a body falling from an infinite height towards it, would have acquired at its
surface greater velocity than that of light, and consequently supposing light to be attracted
by the same force in proportion to its vis inertiae, with other bodies, all light emitted from
such a body would be made to return to it by its own proper gravity."
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pedestrian p o i n t of view, the " f o r c e " between the star a n d a light b e a m
is so great that its p a t h is b e n t into a circle. Or o n e can take the Einsteinian p o i n t of view, in which case the " s h o r t e s t distance between two
points is a curved l i n e . " B e n d i n g a light b e a m into a full circle m e a n s
that space itself has b e e n b e n t full circle. This can h a p p e n only if the
black h o l e has completely p i n c h e d a piece of s p a c e - t i m e along with it,
so the light b e a m is circulating in a h y p e r s p h e r e . This piece of s p a c e time has now d i s c o n n e c t e d itself from the s p a c e - t i m e a r o u n d it. Space
itself has now " r i p p e d . "
The Einstein-Rosen Bridge
T h e relativistic description of t h e black hole comes from the work of
Karl Schwarzschild. In 1916, barely a few m o n t h s after Einstein wrote
down his celebrated e q u a t i o n s , Schwarzschild was able to solve Einstein's
e q u a t i o n s exactly a n d calculate the gravitational field of a massive, stationary star.
Schwarzschild's solution has several interesting features. First, a
" p o i n t of no r e t u r n " s u r r o u n d s t h e black hole. Any object that comes
closer t h a n this radius will inevitably be sucked into the black hole, with
no possibility of escape. Inexorably, any p e r s o n u n f o r t u n a t e e n o u g h to
c o m e within t h e Schwarzschild radius would be c a p t u r e d by the black
h o l e a n d c r u s h e d to d e a t h . Today, this distance from the black hole is
called t h e Schwarzschild radius, or t h e horizon (the farthest visible p o i n t ) .
S e c o n d , a n y o n e w h o fell within t h e Schwarzschild radius would be
aware of a " m i r r o r u n i v e r s e " on the " o t h e r s i d e " of s p a c e - t i m e (Figure
10.2). Einstein was n o t worried a b o u t t h e existence of this bizarre m i r r o r
universe because c o m m u n i c a t i o n with it was impossible. Any space
p r o b e sent i n t o the c e n t e r of a black h o l e would e n c o u n t e r infinite
curvature; that is, the gravitational field would be infinite, a n d any material object would be c r u s h e d . T h e electrons would be r i p p e d off atoms,
a n d even the p r o t o n s a n d n e u t r o n s within the nuclei themselves would
be t o r n apart. Also, to p e n e t r a t e t h r o u g h to the alternative universe, the
p r o b e would have to go faster t h a n t h e speed of light, which is n o t
possible. T h u s a l t h o u g h this m i r r o r universe is mathematically necessary
to m a k e sense of t h e Schwarzschild solution, it could never be observed
physically.
Consequently, the celebrated Einstein-Rosen bridge c o n n e c t i n g these
two universes ( n a m e d after Einstein a n d his collaborator, N a t h a n
Rosen) was c o n s i d e r e d a m a t h e m a t i c a l quirk. T h e b r i d g e was necessary
225
Figure 10.2. The Einstein-Rosen bridge connects two different universes. Einstein believed that any rocket that entered the bridge would be crushed, thereby
making communication between these two universes impossible. However, more
recent calculations show that travel through the bridge might be very difficult, but
perhaps possible.
to have a mathematically consistent theory of the black hole, b u t it was
impossible to reach the m i r r o r universe by traveling t h r o u g h t h e Eins t e i n - R o s e n bridge. E i n s t e i n - R o s e n bridges were s o o n f o u n d in o t h e r
solutions of the gravitational e q u a t i o n s , such as t h e R e i s s n e r - N o r d s t r o m
solution describing an electrically c h a r g e d black hole. However, t h e Ein-
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s t e i n - R o s e n b r i d g e r e m a i n e d a curious b u t forgotten footnote in the
lore of relativity.
T h i n g s b e g a n to c h a n g e with the work of New Zealand mathematician Roy Kerr, w h o in 1963 f o u n d a n o t h e r exact solution to Einstein's
e q u a t i o n s . Kerr assumed that any collapsing star would be rotating. Like
a s p i n n i n g skater w h o speeds up w h e n b r i n g i n g in his or h e r h a n d s , a
rotating star would necessarily accelerate as it b e g a n to collapse. T h u s
the stationary Schwarzschild solution for a black h o l e was n o t the most
physically relevant solution of Einstein's equations.
Kerr's solution created a sensation in the field of relativity when it
was p r o p o s e d . Astrophysicist S u b r a h m a n y a n C h a n d r a s e k h a r o n c e said,
I n m y e n t i r e s c i e n t i f i c life, e x t e n d i n g o v e r forty-five y e a r s , t h e m o s t shattering e x p e r i e n c e has b e e n the realization that an e x a c t solution of Eins t e i n ' s e q u a t i o n s o f g e n e r a l relativity, d i s c o v e r e d b y t h e N e w Z e a l a n d m a t h ematician
R o y Kerr,
provides
the
absolutely exact representation of u n t o l d
n u m b e r s of massive black h o l e s that p o p u l a t e t h e universe. This "shudd e r i n g b e f o r e t h e b e a u t i f u l , " this i n c r e d i b l e fact that a discovery m o t i v a t e d
b y a s e a r c h a f t e r t h e b e a u t i f u l i n m a t h e m a t i c s s h o u l d f i n d its e x a c t r e p l i c a
i n N a t u r e , p e r s u a d e s m e t o say t h a t b e a u t y i s t h a t t o w h i c h t h e h u m a n
m i n d r e s p o n d s a t its d e e p e s t a n d m o s t p r o f o u n d l e v e l .
3
Kerr f o u n d , however, that a massive rotating star does n o t collapse
into a point. Instead, t h e s p i n n i n g star flattens until it eventually is compressed into a ring, which has interesting p r o p e r t i e s . If a p r o b e were
shot into t h e black h o l e from t h e side, it would hit t h e ring a n d be totally
d e m o l i s h e d . T h e curvature of s p a c e - t i m e is still infinite when approaching t h e r i n g from t h e side. T h e r e is still a " r i n g of d e a t h , " so to speak,
s u r r o u n d i n g the center. However, if a space p r o b e were shot into the
r i n g from t h e t o p or b o t t o m , it would e x p e r i e n c e a large b u t finite curvature; that is, the gravitational force would n o t be infinite.
This r a t h e r surprising conclusion from Kerr's solution m e a n s that
any space p r o b e shot t h r o u g h a s p i n n i n g black h o l e a l o n g its axis of
rotation might, in principle, survive t h e e n o r m o u s b u t finite gravitational fields at the center, a n d go right on t h r o u g h to t h e m i r r o r universe
without b e i n g destroyed by infinite curvature. T h e Einstein-Rosen
b r i d g e acts like a t u n n e l c o n n e c t i n g two regions of s p a c e - t i m e ;
it is a w o r m h o l e . T h u s t h e Kerr black h o l e is a gateway to a n o t h e r
universe.
Now i m a g i n e t h a t your r o c k e t has e n t e r e d t h e Einstein-Rosen
b r i d g e . As your rocket a p p r o a c h e s the s p i n n i n g black hole, it sees a
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ring-shaped s p i n n i n g star. At first, it a p p e a r s that t h e rocket is h e a d e d
for a disastrous crash l a n d i n g as it descends toward the black h o l e from
the n o r t h pole. However, as we get closer to the ring, light from the
m i r r o r universe reaches o u r sensors. Since all e l e c t r o m a g n e t i c radiation,
including radar, orbits t h e black hole, o u r r a d a r screens are d e t e c t i n g
signals that have b e e n circulating a r o u n d the black h o l e a n u m b e r of
times. This effect resembles a hall of mirrors, in which we are fooled by
the multiple images that s u r r o u n d us. Light goes r i c o c h e t i n g across
n u m e r o u s mirrors, creating the illusion that t h e r e are n u m e r o u s copies
of ourselves in the hall.
T h e same effect occurs as we pass t h r o u g h the Kerr black h o l e .
Because the same light b e a m orbits the black h o l e n u m e r o u s times, o u r
rocket's r a d a r detects images that have g o n e s p i n n i n g a r o u n d t h e black
hole, creating the illusion of objects t h a t a r e n ' t really t h e r e .
Warp Factor 5
Does this m e a n that black holes can be used for travel t h r o u g h o u t the
galaxy, as in Star Trek a n d o t h e r science-fiction movies?
As we saw earlier, the curvature in a certain space is d e t e r m i n e d by
the a m o u n t of m a t t e r - e n e r g y c o n t a i n e d in that space (Mach's princip l e ) . Einstein's famous e q u a t i o n gives us t h e precise d e g r e e of s p a c e time b e n d i n g caused by t h e p r e s e n c e of m a t t e r - e n e r g y .
W h e n Captain Kirk takes us soaring t h r o u g h hyperspace at " w a r p
factor 5 , " the " d i l i t h i u m crystals" that power the Enterprise must p e r f o r m
miraculous feats of warping space a n d time. This m e a n s that t h e dilit h i u m crystals have the magical power of b e n d i n g t h e s p a c e - t i m e cont i n u u m into pretzels; that is, they are t r e m e n d o u s storehouses of m a t t e r
a n d energy.
If the Enterprise travels from t h e e a r t h to the n e a r e s t star, it does n o t
physically move to Alpha C e n t a u r i — r a t h e r , Alpha C e n t a u r i c o m e s to
the Enterprise. Imagine sitting on a r u g a n d lassoing a table several feet
away. If we are strong e n o u g h a n d the floor is slick e n o u g h , we can pull
the lasso until t h e carpet begins to fold u n d e r n e a t h us. If we pull h a r d
e n o u g h , the table comes to us, a n d the " d i s t a n c e " between t h e table
a n d us disappears into a mass of c r u m p l e d carpeting. T h e n we simply
h o p across this " c a r p e t w a r p . " In o t h e r words, we have hardly moved;
the space between us a n d the table has c o n t r a c t e d , a n d we j u s t step
across this c o n t r a c t e d distance. Similarly, the Enterprise does n o t really
cross the entire space to Alpha C e n t a u r i ; it simply moves across t h e crum-
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UNIVERSE?
pled s p a c e - t i m e — t h r o u g h a w o r m h o l e . To b e t t e r u n d e r s t a n d what happ e n s w h e n o n e falls down t h e E i n s t e i n - R o s e n bridge, let us now discuss
t h e topology of wormholes.
To visualize these multiply c o n n e c t e d spaces, imagine that we are
strolling down New York's Fifth Avenue o n e b r i g h t afternoon, m i n d i n g
o u r own business, w h e n a strange floating window o p e n s up in front of
us, m u c h like Alice's looking glass. (Never m i n d for t h e m o m e n t that
t h e energy necessary to o p e n this window m i g h t be e n o u g h to shatter
the e a r t h . This is a purely hypothetical example.)
We step up to t h e h o v e r i n g window to take a closer look, a n d are
horrified to find ourselves staring at the h e a d of a nasty-looking Tyrannosaurus rex. We are a b o u t to r u n for o u r lives, w h e n we notice that the
tyrannosaur has no body. He c a n ' t h u r t us because his entire body is
clearly on t h e o t h e r side of the window. W h e n we look below the window
to find t h e d i n o s a u r ' s body, we can see all t h e way down the street, as
t h o u g h the d i n o s a u r a n d t h e window w e r e n ' t t h e r e at all. Puzzled, we
slowly circle t h e window a n d are relieved to find that the tyrannosaur is
n o w h e r e to be found. However, w h e n we p e e r into the window from the
back side, we see the h e a d of a b r o n t o s a u r staring us in the face (Figure
10.3)!
F r i g h t e n e d , we walk a r o u n d t h e window o n c e m o r e , staring at the
window sideways. M u c h to o u r surprise, all traces of the window, the
tyrannosaur, a n d the b r o n t o s a u r are g o n e . We now take a few m o r e turns
a r o u n d the floating window. F r o m o n e direction, we see the h e a d of the
tyrannosaur. F r o m t h e o t h e r direction, we see the h e a d of the b r o n t o saur. A n d w h e n we look from t h e side, we find that b o t h t h e m i r r o r a n d
the dinosaurs have d i s a p p e a r e d .
What's happening?
In s o m e faraway universe, t h e tyrannosaur a n d the b r o n t o s a u r have
s q u a r e d off in a life-and-death confrontation. As they face each other, a
floating window suddenly a p p e a r s between t h e m . W h e n the tyrannosaur
p e e r s into t h e floating m i r r o r , he is startled to see t h e h e a d of a puny,
skinny-looking m a m m a l , with frizzy hair a n d a tiny face: a h u m a n . T h e
h e a d is clearly visible, b u t it has no body. However, when the b r o n t o s a u r
stares i n t o t h e same window from t h e o t h e r direction, he sees Fifth Aven u e , with its shops a n d traffic. T h e n t h e tyrannosaur finds that this
h u m a n c r e a t u r e in t h e window has d i s a p p e a r e d , only to a p p e a r on the
side of t h e window facing t h e b r o n t o s a u r .
Now let us say that suddenly the wind blows o u r h a t into the window.
We see t h e h a t sailing into the sky of t h e o t h e r universe, b u t it is n o w h e r e
to be seen a l o n g Fifth Avenue. We take o n e long gulp, a n d t h e n , in
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Figure 10.3. In this purely hypothetical example, a "window" or wormhole has
opened up in our universe. If we look into the window from one direction, we see
one dinosaur. If we look into the other side of the window, we see another dinosaur.
As seen from the other universe, a window has opened up between the two dinosaurs. Inside the window, the dinosaurs see a strange small animal (us).
desperation, we stick o u r h a n d i n t o t h e window to retrieve t h e hat. As
seen by the tyrannosaur, a h a t blows o u t t h e window, a p p e a r i n g from
n o w h e r e . T h e n he sees a d i s e m b o d i e d h a n d r e a c h i n g o u t t h e window,
desperately g r o p i n g for t h e hat.
T h e wind now c h a n g e s direction, a n d t h e h a t is carried in t h e o t h e r
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TO A N O T H E R
UNIVERSE?
Figure 10.4. If we insert our hands into the window from two different directions,
then it appears as though our hands have disappeared. We have a body, but no
hands. In the alternative universe, two hands have emerged from either side of
the window but they are not attached to a body.
direction. W e stick o u r o t h e r h a n d into t h e window, b u t from the o t h e r
side. We are now in an awkward position. Both o u r h a n d s are sticking
into t h e window, b u t from different sides. But we c a n ' t see o u r fingers.
Instead, it a p p e a r s to us that b o t h h a n d s have d i s a p p e a r e d .
H o w d o e s this a p p e a r to t h e dinosaurs? They see two wiggling, tiny
h a n d s d a n g l i n g from t h e window, from either side. But t h e r e is no body
(Figure 10.4).
This e x a m p l e illustrates s o m e of t h e delicious distortions of space
a n d time that o n e can invent with multiply c o n n e c t e d spaces.
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Closing the Wormhole
It seems r e m a r k a b l e t h a t such a simple i d e a — t h a t h i g h e r d i m e n s i o n s
can unify space with time, a n d that a " f o r c e " can be e x p l a i n e d by t h e
warping of that s p a c e - t i m e — w o u l d lead to such a rich diversity of physical consequences. However, with the w o r m h o l e a n d multiply c o n n e c t e d
spaces, we are p r o b i n g t h e very limits of Einstein's theory of g e n e r a l
relativity. In fact, the a m o u n t of m a t t e r - e n e r g y necessary to create a
w o r m h o l e or d i m e n s i o n a l gateway is so large that we expect q u a n t u m
effects to d o m i n a t e . Q u a n t u m corrections, in t u r n , may actually close
the o p e n i n g of t h e w o r m h o l e , m a k i n g travel t h r o u g h the gateway impossible.
Since n e i t h e r q u a n t u m theory n o r relativity is powerful e n o u g h to
settle this question, we will have to wait until t h e ten-dimensional theory
is c o m p l e t e d to decide w h e t h e r these w o r m h o l e s are physically relevant
or j u s t a n o t h e r crazy idea. However, before we discuss t h e question of
q u a n t u m corrections a n d the ten-dimensional theory, let us n o w pause
a n d consider p e r h a p s the most bizarre c o n s e q u e n c e of w o r m h o l e s . J u s t
as physicists can show that w o r m h o l e s allow for multiply c o n n e c t e d
spaces, we can also show that they allow for time travel as well.
Let us now consider p e r h a p s the most fascinating, a n d speculative,
c o n s e q u e n c e of multiply c o n n e c t e d universes: b u i l d i n g a time m a c h i n e .
II
To Build a Time Machine
P e o p l e like us, w h o b e l i e v e in physics, k n o w that the distinct i o n b e t w e e n past, p r e s e n t , a n d future is o n l y a s t u b b o r n l y
persistent illusion.
Albert Einstein
Time Travel
C
AN we go backward in time?
Like the protagonist in H. G. Wells's The Time Machine, can we
spin t h e dial of a m a c h i n e a n d leap h u n d r e d s of t h o u s a n d s of years to
the year 802,701? O r , like Michael J. Fox, can we h o p into o u r plutonium-fired cars a n d go back to the future?
T h e possibility of time travel o p e n s up a vast world of interesting
possibilities. Like Kathleen T u r n e r in Peggy Sue Got Married, everyone
h a r b o r s a secret wish s o m e h o w to relive t h e past a n d correct some small
b u t vital mistake in o n e ' s life. In R o b e r t Frost's p o e m " T h e Road N o t
T a k e n , " we w o n d e r what m i g h t have h a p p e n e d , at key j u n c t u r e s in o u r
lives, if we h a d m a d e different choices a n d taken a n o t h e r p a t h . With
time travel, we could go back to o u r youth a n d erase embarrassing events
from o u r past, choose a different m a t e , or e n t e r different careers; or we
c o u l d even c h a n g e t h e o u t c o m e of key historical events a n d alter t h e
fate of h u m a n i t y .
For e x a m p l e , in the climax of Superman, o u r h e r o is emotionally devastated w h e n an e a r t h q u a k e ravages most of California a n d crushes his
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lover u n d e r h u n d r e d s o f tons o f rock a n d debris. M o u r n i n g h e r h o r r i b l e
d e a t h , he is so o v e r c o m e by anguish that he rockets into space a n d violates his oath n o t to t a m p e r with the course of h u m a n history. He
increases his velocity until he shatters t h e light barrier, d i s r u p t i n g the
fabric of space a n d time. By traveling at t h e s p e e d of light, he forces
time to slow down, t h e n to stop, a n d finally to go backward, to a time
before Lois L a n e was c r u s h e d to d e a t h .
This trick, however, is clearly n o t possible. A l t h o u g h time does slow
down when you increase your velocity, you c a n n o t go faster t h a n t h e
speed of light (and h e n c e m a k e time go backward) because special relativity states that your mass would b e c o m e infinite in t h e process. T h u s
the faster-than-light travel m e t h o d p r e f e r r e d by most science-fiction writers contradicts the special theory of relativity.
Einstein h i m s e l f was well aware of this impossibility, as was
A. H. R. Buller w h e n he p u b l i s h e d t h e following limerick in Punch :
1
T h e r e w a s a y o u n g l a d y girl n a m e d B r i g h t ,
W h o s e s p e e d w a s far f a s t e r t h a n l i g h t ,
S h e t r a v e l e d o n e day,
In a r e l a t i v e way,
A n d returned on the previous night.
Most scientists, who have n o t seriously studied Einstein's e q u a t i o n s ,
dismiss time travel as poppycock, with as m u c h validity as lurid accounts
of kidnappings by space aliens. However, t h e situation is actually quite
complex.
To resolve the question, we must leave t h e simpler theory of special
relativity, which forbids time travel, a n d e m b r a c e the full power of the
general theory of relativity, which may p e r m i t it. G e n e r a l relativity has
m u c h wider validity t h a n special relativity. While special relativity
describes only objects moving at constant velocity far away from any stars,
the general theory of relativity is m u c h m o r e powerful, capable of
describing rockets accelerating n e a r supermassive stars a n d black holes.
T h e general theory therefore supplants s o m e of the simpler conclusions
of the special theory. For any physicist w h o has seriously analyzed t h e
mathematics of time travel within Einstein's general theory of relativity,
the final conclusion is, surprisingly e n o u g h , far from clear.
P r o p o n e n t s of time travel p o i n t o u t that Einstein's e q u a t i o n s for general relativity do allow s o m e forms of time travel. T h e y acknowledge,
however, that the energies necessary to twist time i n t o a circle a r e so
great that Einstein's e q u a t i o n s b r e a k down. In t h e physically interesting
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UNIVERSE?
r e g i o n w h e r e time travel b e c o m e s a serious possibility, q u a n t u m theory
takes over from g e n e r a l relativity.
Einstein's e q u a t i o n s , we recall, state that the curvature or b e n d i n g
of space a n d time is d e t e r m i n e d by t h e m a t t e r - e n e r g y c o n t e n t of the
universe. It is, in fact, possible to find configurations of m a t t e r - e n e r g y
powerful e n o u g h to force the b e n d i n g of time a n d allow for time travel.
However, the c o n c e n t r a t i o n s of m a t t e r - e n e r g y necessary to b e n d time
backward are so vast that g e n e r a l relativity breaks down a n d q u a n t u m
corrections b e g i n to d o m i n a t e over relativity. T h u s the final verdict on
time travel c a n n o t be answered within the framework of Einstein's equations, which b r e a k down in extremely large gravitational fields, where
we expect q u a n t u m theory to b e c o m e d o m i n a n t .
This is w h e r e t h e hyperspace theory can settle t h e question. Because
b o t h q u a n t u m theory a n d Einstein's theory of gravity are united in tend i m e n s i o n a l space, we expect that t h e question of time travel will be
settled decisively by the hyperspace theory. As in the case of wormholes
a n d d i m e n s i o n a l windows, the final c h a p t e r will be written when we
i n c o r p o r a t e t h e full power of t h e hyperspace theory.
Let us now describe t h e controversy s u r r o u n d i n g time travel a n d the
delicious p a r a d o x e s that inevitably arise.
Collapse of Causality
Science-fiction writers have often w o n d e r e d what m i g h t h a p p e n if a single individual went back in time. Many of these stories, on the surface,
a p p e a r plausible. But imagine the chaos that would arise if time
m a c h i n e s were as c o m m o n as automobiles, with tens of millions of t h e m
commercially available. Havoc would soon break loose, tearing at the
fabric of o u r universe. Millions of p e o p l e would go back in time to meddle with their own past a n d the past of others, rewriting history in the
process. A few m i g h t even go back in time a r m e d with guns to shoot
down t h e p a r e n t s of their e n e m i e s before they were b o r n . It would thus
be impossible to take a simple census to see how m a n y p e o p l e t h e r e were
at any given time.
If time travel is possible, t h e n the laws of causality c r u m b l e . In fact,
all of history as we know it m i g h t collapse as well. Imagine the chaos
caused by t h o u s a n d s of p e o p l e going back in time to alter key events
that c h a n g e d t h e course of history. All of a s u d d e n , the a u d i e n c e at
F o r d ' s T h e a t e r would be c r a m m e d with p e o p l e from t h e future bickering a m o n g themselves to see w h o would have the h o n o r of preventing
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Lincoln's assassination. T h e l a n d i n g at N o r m a n d y would be b o t c h e d as
t h o u s a n d s of thrill seekers with cameras arrived to take pictures.
T h e key battlefields of history would be c h a n g e d b e y o n d r e c o g n i t i o n .
Consider A l e x a n d e r t h e Great's decisive victory over t h e Persians, led by
Darius III, in 331 B.C. at t h e Battle of Gaugamela. This battle led to the
collapse of the Persian forces a n d e n d e d their rivalry with t h e West,
which h e l p e d allow t h e flourishing of Western civilization a n d culture
over the world for t h e n e x t 1,000 years. But consider what would h a p p e n
if a small b a n d of a r m e d m e r c e n a r i e s e q u i p p e d with small rockets a n d
m o d e r n artillery were to e n t e r t h e battle. T h e slightest display of m o d e r n
firepower would r o u t A l e x a n d e r ' s terrified soldiers. This m e d d l i n g in
the past would cripple t h e e x p a n s i o n of Western influence in t h e world.
T i m e travel would m e a n that any historical event could never be
completely resolved. History books could never be written. S o m e dieh a r d would always be trying to assassinate G e n e r a l Ulysses S. G r a n t or
give the secret of the atomic b o m b to t h e G e r m a n s in the 1930s.
W h a t would h a p p e n if history could be rewritten as casually as erasing
a blackboard? O u r past would be like the shifting sands at the seashore,
constantly blown this way or t h a t by t h e slightest b r e e z e . History would
be constantly c h a n g i n g every time s o m e o n e s p u n t h e dial of a time
m a c h i n e a n d b l u n d e r e d his or h e r way into t h e past. History, as we know
it, would be impossible. It would cease to exist.
Most scientists obviously do n o t relish this u n p l e a s a n t possibility. N o t
only would it be impossible for historians to m a k e any sense o u t of "history," b u t g e n u i n e p a r a d o x e s immediately arise whenever we e n t e r t h e
past or future. Cosmologist S t e p h e n Hawking, in fact, has u s e d this situation to provide " e x p e r i m e n t a l " evidence that time travel is n o t
possible. He believes t h a t time travel is n o t possible by " t h e
fact that we have n o t b e e n invaded by h o a r d e s of tourists from t h e
future."
Time Paradoxes
To u n d e r s t a n d t h e p r o b l e m s with time travel, it is first necessary to classify the various p a r a d o x e s . In general, most can be b r o k e n down into
o n e of two principal types:
1. Meeting your p a r e n t s before you are b o r n
2. T h e m a n with no past
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UNIVERSE?
T h e first type of time travel d o e s t h e most d a m a g e to the fabric of
s p a c e - t i m e because it alters previously r e c o r d e d events. For example,
r e m e m b e r that in Back to the Future, o u r y o u n g h e r o goes back in time
a n d m e e t s his m o t h e r as a y o u n g girl, j u s t before she falls in love with
his father. To his shock a n d dismay, he finds that he has inadvertently
p r e v e n t e d t h e fateful e n c o u n t e r between his p a r e n t s . To make matters
worse, his y o u n g m o t h e r has now b e c o m e amorously attracted to him!
If he unwittingly prevents his m o t h e r a n d father from falling in love a n d
is u n a b l e to divert his m o t h e r ' s misplaced affections, he will disappear
because his birth will never h a p p e n .
T h e s e c o n d p a r a d o x involves events without any b e g i n n i n g . For
e x a m p l e , let's say that an impoverished, struggling inventor is trying to
construct the world's first time m a c h i n e in his cluttered basement. O u t
of n o w h e r e , a wealthy, elderly g e n t l e m a n a p p e a r s a n d offers h i m ample
funds a n d t h e c o m p l e x e q u a t i o n s a n d circuitry to m a k e a time m a c h i n e .
T h e inventor subsequently e n r i c h e s himself with t h e knowledge of time
travel, knowing b e f o r e h a n d exactly w h e n stock-market b o o m s a n d busts
will occur before they h a p p e n . He makes a fortune betting on the stock
market, h o r s e races, a n d o t h e r events. Decades later, as a wealthy, aging
m a n , he goes back in time to fulfill his destiny. He meets himself as a
y o u n g m a n working in his b a s e m e n t , a n d gives his y o u n g e r self the secret
of time travel a n d t h e m o n e y to exploit it. T h e question is: W h e r e did
t h e idea of time travel c o m e from?
P e r h a p s t h e craziest of these time travel p a r a d o x e s of the second type
was c o o k e d up by R o b e r t H e i n l e i n in his classic short story "All You
Zombies—."
A baby girl is mysteriously d r o p p e d off at an o r p h a n a g e in Cleveland
in 1945. " J a n e " grows up lonely a n d dejected, n o t knowing who h e r
p a r e n t s a r e , until o n e day in 1963 she is strangely attracted to a drifter.
She falls in love with h i m . But j u s t w h e n things are finally looking up for
J a n e , a series of disasters strike. First, she b e c o m e s p r e g n a n t by the
drifter, w h o t h e n disappears. Second, d u r i n g t h e complicated delivery,
d o c t o r s find that J a n e has b o t h sets of sex organs, a n d to save h e r life,
they are forced to surgically convert " h e r " to a " h i m . " Finally, a mysterious stranger kidnaps h e r baby from t h e delivery r o o m .
Reeling from these disasters, rejected by society, scorned by fate,
" h e " b e c o m e s a d r u n k a r d a n d drifter. N o t only has J a n e lost h e r parents
a n d h e r lover, b u t he has lost his only child as well. Years later, in 1970,
he stumbles into a lonely bar, called P o p ' s Place, a n d spills o u t his
p a t h e t i c story to an elderly b a r t e n d e r . T h e sympathetic b a r t e n d e r offers
the drifter t h e c h a n c e to avenge t h e stranger w h o left h e r p r e g n a n t a n d
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a b a n d o n e d , o n the c o n d i t i o n t h a t h e j o i n t h e " t i m e travelers c o r p s . "
Both of t h e m e n t e r a time m a c h i n e , a n d t h e b a r t e n d e r d r o p s off t h e
drifter in 1963. T h e drifter is strangely attracted to a y o u n g o r p h a n
w o m a n , w h o subsequently b e c o m e s p r e g n a n t .
T h e b a r t e n d e r t h e n goes forward 9 m o n t h s , kidnaps t h e baby girl
from the hospital, a n d d r o p s off t h e baby in an o r p h a n a g e back in 1945.
T h e n the b a r t e n d e r d r o p s off the t h o r o u g h l y confused drifter in 1985,
to enlist in the time travelers corps. T h e drifter eventually gets his life
together, b e c o m e s a respected a n d elderly m e m b e r of the time travelers
corps, a n d t h e n disguises himself as a b a r t e n d e r a n d has his most difficult mission: a d a t e with destiny, m e e t i n g a certain drifter at P o p ' s Place
in 1970.
T h e question is: W h o is J a n e ' s m o t h e r , father, g r a n d f a t h e r , g r a n d m o t h e r , son, d a u g h t e r , g r a n d d a u g h t e r , a n d g r a n d s o n ? T h e girl, t h e
drifter, a n d the b a r t e n d e r , of course, are all the same p e r s o n . T h e s e
p a r a d o x e s can m a d e your h e a d spin, especially if you try to u n t a n g l e
J a n e ' s twisted p a r e n t a g e . If we draw J a n e ' s family tree, we find that all
the b r a n c h e s are curled inward back on themselves, as in a circle. We
c o m e to the astonishing conclusion that she is h e r own m o t h e r a n d
father! She is an e n t i r e family tree u n t o herself.
World Lines
Relativity gives us a simple m e t h o d to sort t h r o u g h t h e thorniest of these
paradoxes. We will m a k e use of the "world l i n e " m e t h o d , p i o n e e r e d by
Einstein.
For example, say o u r alarm clock wakes us up o n e day at 8:00 A.M.,
a n d we decide to s p e n d t h e m o r n i n g in b e d instead of g o i n g to work.
Although it a p p e a r s that we a r e d o i n g n o t h i n g by loafing in bed, we a r e
actually tracing o u t a "world l i n e . "
Take a sheet of g r a p h p a p e r , a n d on the horizontal scale p u t "dist a n c e " a n d on the vertical scale p u t " t i m e . " If we simply lie in b e d from
8:00 to 12:00, o u r world line is a straight vertical line. We went 4 h o u r s
into the future, b u t traveled no distance. Even e n g a g i n g in o u r favorite
pastime, d o i n g n o t h i n g , creates a world line. (If s o m e o n e ever criticizes
us for loafing, we can truthfully claim that, a c c o r d i n g to Einstein's theory
of relativity, we are tracing o u t a world line in four-dimensional s p a c e time.)
Now let's say that we finally get o u t of b e d at n o o n a n d arrive at work
at 1:00 P.M. O u r world line b e c o m e s slanted because we are moving in
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space as well as time. In t h e lower left c o r n e r is o u r h o m e , a n d on the
u p p e r right is o u r office (Figure 11.1) If we take the car to work, t h o u g h ,
we arrive at t h e office earlier, at 12:30. This m e a n s that the faster we
travel, t h e m o r e o u r world line deviates from t h e vertical. (Notice that
t h e r e is also a " f o r b i d d e n r e g i o n " in the d i a g r a m that o u r world line
c a n ' t e n t e r because we would have to be going faster t h a n the speed of
light.)
O n e conclusion is i m m e d i a t e . O u r world line never really begins or
e n d s . Even w h e n we die, t h e world lines of t h e molecules in o u r bodies
k e e p going. T h e s e molecules may disperse into t h e air or soil, b u t they
will trace o u t their own never-ending world lines. Similarly, w h e n we are
b o r n , t h e world lines of t h e molecules c o m i n g from o u r m o t h e r coalesce
into a baby. At no p o i n t do world lines b r e a k off or a p p e a r from n o t h i n g .
To see how this all fits t o g e t h e r , take the simple e x a m p l e of o u r own
p e r s o n a l world line. In 1950, say, o u r m o t h e r a n d father met, fell in love,
a n d p r o d u c e d a baby (us). T h u s t h e world lines of o u r m o t h e r a n d father
collided a n d p r o d u c e d a third world line ( o u r s ) . Eventually, when someo n e dies, the world lines forming t h e p e r s o n disperse into billions of
world lines of o u r molecules. F r o m this p o i n t of view, a h u m a n being
can be defined as a t e m p o r a r y collection of world lines of molecules.
T h e s e world lines were scattered before we were b o r n , c a m e t o g e t h e r to
form o u r bodies, a n d will rescatter after we die. T h e Bible says, "from
dust to d u s t . " In this relativistic p i c t u r e , we m i g h t say, "from world lines
to world l i n e s . "
O u r world line thus contains the entire body of information conc e r n i n g o u r history. Everything that ever h a p p e n e d to us—from o u r first
bicycle, to o u r first date, to o u r first j o b — i s r e c o r d e d in o u r world line.
In fact, t h e great Russian cosmologist G e o r g e Gamow, who was famous
for a p p r o a c h i n g Einstein's work with wit a n d whimsy, aptly titled his
a u t o b i o g r a p h y My World Line.
With t h e aid of t h e world line, we can now picture what h a p p e n s
w h e n we go back in time. Let's say we e n t e r a time m a c h i n e a n d m e e t
o u r m o t h e r before we are b o r n . Unfortunately, she falls in love with us
a n d jilts o u r father. Do we really disappear, as depicted in Back to the
Future? On a world line, we n o w see why this is impossible. W h e n we
disappear, o u r world line disappears. However, a c c o r d i n g to Einstein,
world lines c a n n o t be cut. T h u s altering the past is n o t possible in relativity.
T h e s e c o n d p a r a d o x , involving re-creating t h e past, poses interesting
p r o b l e m s , however. For e x a m p l e , by g o i n g back in time, we are fulfilling
t h e past, n o t destroying it. T h u s t h e world line of t h e inventor of time
Figure 11.1. Our world line summarizes our entire history, from birth to death.
For example, if we lie in bed from 8:00 A.M. to 12:00, our world line is a vertical
line. If we travel by car to work, then our world becomes a slanted line. The faster
we move, the more slanted our world line becomes. The fastest we can travel,
however, is the speed of light. Thus part of this space-time diagram is "forbidden ";
that is, we would have to go faster than the speed of light to enter into this
forbidden zone.
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travel is a closed loop. His world line fulfills, r a t h e r t h a n changes, the
past.
M u c h m o r e complicated is t h e world line of " J a n e , " t h e w o m a n who
is h e r own m o t h e r a n d father a n d son a n d d a u g h t e r (Figure 11.2).
Notice, o n c e again, that we c a n n o t alter t h e past. W h e n o u r world
line goes back in time, it simply fulfills what is already known. In such a
universe, therefore, it is possible to m e e t yourself in the past. If we live
t h r o u g h o n e cycle, t h e n s o o n e r or later we m e e t a y o u n g m a n or woman
w h o h a p p e n s to be ourselves w h e n we were younger. We tell this y o u n g
p e r s o n that he or she looks suspiciously familiar. T h e n , thinking a bit,
we r e m e m b e r that w h e n we were young, we m e t a curious, o l d e r person
w h o claimed that we looked familiar.
T h u s p e r h a p s we can fulfill the past, b u t never alter it. World lines,
as we have stressed, c a n n o t be cut a n d c a n n o t e n d . They can p e r h a p s
p e r f o r m loops in time, b u t never alter it.
T h e s e light c o n e diagrams, however, have b e e n p r e s e n t e d only in the
framework of special relativity, which can describe what h a p p e n s if we
e n t e r t h e past, b u t is too primitive to settle the question of w h e t h e r time
travel makes any sense. To answer this larger question, we must turn to
t h e g e n e r a l theory of relativity, w h e r e t h e situation b e c o m e s m u c h m o r e
delicate.
With t h e full power of g e n e r a l relativity, we see that these twisted
world lines m i g h t be physically allowed. T h e s e closed loops go by the
scientific n a m e closed timelike curves (CTCs). T h e d e b a t e in scientific circles is w h e t h e r CTCs are allowed by g e n e r a l relativity a n d q u a n t u m
theory.
Spoiler of Arithmetic and General Relativity
In 1949, Einstein was c o n c e r n e d a b o u t a discovery by o n e of his close
colleagues a n d friends, t h e Viennese m a t h e m a t i c i a n Kurt Godel, also at
the Institute for Advanced Study at P r i n c e t o n , w h e r e Einstein worked.
Godel f o u n d a d i s t u r b i n g solution to Einstein's e q u a t i o n s that allowed
for violations of t h e basic tenets of c o m m o n sense: His solution allowed
for certain forms of time travel. For t h e first time in history, time travel
was given a m a t h e m a t i c a l f o u n d a t i o n .
In s o m e quarters, Godel was known as a spoiler. In 1931, he b e c a m e
famous (or, actually, infamous) w h e n he proved, contrary to every expectation, that you c a n n o t prove t h e self-consistency of arithmetic. In the
process, he r u i n e d a 2,000-year-old d r e a m , d a t i n g back to Euclid a n d
Figure 11.2. If time travel is possible, then our world line becomes a closed loop.
In 1945, the girl is born. In 1963, she has a baby. In 1970, he is a drifter, who
goes back to 1945 to meet himself. In 1985, he is a time traveler, who picks himself
up in a bar in 1970, takes himself back to 1945, kidnaps the baby and takes her
back to 1945, to start all over again. The girl is her own mother, father, grandfather, grandmother, son, daughter, and so on.
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the Greeks, which was to have b e e n the crowning a c h i e v e m e n t of mathematics: to r e d u c e all of m a t h e m a t i c s to a small, self-consistent set of
axioms from which everything could be derived.
In a m a t h e m a t i c a l t o u r de force, Godel showed that t h e r e will always
be t h e o r e m s in arithmetic whose correctness or incorrectness can never
be d e m o n s t r a t e d from the axioms of arithmetic; that is, arithmetic will
always be i n c o m p l e t e . G o d e l ' s result was t h e most startling, u n e x p e c t e d
d e v e l o p m e n t in m a t h e m a t i c a l logic in p e r h a p s a t h o u s a n d years.
Mathematics, o n c e t h o u g h t to be t h e p u r e s t of all sciences because
it was precise a n d certain, u n t a r n i s h e d by the u n p l e a s a n t crudeness of
o u r material world, now b e c a m e u n c e r t a i n . After Godel, the fundamental basis for m a t h e m a t i c s s e e m e d to be left adrift. (Crudely speaking,
G o d e l ' s r e m a r k a b l e p r o o f b e g a n by showing that t h e r e are curious paradoxes in logic. For e x a m p l e , consider the s t a t e m e n t " T h i s sentence is
false." If t h e s e n t e n c e is true, t h e n it follows that it is false. If the sentence
is false, t h e n the s e n t e n c e is true. Or consider the s t a t e m e n t "I am a
liar." T h e n I am a liar only if I tell t h e truth. Godel t h e n formulated the
s t a t e m e n t " T h i s s e n t e n c e c a n n o t be proved t r u e . " If the sentence is
correct, t h e n it c a n n o t be proved to be correct. By carefully building a
c o m p l e x web of such p a r a d o x e s , G o d e l showed that t h e r e are true statem e n t s that c a n n o t b e proved using arithmetic.)
After d e m o l i s h i n g o n e of the most c h e r i s h e d d r e a m s of all of mathematics, G o d e l n e x t shattered t h e conventional wisdom s u r r o u n d i n g
Einstein's equations. He showed that Einstein's theory contains some
surprising pathologies, i n c l u d i n g time travel.
He first assumed that t h e universe was filled with gas or dust that was
slowly rotating. This s e e m e d reasonable, since the far reaches of the
universe do s e e m to be filled with gas a n d dust. However, Godel's solution caused great c o n c e r n for two reasons.
First, his solution violated Mach's principle. He showed that two solutions of Einstein's e q u a t i o n s were possible with the same distribution of
dust a n d gas. (This m e a n t that M a c h ' s principle was somehow incomplete, that h i d d e n assumptions were present.)
M o r e i m p o r t a n t , he showed t h a t certain forms of time travel were
p e r m i t t e d . If o n e followed t h e p a t h of a particle in a Godel universe,
eventually it would c o m e back a n d m e e t itself in t h e past. He wrote, "By
m a k i n g a r o u n d trip on a r o c k e t ship in a sufficiently wide curve, it is
possible in these worlds to travel i n t o any region of t h e past, present,
a n d future, a n d back a g a i n . " T h u s Godel f o u n d t h e f i r s t CTC i n general
relativity.
Previously, Newton c o n s i d e r e d time to be moving like a straight
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arrow, which unerringly flies forward toward its target. N o t h i n g could
deflect or c h a n g e the course of this arrow o n c e it was shot. Einstein,
however, showed that time was m o r e like a mighty river, moving forward
b u t often m e a n d e r i n g t h r o u g h twisting valleys a n d plains. T h e p r e s e n c e
of matter or energy m i g h t m o m e n t a r i l y shift t h e direction of t h e river,
b u t overall the river's course was s m o o t h : It never abruptly e n d e d or
j e r k e d backward. However, Godel showed that t h e river of time could
be smoothly b e n t backward into a circle. Rivers, after all, have eddy
currents a n d whirlpools. In t h e m a i n , a river may flow forward, b u t at
the edges t h e r e are always side pools w h e r e water flows in a circular
motion.
Godel's solution could n o t be dismissed as t h e work of a crackpot
because Godel h a d used Einstein's own field e q u a t i o n s to find strange
solutions in which time b e n t into a circle. Because Godel h a d played by
the rules a n d discovered a legitimate solution to his e q u a t i o n s , Einstein
was forced to take t h e evasive r o u t e a n d dismiss it because it did n o t fit
the experimental data.
T h e weak spot in Godel's universe was the assumption that t h e gas
a n d dust in the universe were slowly rotating. Experimentally, we do n o t
see any rotation of the cosmic dust a n d gas in space. O u r i n s t r u m e n t s
have verified that the universe is e x p a n d i n g , b u t it does n o t a p p e a r to
be rotating. T h u s the Godel universe can be safely ruled out. (This leaves
us with the r a t h e r disturbing, a l t h o u g h plausible, possibility that if o u r
universe did rotate, as Godel speculated, t h e n CTCs a n d time travel
would be physically possible.)
Einstein died in 1955, c o n t e n t that disturbing solutions to his equations could be swept u n d e r the r u g for e x p e r i m e n t a l reasons a n d that
people could n o t m e e t their p a r e n t s before they were b o r n .
Living in the Twilight Zone
T h e n , in 1963, Ezra Newman, T h e o d o r e Unti, a n d Louis T a m b u r i n o
discovered a new solution to Einstein's e q u a t i o n s that was even crazier
t h a n Godel's. Unlike the Godel universe, their solution was n o t based
on a rotating dust-filled universe. On t h e surface, it r e s e m b l e d a typical
black hole.
As in the Godel solution, their universe allowed for CTCs a n d time
travel. Moreover, when -going 360 d e g r e e s a r o u n d the black h o l e , you
would n o t wind up w h e r e you originally started. Instead, like living on
a universe with a R i e m a n n cut, you would wind up on a n o t h e r s h e e t of
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the universe. T h e topology of a N e w m a n - U n t i - T a m b u r i n o universe
m i g h t be c o m p a r e d to living on a spiral staircase. If we move 360 degrees
a r o u n d t h e staircase, we do n o t arrive at t h e same p o i n t at which we
started, b u t on a n o t h e r l a n d i n g of t h e staircase. Living in such a universe
would surpass o u r worst n i g h t m a r e , with c o m m o n sense being completely thrown o u t the window. In fact, this bizarre universe was so pathological t h a t it was quickly c o i n e d t h e N U T universe, after the initials of
its creators.
At first, relativists dismissed the N U T solution in the same way they
h a d dismissed t h e Godel solution; that is, o u r universe d i d n ' t seem to
evolve in t h e way p r e d i c t e d by these solutions, so they were arbitrarily
discarded for e x p e r i m e n t a l reasons. However, as the decades went by,
t h e r e was a flood of such bizarre solutions to Einstein's equations that
allowed for time travel. In t h e early 1970s, Frank J. Tipler at T u l a n e
University in New O r l e a n s reanalyzed an old solution to Einstein's equations f o u n d by W . J . van Stockum in 1936, even before Godel's solution.
This solution assumed t h e existence of an infinitely long, rotating cyli n d e r . Surprisingly e n o u g h , Tipler was able to show that this solution
also violated causality.
Even t h e Kerr solution (which r e p r e s e n t s the most physically realistic
description of black holes in o u t e r space) was shown to allow for time
travel. Rocket ships that pass t h r o u g h t h e c e n t e r of the Kerr black hole
(assuming they are n o t c r u s h e d in t h e process) could violate causality.
Soon, physicists f o u n d that NUT-type singularities could be inserted
into any black h o l e or e x p a n d i n g universe. In fact, it now b e c a m e possible to cook up an infinite n u m b e r of pathological solutions to Einstein's e q u a t i o n s . For e x a m p l e , every w o r m h o l e solution to Einstein's
e q u a t i o n s could be shown to allow s o m e form of time travel.
A c c o r d i n g to relativist Frank Tipler, "solutions to the field equations
can be f o u n d which exhibit virtually any type of bizarre b e h a v i o r . " T h u s
an explosion of pathological solutions to Einstein's equations was discovered that certainly would have horrified Einstein h a d he still b e e n
alive.
Einstein's e q u a t i o n s , in some sense, were like a Trojan horse. On the
surface, t h e h o r s e looks like a perfectly acceptable gift, giving us the
observed b e n d i n g of starlight u n d e r gravity a n d a compelling explanation of t h e origin of t h e universe. However, inside lurk all sorts of strange
d e m o n s a n d goblins, which allow for the possibility of interstellar travel
t h r o u g h w o r m h o l e s a n d time travel. T h e price we h a d to pay for p e e r i n g
i n t o the darkest secrets of the universe was t h e potential downfall of
s o m e of o u r most c o m m o n l y h e l d beliefs a b o u t o u r world—that its space
is simply c o n n e c t e d a n d its history is u n a l t e r a b l e .
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But the question still r e m a i n e d : Could these CTCs be dismissed on
purely e x p e r i m e n t a l g r o u n d s , as Einstein did, or could s o m e o n e show
that they were theoretically possible a n d t h e n actually build a time
machine?
In J u n e 1988, t h r e e physicists (Kip T h o m e a n d Michael Morris at t h e
California Institute of T e c h n o l o g y a n d Ulvi Yurtsever at the University
of Michigan) m a d e t h e first serious proposal for a time m a c h i n e . T h e y
convinced t h e editors of Physical Review Letters, o n e of the most distinguished publications in t h e world, that their work m e r i t e d serious consideration. (Over t h e decades, scores of crackpot proposals for time
travel have b e e n s u b m i t t e d to m a i n s t r e a m physics j o u r n a l s , b u t all have
b e e n rejected because they were n o t based on s o u n d physical principles
or Einstein's equations.) Like e x p e r i e n c e d scientists, they p r e s e n t e d
their a r g u m e n t s in accepted field theoretical l a n g u a g e a n d t h e n carefully explained where their weakest assumptions were.
To overcome the skepticism of t h e scientific c o m m u n i t y , T h o m e a n d
his colleagues realized that they would have to overcome t h e s t a n d a r d
objections to using w o r m h o l e s as time m a c h i n e s . First, as m e n t i o n e d
earlier, Einstein himself realized that the gravitational forces at t h e center of a black h o l e would be so e n o r m o u s that any spacecraft would be
torn apart. A l t h o u g h wormholes were mathematically possible, they
were, in practice, useless.
Second, wormholes m i g h t be unstable. O n e could show that small
disturbances in w o r m h o l e s would cause the E i n s t e i n - R o s e n b r i d g e to
collapse. T h u s a spaceship's p r e s e n c e inside a black hole would be sufficient to cause a disturbance that would close t h e e n t r a n c e to the wormhole.
Third, o n e would have to go faster t h a n the speed of light actually
to p e n e t r a t e the w o r m h o l e to the o t h e r side.
F o u r t h , q u a n t u m effects would be so large that t h e w o r m h o l e m i g h t
close by itself. For e x a m p l e , the intense radiation e m i t t e d by t h e
e n t r a n c e to the black h o l e n o t only would kill a n y o n e w h o tried to e n t e r
the black hole, b u t also m i g h t close t h e e n t r a n c e .
Fifth, time slows down in a w o r m h o l e a n d comes to a c o m p l e t e stop
at the center. T h u s wormholes have t h e u n d e s i r a b l e feature that as seen
by s o m e o n e on the e a r t h , a space traveler a p p e a r s to slow down a n d
c o m e to a total halt at t h e c e n t e r of t h e black h o l e . T h e space traveler
looks like he or she is frozen in time. In o t h e r words, it takes an infinite
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a m o u n t of time for a space traveler to go t h r o u g h a w o r m h o l e . Assuming, for t h e m o m e n t , t h a t o n e could s o m e h o w go t h r o u g h t h e c e n t e r of
t h e w o r m h o l e a n d r e t u r n to e a r t h , the distortion of time would still be
so great that millions or even billions of years may have passed on the
earth.
For all these reasons, the w o r m h o l e solutions were never taken seriously.
T h o r n e is a serious cosmologist, o n e w h o m i g h t normally view time
m a c h i n e s with e x t r e m e skepticism or even derision. However, T h o r n e
was gradually drawn into this quest in the most curious way. In the summ e r of 1985, Carl Sagan sent to T h o r n e the prepublication draft of his
n e x t book, a novel called Contact, which seriously explores the scientific
a n d political questions s u r r o u n d i n g an e p o c h - m a k i n g event: m a k i n g
contact with t h e first extraterrestrial life in o u t e r space. Every scientist
p o n d e r i n g t h e question of life in o u t e r space must confront the question
of how to b r e a k t h e light barrier. Since Einstein's special theory of relativity explicitly forbids travel faster t h a n the speed of light, traveling to
t h e distant stars in a conventional spaceship may take thousands of years,
thereby m a k i n g interstellar travel impractical. Since Sagan wanted to
m a k e his b o o k as scientifically accurate as possible, he wrote to T h o r n e
asking w h e t h e r t h e r e was any scientifically acceptable way of evading the
light barrier.
Sagan's request p i q u e d T h o m e ' s intellectual curiosity. H e r e was an
honest, scientifically relevant request m a d e by o n e scientist to a n o t h e r
that d e m a n d e d a serious reply. Fortunately, because of the u n o r t h o d o x
n a t u r e of t h e request, T h o r n e a n d his colleagues a p p r o a c h e d the question in a most u n u s u a l way: T h e y worked backward. Normally, physicists
start with a certain known astronomical object (a n e u t r o n star, a black
hole, t h e Big Bang) a n d t h e n solve Einstein's equations to find the curvature of t h e s u r r o u n d i n g space. T h e essence of Einstein's equations,
we recall, is that t h e m a t t e r a n d energy c o n t e n t of an object d e t e r m i n e s
t h e a m o u n t of curvature in the s u r r o u n d i n g space a n d time. P r o c e e d i n g
in this way, we are g u a r a n t e e d to find solutions to Einstein's equations
for astronomically relevant objects that we expect to find in o u t e r space.
However, because of Sagan's strange request, T h o r n e a n d his colleagues a p p r o a c h e d t h e question backward. T h e y started with a r o u g h
idea of what they w a n t e d to find. T h e y wanted a solution to Einstein's
e q u a t i o n s in which a space traveler would n o t be torn a p a r t by the tidal
effects of t h e i n t e n s e gravitational field. They wanted a w o r m h o l e that
would be stable a n d n o t suddenly close up in the m i d d l e of the trip.
T h e y wanted a w o r m h o l e in which the time it takes for a r o u n d trip
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would be m e a s u r e d in days, n o t millions or billions of e a r t h years, a n d
so on. In fact, their g u i d i n g principle was that they w a n t e d a time traveler
to have a reasonably comfortable ride back t h r o u g h time after e n t e r i n g
the w o r m h o l e . O n c e they d e c i d e d what their w o r m h o l e would look like,
t h e n , a n d only t h e n , did they begin to calculate t h e a m o u n t of energy
necessary to create such a w o r m h o l e .
From their u n o r t h o d o x p o i n t of view, they did n o t particularly care
if the energy r e q u i r e m e n t s were well beyond twentieth-century science.
To t h e m , it was an e n g i n e e r i n g p r o b l e m for s o m e future civilization
actually to construct t h e time m a c h i n e . T h e y w a n t e d to prove that it was
scientifically feasible, n o t that it was e c o n o m i c a l or within t h e b o u n d s of
present-day earth science:
N o r m a l l y , t h e o r e t i c a l p h y s i c i s t s ask, " W h a t a r e t h e laws o f p h y s i c s ? " a n d /
o r " W h a t d o t h o s e laws p r e d i c t a b o u t t h e U n i v e r s e ? " I n t h i s L e t t e r , w e
ask, i n s t e a d , " W h a t c o n s t r a i n t s d o t h e laws o f p h y s i c s p l a c e o n t h e a c t i v i t i e s
o f a n arbitrarily a d v a n c e d c i v i l i z a t i o n ? " T h i s will l e a d t o s o m e i n t r i g u i n g
q u e r i e s a b o u t t h e laws t h e m s e l v e s . W e b e g i n b y a s k i n g w h e t h e r t h e laws o f
p h y s i c s p e r m i t a n arbitrarily a d v a n c e d c i v i l i z a t i o n t o c o n s t r u c t a n d m a i n tain w o r m h o l e s f o r i n t e r s t e l l a r t r a v e l .
4
T h e key p h r a s e , of course, is "arbitrarily advanced civilization." T h e
laws of physics tell us what is possible, n o t what is practical. T h e laws of
physics are i n d e p e n d e n t of what it m i g h t cost to test t h e m . T h u s what is
theoretically possible may exceed t h e gross national p r o d u c t of the
p l a n e t earth. T h o r n e a n d his colleagues were careful to state that this
mythical civilization that can h a r n e s s the power of w o r m h o l e s m u s t be
"arbitrarily a d v a n c e d " — t h a t is, capable of p e r f o r m i n g all e x p e r i m e n t s
that are possible (even if they are n o t practical for earthlings).
Much to their delight, with r e m a r k a b l e ease they soon f o u n d a surprisingly simple solution that satisfied all their rigid constraints. It was
not a typical black h o l e solution at all, so they d i d n ' t have to worry a b o u t
all the p r o b l e m s of b e i n g r i p p e d a p a r t by a collapsed star. T h e y christ e n e d their solution t h e "transversible w o r m h o l e , " to distinguish it from
the o t h e r w o r m h o l e solutions that a r e n o t transversible by spaceship.
They were so excited by their solution that they wrote back to Sagan,
who t h e n i n c o r p o r a t e d s o m e of their ideas in his novel. In fact, they
were so surprised by t h e simplicity of their solution that they were convinced that a b e g i n n i n g g r a d u a t e s t u d e n t in physics would be able to
u n d e r s t a n d their solution. In t h e a u t u m n of 1985, on t h e final e x a m in
a course on general relativity given at Caltech, T h o r n e gave t h e worm-
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h o l e solution to the students without telling t h e m what it was, a n d they
were asked to d e d u c e its physical p r o p e r t i e s . (Most students gave
detailed m a t h e m a t i c a l analyses of the solution, b u t they failed to grasp
that they were looking at a solution that p e r m i t t e d time travel.)
If t h e s t u d e n t s h a d b e e n a bit m o r e observant on that final exam,
they would have b e e n able to d e d u c e s o m e r a t h e r astonishing properties
of the w o r m h o l e . In fact, they would have f o u n d that a trip t h r o u g h this
transversible w o r m h o l e would be as comfortable as a trip on an airplane.
T h e m a x i m u m gravitational forces e x p e r i e n c e d by the travelers would
n o t e x c e e d 1 g. In o t h e r words, their a p p a r e n t weight would n o t exceed
their weight on t h e e a r t h . F u r t h e r m o r e , the travelers would never have
to worry a b o u t the e n t r a n c e of t h e w o r m h o l e closing up d u r i n g the
j o u r n e y . T h o m e ' s w o r m h o l e is, in fact, p e r m a n e n t l y o p e n . Instead of
taking a million or a billion years, a trip t h r o u g h t h e transversible wormh o l e would be m a n a g e a b l e . Morris a n d T h o r n e write that " t h e trip will
be fully comfortable a n d will r e q u i r e a total of a b o u t 200 days," or less.
So far, T h o r n e n o t e s that t h e time p a r a d o x e s that o n e usually
e n c o u n t e r s in the movies are n o t to be found: " F r o m e x p o s u r e to scie n c e fiction scenarios (for e x a m p l e , those in which o n e goes back in
time a n d kills oneself) o n e m i g h t expect CTCs to give rise to initial
trajectories with zero multiplicities" (that is, trajectories that are imposs i b l e ) . However, he has shown that t h e CTCs that a p p e a r in his wormh o l e seem to fulfill t h e past, r a t h e r t h a n c h a n g e it or initiate time paradoxes.
Finally, in p r e s e n t i n g these surprising results to the scientific community, T h o r n e wrote, "A new class of solutions of the Einstein field
e q u a t i o n s is p r e s e n t e d , which describe w o r m h o l e s that, in principle,
could be traversed by h u m a n b e i n g s . "
T h e r e is, of course, a catch to all this, which is o n e reason why we
do n o t have time m a c h i n e s today. T h e last step in T h o m e ' s calculation
was to d e d u c e t h e precise n a t u r e of t h e m a t t e r a n d energy necessary to
create this marvelous transversible w o r m h o l e . T h o r n e a n d his colleagues
f o u n d that at t h e c e n t e r of t h e w o r m h o l e , t h e r e must be an " e x o t i c "
form of m a t t e r that has u n u s u a l p r o p e r t i e s . T h o r n e is quick to p o i n t o u t
that this " e x o t i c " form of matter, a l t h o u g h u n u s u a l , does n o t seem to
violate any of t h e known laws of physics. He cautions that, at some future
p o i n t , scientists may prove that exotic m a t t e r does n o t exist. However,
at p r e s e n t , exotic m a t t e r seems to be a perfectly acceptable form of
m a t t e r if o n e has access to sufficiently advanced technology. T h o r n e
writes confidently t h a t " f r o m a single w o r m h o l e an arbitrarily advanced
civilization can construct a m a c h i n e for backward time travel."
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Blueprint for a Time Machine
Anyone w h o has r e a d H. G. Wells's The Time Machine, however, may be
disappointed with T h o m e ' s b l u e p r i n t for a time m a c h i n e . You do n o t
sit in a chair in your living r o o m , t u r n a few dials, see blinking lights,
a n d witness t h e vast p a n o r a m a of history, i n c l u d i n g destructive world
wars, the rise a n d fall of great civilizations, or t h e fruits of futuristic
scientific marvels.
O n e version of T h o m e ' s time m a c h i n e consists of two c h a m b e r s ,
each c o n t a i n i n g two parallel metal plates. T h e i n t e n s e electric fields
created between each pair of plates (larger t h a n a n y t h i n g possible with
today's technology) rips t h e fabric of s p a c e - t i m e , creating a h o l e in
space that links the two c h a m b e r s . O n e c h a m b e r is t h e n placed in a
rocket ship a n d is accelerated to near-light velocities, while t h e o t h e r
c h a m b e r stays on the e a r t h . Since a w o r m h o l e can c o n n e c t two regions
of space with different times, a clock in t h e first c h a m b e r ticks slower
t h a n a clock in the s e c o n d c h a m b e r . Because time would pass at different rates at the two e n d s of t h e w o r m h o l e , a n y o n e falling i n t o o n e
e n d of the w o r m h o l e would be instantly h u r l e d into t h e past or t h e
future.
A n o t h e r time m a c h i n e m i g h t look like t h e following. If exotic m a t t e r
can be found a n d s h a p e d like metal, t h e n presumably the ideal s h a p e
would be a cylinder. A h u m a n stands in t h e c e n t e r of t h e cylinder. T h e
exotic matter t h e n warps t h e space a n d time s u r r o u n d i n g it, creating a
w o r m h o l e that c o n n e c t s to a distant p a r t of t h e universe in a different
time. At the c e n t e r of the vortex is t h e h u m a n , w h o t h e n e x p e r i e n c e s
no m o r e t h a n 1 g of gravitational stress as he or she is t h e n sucked into
the w o r m h o l e a n d f i n d s himself o r herself o n t h e o t h e r e n d o f t h e universe.
On the surface, T h o m e ' s m a t h e m a t i c a l r e a s o n i n g is impeccable. Einstein's equations i n d e e d show that w o r m h o l e solutions allow for time to
pass at different rates on e i t h e r side of the w o r m h o l e , so t h a t time travel,
in principle, is possible. T h e trick, of course, is to create the w o r m h o l e
in the first place. As T h o r n e a n d his collaborators are quick to p o i n t
out, the main p r o b l e m is how to harness e n o u g h energy to create a n d
maintain a w o r m h o l e with exotic matter.
Normally, o n e of t h e basic tenets of e l e m e n t a r y physics is t h a t all
objects have positive energy. Vibrating molecules, moving cars, flying
birds, a n d soaring rockets all have positive energy. (By definition, t h e
empty vacuum of space has zero energy.) However, if we can p r o d u c e
objects with "negative e n e r g i e s " (that is, s o m e t h i n g that has an energy
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c o n t e n t less t h a n the v a c u u m ) , t h e n we m i g h t be able to g e n e r a t e exotic
configurations of space a n d time in which time is b e n t into a circle.
This r a t h e r simple c o n c e p t goes by a complicated-sounding title: the
averaged weak energy condition (AWEC). As T h o r n e is careful to p o i n t out,
the AWEC m u s t be violated; energy must b e c o m e temporarily negative
for time travel to be successful. However, negative energy has historically
b e e n a n a t h e m a to relativists, w h o realize that negative energy would
m a k e possible antigravity a n d a host of o t h e r p h e n o m e n a that have
never b e e n seen experimentally.
But T h o r n e is quick to p o i n t o u t that t h e r e is a way to obtain negative
energy, a n d this is t h r o u g h q u a n t u m theory. In 1948, the Dutch physicist
H e n r i k Casimir d e m o n s t r a t e d that q u a n t u m theory can create negative
energy: J u s t take two large, u n c h a r g e d parallel metal plates. Ordinarily,
c o m m o n sense tells us that these two plates, because they are electrically
n e u t r a l , have no force between t h e m . But Casimir proved that the vacu u m separating these two plates, because of the H e i s e n b e r g Uncertainty
Principle, is actually t e e m i n g with activity, with trillions of particles a n d
antiparticles constantly a p p e a r i n g a n d disappearing. They a p p e a r o u t of
n o w h e r e a n d d i s a p p e a r back into the vacuum. Because they are so fleeting, they are, for the most part, unobservable, a n d they do n o t violate
any of t h e laws of physics. T h e s e "virtual particles" create a n e t attractive
force between these two plates that Casimir p r e d i c t e d was measurable.
W h e n Casimir first p u b l i s h e d his p a p e r , it m e t with e x t r e m e skepticism. After all, how can two electrically n e u t r a l objects attract each other,
thereby violating t h e usual laws of classical electricity? This was u n h e a r d
of. However, in 1958 physicist M . J . Sparnaay observed this effect in the
laboratory, exactly as Casimir h a d p r e d i c t e d . Since t h e n , it has b e e n
c h r i s t e n e d t h e Casimir effect.
O n e way of h a r n e s s i n g the Casimir effect is to place two large cond u c t i n g parallel plates at t h e e n t r a n c e of each w o r m h o l e , thereby creating negative energy at each e n d . As T h o r n e a n d his colleagues conclude, " I t may t u r n o u t that t h e average weak energy condition can never
be violated, in which case t h e r e could be no such things as transversible
w o r m h o l e s , time travel, or a failure of causality. It's p r e m a t u r e to try to
cross a b r i d g e before you c o m e to i t . "
At present, t h e j u r y is still o u t on T h o m e ' s time m a c h i n e . T h e decisive factor, all agree, is to have a fully q u a n t i z e d theory of gravity settle
t h e m a t t e r o n c e a n d for all. For e x a m p l e , S t e p h e n Hawking has p o i n t e d
o u t that t h e radiation e m i t t e d at the w o r m h o l e e n t r a n c e will be quite
large a n d will c o n t r i b u t e back into the m a t t e r - e n e r g y c o n t e n t of Einstein's equations. This feedback into Einstein's e q u a t i o n s will distort the
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e n t r a n c e to t h e w o r m h o l e , p e r h a p s even closing it forever. T h o r n e , however, disagrees that the radiation will be sufficient to close the e n t r a n c e .
This is w h e r e superstring theory c o m e s in. Because s u p e r s t r i n g theory is a fully q u a n t u m - m e c h a n i c a l theory t h a t includes Einstein's theory
of general relativity as a subset, it can be used to calculate corrections
to the original w o r m h o l e theory. In principle, it will allow us to determ i n e w h e t h e r the AWEC condition is physically realizable, a n d w h e t h e r
the w o r m h o l e e n t r a n c e stays o p e n for time travelers to enjoy a trip to
the past.
Hawking has expressed reservations a b o u t T h o r n e ' s w o r m h o l e s .
However, this is ironic because Hawking himself has p r o p o s e d a new
theory of wormholes that is even m o r e fantastic. Instead of c o n n e c t i n g
the p r e s e n t with the past, Hawking p r o p o s e s to use w o r m h o l e s to connect o u r universe with an infinite n u m b e r of parallel universes!
12
Colliding Universes
[Nature is] not only queerer than we suppose, it is queerer
than we can suppose.
J. B. S. H a l d a n e
C
O S M O L O G I S T S t e p h e n Hawking is o n e of the most tragic figures
in science. Dying of an incurable, degenerative disease, he has
relentlessly p u r s u e d his research activities in t h e face of almost insurm o u n t a b l e obstacles. A l t h o u g h he has lost control of his h a n d s , legs,
t o n g u e , a n d finally his vocal cords, he has s p e a r h e a d e d new avenues of
research while confined to a wheelchair. Any lesser physicist would have
l o n g ago given up the struggle to tackle t h e great p r o b l e m s of science.
U n a b l e to grasp a pencil or p e n , he performs all his calculations in
his h e a d , occasionally aided by an assistant. Bereft of vocal cords, he uses
mechanical devices to c o m m u n i c a t e with t h e outside world. But he n o t
only maintains a vigorous research p r o g r a m , b u t still took time to write
a best-selling book, A Brief History of Time, a n d to lecture a r o u n d the
world.
I o n c e visited Hawking in his h o m e j u s t outside C a m b r i d g e University
w h e n I was invited to speak at a physics conference he was organizing.
Walking t h r o u g h his living r o o m , I was surprised by t h e impressive array
of i n g e n i o u s gadgets that he uses to c o n t i n u e his research. For example,
I saw on his desk a device m u c h like those used by musicians to hold
music sheets. This o n e , however, was m u c h m o r e elaborate a n d h a d the
ability to g r a b e a c h page a n d carefully t u r n it for r e a d i n g a book. (I
shivered to p o n d e r , as I think many physicists have, w h e t h e r I would
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have the stamina a n d s h e e r willpower to c o n t i n u e research witho u t arms, legs, or a voice even if I h a d t h e finest m e c h a n i c a l aids
available.)
Hawking is the Lucasian Professor of Physics at C a m b r i d g e University, the same chair h e l d by Isaac Newton. A n d like his illustrious predecessor, Hawking has e m b a r k e d on the greatest quest of t h e century,
the final unification of Einstein's theory of gravity a n d q u a n t u m theory.
As a result, h e , too, has marveled at the elegant, self-consistency of the
ten-dimensional theory, a n d in fact closes his best-selling b o o k with a
discussion of it.
Hawking no l o n g e r s p e n d s t h e bulk of his creative energy on the
field that m a d e h i m world-famous—black holes—which are by now
passe. He is h u n t i n g bigger g a m e — t h e unified field theory. String theory, we recall, b e g a n as a q u a n t u m theory a n d t h e n later a b s o r b e d Einstein's theory of gravity. Hawking, starting as a p u r e classical relativist
r a t h e r t h a n a q u a n t u m theorist, a p p r o a c h e s the p r o b l e m from the o t h e r
p o i n t of view. He a n d his colleague J a m e s Hartle start with Einstein's
classical universe, a n d t h e n quantize t h e e n t i r e universe!
Wave Function of the Universe
Hawking is o n e of t h e f o u n d e r s of a new scientific discipline, called
quantum cosmology. At first, this seems like a c o n t r a d i c t i o n in terms. T h e
word quantum applies to t h e infinitesimally small world of quarks a n d
n e u t r i n o s , while cosmology signifies t h e almost limitless e x p a n s e of o u t e r
space. However, Hawking a n d o t h e r s now believe that t h e ultimate questions of cosmology can be answered only by q u a n t u m theory. Hawking
takes q u a n t u m cosmology to its ultimate q u a n t u m conclusion, allowing
the existence of infinite n u m b e r s of parallel universes.
T h e starting p o i n t of q u a n t u m theory, we recall, is a wave function
that describes all the various possible states of a particle. For e x a m p l e ,
imagine a large, irregular t h u n d e r c l o u d that fills up t h e sky. T h e d a r k e r
the t h u n d e r c l o u d , t h e greater the c o n c e n t r a t i o n of water vapor a n d dust
at that point. T h u s by simply looking at a t h u n d e r c l o u d , we can rapidly
estimate the probability of finding large c o n c e n t r a t i o n s of water a n d
dust in certain parts of the sky.
T h e t h u n d e r c l o u d may be c o m p a r e d to a single e l e c t r o n ' s wave function. Like a t h u n d e r c l o u d , it fills up all space. Likewise, t h e g r e a t e r its
value at a point, t h e greater t h e probability of finding t h e e l e c t r o n t h e r e .
Similarly, wave functions can be associated with large objects, like peo-
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ple. As I sit in my chair in P r i n c e t o n , I know that I have a Schodinger
probability wave function. If I could s o m e h o w see my own wave function,
it would resemble a cloud very m u c h in t h e shape of my body. However,
s o m e of t h e cloud would spread o u t over all space, o u t to Mars a n d even
b e y o n d t h e solar system, a l t h o u g h it would be vanishingly small there.
This m e a n s that t h e r e is very large likelihood that I am, in fact, sitting
in my chair a n d n o t on the p l a n e t Mars. A l t h o u g h p a r t of my wave function has spread even b e y o n d t h e Milky Way galaxy, t h e r e is only an infinitesimal c h a n c e that I am sitting in a n o t h e r galaxy.
Hawking's new idea was to treat the entire universe as t h o u g h it were
a q u a n t u m particle. By r e p e a t i n g s o m e simple steps, we are led to some
eye-opening conclusions.
We begin with a wave function describing the set of all possible universes. This m e a n s t h a t t h e starting p o i n t of Hawking's theory must be
an infinite set of parallel universes, t h e wave function of the universe. Hawking's r a t h e r simple analysis, replacing t h e word particle with universe, has
led to a c o n c e p t u a l revolution in o u r t h i n k i n g a b o u t cosmology.
A c c o r d i n g to this picture, t h e wave function of the universe spreads
o u t over all possible universes. T h e wave function is assumed to be quite
large n e a r o u r own universe, so t h e r e is a good c h a n c e that o u r universe
is t h e c o r r e c t o n e , as we expect. However, the wave function spreads out
over all o t h e r universes, even those that are lifeless a n d incompatible
with t h e familiar laws of physics. Since the wave function is supposedly
vanishingly small for these o t h e r universes, we do n o t expect that o u r
universe will m a k e a q u a n t u m leap to t h e m in t h e n e a r future.
T h e goal facing q u a n t u m cosmologists is to verify this conjecture
mathematically, to show that the wave function of the universe is large
for o u r p r e s e n t universe a n d vanishingly small for o t h e r universes. This
would t h e n prove that o u r familiar universe is in some sense u n i q u e a n d
also stable. (At present, q u a n t u m cosmologists are u n a b l e to solve this
important problem.)
If we take Hawking seriously, it m e a n s t h a t we must begin o u r analysis
with an infinite n u m b e r of all possible universes, coexisting with o n e
a n o t h e r . To p u t it bluntly, t h e definition of the word universes no longer
"all that exists." It now m e a n s "all that can exist." For example, in
Figure 12.1 we see how t h e wave function of t h e universe can spread o u t
over several possible universes, with o u r universe b e i n g the most likely
o n e b u t certainly n o t t h e only o n e . Hawking's q u a n t u m cosmology also
assumes t h a t t h e wave function of t h e universe allows these universes to
collide. W o r m h o l e s can develop a n d link these universes. However,
these w o r m h o l e s are n o t like the o n e s we e n c o u n t e r e d in the previous
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Wave
function
of the
universe
Our
universe
Other
universes
Figure 12.1. In Hawking's wave function of the universe, the wave function is
most likely concentrated around own universe. We live in our universe because it
is the most likely, with the largest probability. However, there is a small but nonvanishing probability that the wave function prefers neighboring, parallel universes. Thus transitions between universes may be possible (although with very
low probability).
chapters, which c o n n e c t different parts of three-dimensional space with
itself—these wormholes c o n n e c t different universes with o n e a n o t h e r .
Think, for e x a m p l e , of a large collection of soap b u b b l e s , s u s p e n d e d
in air. Normally, each soap b u b b l e is like a universe u n t o itself, e x c e p t
that periodically it b u m p s into a n o t h e r b u b b l e , forming a larger o n e , or
splits into two smaller bubbles. T h e difference is that each soap b u b b l e
is now an entire ten-dimensional universe. Since space a n d time can exist
only on each b u b b l e , t h e r e is no such t h i n g as space a n d time between
the bubbles. Each universe has its own self-contained " t i m e . " It is m e a n ingless to say that time passes at t h e same rate in all these universes. (We
should, however, stress that travel between these universes is n o t o p e n
to us because of o u r primitive technological level. F u r t h e r m o r e ,
Figure 12.2. Our universe may be one of an infinite number of parallel universes,
each connected to the others by an infinite series of wormholes. Travel between
these wormholes is possible but extremely unlikely.
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we should also stress that large q u a n t u m transitions on this scale are
extremely rare, probably m u c h larger t h a n the lifetime of o u r universe.)
Most of these universes are d e a d universes, devoid of any life. On these
universes, the laws of physics were different, a n d h e n c e t h e physical conditions that m a d e life possible were n o t satisfied. P e r h a p s , a m o n g t h e
billions of parallel universes, only o n e (ours) h a d the right set of physical
laws to allow life (Figure 12.2).
Hawking's " b a b y u n i v e r s e " theory, a l t h o u g h n o t a practical m e t h o d
of transportation, certainly raises philosophical a n d p e r h a p s even religions questions. Already, it has stimulated two long-simmering d e b a t e s
a m o n g cosmologists.
Putting God Back in the Universe?
T h e first d e b a t e c o n c e r n s t h e anthropic principle. Over the centuries, scientists have l e a r n e d to view t h e universe largely i n d e p e n d e n t of h u m a n
bias. We no longer project o u r h u m a n prejudices a n d whims o n t o every
scientific discovery. Historically, however, early scientists often committed the fallacy of a n t h r o p o m o r p h i s m , which assumes that objects a n d
animals have h u m a n l i k e qualities. This e r r o r is c o m m i t t e d by a n y o n e
w h o sees h u m a n e m o t i o n s a n d feelings b e i n g exhibited by their pets. (It
is also c o m m i t t e d by Hollywood scriptwriters w h o regularly assume that
beings similar to us must p o p u l a t e planets orbiting t h e stars in t h e heavens.)
A n t h r o p o m o r p h i s m is an age-old p r o b l e m . T h e I o n i a n p h i l o s o p h e r
X e n o p h a n e s o n c e l a m e n t e d , " M e n imagine gods t o b e b o r n , a n d t o
have clothes a n d voices a n d shapes like theirs. . . . Yea, the gods of t h e
Ethiopians are black a n d flat-nosed, a n d t h e gods of the T h r a c i a n s are
red-haired a n d blue-eyed." Within the past few decades, s o m e cosmologists have b e e n horrified to find a n t h r o p o m o r p h i s m c r e e p i n g back into
science, u n d e r the guise of the a n t h r o p i c principle, s o m e of whose advocates openly declare that they would like to p u t G o d back into science.
Actually, t h e r e is s o m e scientific m e r i t to this strange d e b a t e over the
a n t h r o p i c principle, which revolves a r o u n d t h e indisputable fact that if
the physical constants of the universe were altered by t h e smallest
a m o u n t , life in t h e universe would be impossible. Is this r e m a r k a b l e fact
j u s t a fortunate coincidence, or d o e s it show the work of some S u p r e m e
Being?
T h e r e are two versions of t h e a n t h r o p i c principle. T h e " w e a k " version states that t h e fact that intelligent life (us) exists in t h e universe
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s h o u l d be taken as an e x p e r i m e n t a l fact that helps us u n d e r s t a n d the
constants of t h e universe. As Nobel laureate Steven W e i n b e r g explains
it, " t h e world is t h e way it is, at least in part, because otherwise there
would be no o n e to ask why it is the way it i s . " Stated in this way, the
weak version of the a n t h r o p i c principle is h a r d to a r g u e with.
To have life in t h e universe, you n e e d a rare conjunction of many
coincidences. Life, which d e p e n d s on a variety of c o m p l e x biochemical
reactions, can easily be r e n d e r e d impossible if we c h a n g e some of the
constants of chemistry a n d physics by a small a m o u n t . For example, if
the constants that govern n u c l e a r physics were c h a n g e d even slightly,
t h e n nucleosynthesis a n d the creation of the heavy elements in the stars
a n d supernovae m i g h t b e c o m e impossible. T h e n atoms might b e c o m e
unstable or impossible to create in supernovae. Life d e p e n d s on the
heavy e l e m e n t s (elements b e y o n d iron) for t h e creation of DNA a n d
p r o t e i n molecules. T h u s t h e smallest c h a n g e in n u c l e a r physics would
m a k e the heavy e l e m e n t s of the universe impossible to manufacture in
the stars. We are c h i l d r e n of t h e stars; however, if the laws of nuclear
physics c h a n g e in t h e slightest, t h e n o u r " p a r e n t s " are incapable of
having " c h i l d r e n " (us). As a n o t h e r e x a m p l e , it is safe to say that the
creation of life in t h e early oceans probably took 1 to 2 billion years.
However, if we could s o m e h o w shrink the lifetime of the p r o t o n to several million years, t h e n life would be impossible. T h e r e would n o t be
e n o u g h time to create life o u t of r a n d o m collisions of molecules.
1
In o t h e r words, the very fact that we exist in the universe to ask these
questions a b o u t it m e a n s that a c o m p l e x s e q u e n c e of events must necessarily have h a p p e n e d . It m e a n s that the physical constants of n a t u r e
must have a certain r a n g e of values, so that t h e stars lived long e n o u g h
to create t h e heavy e l e m e n t s in o u r bodies, so that p r o t o n s d o n ' t decay
too rapidly before life has a c h a n c e to g e r m i n a t e , a n d so on. In o t h e r
words, the existence of h u m a n s w h o can ask questions a b o u t the universe
places a h u g e n u m b e r of rigid constraints on the physics of the universe—for e x a m p l e , its age, its chemical composition, its t e m p e r a t u r e ,
its size, a n d its physical processes.
R e m a r k i n g on these cosmic coincidences, physicist F r e e m a n Dyson
o n c e wrote, "As we look o u t into the Universe a n d identify the many
accidents of physics a n d a s t r o n o m y that have worked t o g e t h e r to o u r
benefit, it almost seems as if t h e Universe must in s o m e sense have known
t h a t we were c o m i n g . " This takes us to the " s t r o n g " version of the
a n t h r o p i c principle, which states that all t h e physical constants of the
universe have b e e n precisely c h o s e n (by God or some S u p r e m e Being)
so t h a t life is possible in o u r universe. T h e strong version, because it
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raises questions a b o u t a deity, is m u c h m o r e controversial a m o n g scientists.
Conceivably, it m i g h t have b e e n blind luck if only a few constants of
n a t u r e were r e q u i r e d to assume certain values to m a k e life possible.
However, it appears that a large set of physical constants must assume a
narrow b a n d of values in o r d e r for life to form in o u r universe. Since
accidents of this type are highly i m p r o b a b l e , p e r h a p s a divine intelligence (God) precisely chose those values in o r d e r to create life.
W h e n scientists first h e a r of some version of t h e a n t h r o p i c principle,
they are immediately taken aback. Physicist H e i n z Pagels recalled, " H e r e
was a form of r e a s o n i n g completely foreign to the usual way that theoretical physicists went a b o u t their b u s i n e s s . "
T h e a n t h r o p i c a r g u m e n t is a m o r e sophisticated version of t h e old
a r g u m e n t that God located t h e earth at j u s t t h e right distance from the
sun. If God h a d placed the e a r t h too close, t h e n it would be too h o t to
s u p p o r t life. If God h a d placed the e a r t h too far, t h e n it would be too
cold. T h e fallacy of this a r g u m e n t is that millions of planets in the galaxy
probably are sitting at the incorrect distance from their sun, a n d therefore life on t h e m is impossible. However, some planets will, by p u r e
accident, be at t h e right distance from their sun. O u r p l a n e t is o n e of
t h e m , a n d h e n c e we are h e r e to discuss t h e question.
Eventually, most scientists b e c o m e disillusioned with the a n t h r o p i c
principle because it has no predictive power, n o r can it be tested. Pagels
reluctantly c o n c l u d e d that " u n l i k e the principles of physics, it affords
no way to d e t e r m i n e w h e t h e r it is right or wrong; t h e r e is no way to test
it. Unlike conventional physical principles, t h e a n t h r o p i c principle is n o t
subject to e x p e r i m e n t a l falsification—the sure sign that it is n o t a scientific p r i n c i p l e . " Physicist Alan G u t h says bluntly, "Emotionally, t h e
a n t h r o p i c principle kind of rubs me t h e w r o n g way. . . . T h e a n t h r o p i c
principle is s o m e t h i n g that p e o p l e do if they c a n ' t t h i n k of a n y t h i n g
better t o d o . "
To Richard Feynman, the goal of a theoretical physicist is to " p r o v e
yourself w r o n g as fast as p o s s i b l e . " However, t h e a n t h r o p i c principle is
sterile a n d c a n n o t be disproved. O r , as W e i n b e r g said, " a l t h o u g h science
is clearly impossible without scientists, it is n o t clear that the universe is
impossible without s c i e n c e . "
T h e d e b a t e over t h e a n t h r o p i c principle ( a n d h e n c e , a b o u t G o d )
was d o r m a n t for many years, until it was recently revived by Hawking's
wave function of the universe. If Hawking is correct, t h e n i n d e e d t h e r e
are an infinite n u m b e r of parallel universes, m a n y with different physical
constants. In some of t h e m , p e r h a p s p r o t o n s decay too rapidly, or stars
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3
4
5
6
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c a n n o t m a n u f a c t u r e the heavy e l e m e n t s beyond iron, or the Big C r u n c h
takes place too rapidly before life can begin, a n d so o n . In fact, an infinite n u m b e r of these parallel universes are dead, without the physical
laws that can m a k e life as we know it possible.
On o n e such parallel universe (ours), the laws of physics were compatible with life as we know it. T h e p r o o f is that we are h e r e today to
discuss t h e matter. If this is true, t h e n p e r h a p s G o d does n o t have to be
evoked to explain why life, precious as it is, is possible in o u r universe.
However, this r e o p e n s the possibility of the weak a n t h r o p i c principle—
that is, that we coexist with many d e a d universes, a n d that ours is the
only o n e c o m p a t i b l e with life.
T h e second controversy stimulated by Hawking's wave function of
the universe is m u c h d e e p e r a n d in fact is still unresolved. It is called
t h e S c h r o d i n g e r ' s cat p r o b l e m .
Schrodinger's Cat Revisited
Because Hawking's theory of baby universes a n d wormholes uses the
power of q u a n t u m theory, it inevitably r e o p e n s the still unresolved
d e b a t e s c o n c e r n i n g its foundations. Hawking's wave function of the universe d o e s n o t completely solve these p a r a d o x e s of q u a n t u m theory; it
only expresses t h e m in a startling new light.
Q u a n t u m theory, we recall, states that for every object t h e r e is a wave
function that measures the probability of finding that object at a certain
p o i n t in space a n d time. Q u a n t u m theory also states that you never really
know t h e state of a particle until you have m a d e an observation. Before
a m e a s u r e m e n t is m a d e , the particle can be in o n e of a variety of states,
described by t h e S c h r o d i n g e r wave function. T h u s before an observation
or m e a s u r e m e n t can be m a d e , you c a n ' t really know the state of the
particle. In fact, t h e particle exists in a n e t h e r state, a sum of all possible
states, until a m e a s u r e m e n t is m a d e .
W h e n this idea was first p r o p o s e d by Niels B o h r a n d W e r n e r Heisenb e r g , Einstein revolted against this c o n c e p t . " D o e s the m o o n exist j u s t
because a m o u s e looks at i t ? " he was fond of asking. According to the
strict i n t e r p r e t a t i o n of q u a n t u m theory, the m o o n , before it is observed,
d o e s n ' t really exist as we know it. T h e m o o n can b e , in fact, in any o n e
of an infinite n u m b e r of states, i n c l u d i n g t h e state of b e i n g in the sky,
of b e i n g blown u p , or of n o t b e i n g t h e r e at all. It is the m e a s u r e m e n t
process of looking at it that decides that t h e m o o n is actually circling
the e a r t h .
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Einstein h a d many h e a t e d discussions with Niels B o h r c h a l l e n g i n g
this u n o r t h o d o x world view. (In o n e e x c h a n g e , B o h r said to Einstein in
exasperation, "You are n o t thinking. You are merely b e i n g logical!" )
Even Erwin S c h r o d i n g e r (who initiated the whole discussion with his
celebrated wave e q u a t i o n ) p r o t e s t e d this r e i n t e r p r e t a t i o n of his equation. He o n c e l a m e n t e d , "I d o n ' t like it, a n d I ' m sorry I ever h a d anything to do with i t . "
To challenge this revisionist i n t e r p r e t a t i o n , the critics asked, "Is a
cat d e a d or alive before you look at i t ? "
To show how absurd this question is, S c h r o o d i n g e r placed an imaginary cat in a sealed box. T h e cat faces a g u n , which is c o n n e c t e d to a
Geiger c o u n t e r , which in t u r n is c o n n e c t e d to a piece of u r a n i u m . T h e
u r a n i u m a t o m is unstable a n d will u n d e r g o radioactive decay. If a uran i u m nucleus disintegrates, it will be picked up by t h e Geiger c o u n t e r ,
which will t h e n trigger the g u n , whose bullet will kill the cat.
To d e c i d e w h e t h e r t h e cat is d e a d or alive, we m u s t o p e n the b o x
a n d observe the cat. However, what is t h e state of t h e cat before we o p e n
the box? According to q u a n t u m theory, we can only state t h a t t h e cat is
described by a wave function that describes t h e s u m of a d e a d cat a n d a
live cat.
To Schrodinger, the idea of t h i n k i n g a b o u t cats that a r e n e i t h e r d e a d
n o r alive was t h e h e i g h t of absurdity, yet nevertheless t h e e x p e r i m e n t a l
confirmation of q u a n t u m m e c h a n i c s forces us to this conclusion. At
present, every e x p e r i m e n t has verified q u a n t u m theory.
T h e p a r a d o x of S c h r o d i n g e r ' s cat is so bizarre that o n e is often
r e m i n d e d of how Alice reacted to the vanishing of t h e C h e s h i r e cat in
Lewis Carroll's fable: " 'You'll see me t h e r e , ' said the Cat, a n d vanished.
Alice was n o t m u c h surprised at this, she was getting so well used to
q u e e r things h a p p e n i n g . " Over the years, physicists, too, have g o t t e n
used t o " q u e e r " things h a p p e n i n g i n q u a n t u m m e c h a n i c s .
T h e r e are at least t h r e e major ways that physicists deal with this complexity. First, we can assume that God exists. Because all " o b s e r v a t i o n s "
imply an observer, t h e n t h e r e m u s t be s o m e " c o n s c i o u s n e s s " in the
universe. Some physicists, like N o b e l laureate E u g e n e Wigner, have
insisted that q u a n t u m theory proves t h e existence of s o m e sort of universal cosmic consciousness in the universe.
T h e second way of dealing with t h e p a r a d o x is favored by t h e vast
majority of working physicists—to i g n o r e the p r o b l e m . Most physicists,
p o i n t i n g o u t that a c a m e r a without any consciousness can also m a k e
m e a s u r e m e n t s , simply wish that this sticky, b u t unavoidable, p r o b l e m
would go away.
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T h e physicist Richard Feynman o n c e said, "I think it is safe to say
that n o o n e u n d e r s t a n d s q u a n t u m mechanics. D o n o t k e e p saying t o
yourself, if you can possibly avoid it, 'But how can it be like that?' because
you will go 'down t h e d r a i n ' i n t o a blind alley from which n o b o d y has
yet escaped. N o b o d y knows how it can be like t h a t . " In fact, it is often
stated that of all t h e theories p r o p o s e d in this century, the silliest is
q u a n t u m theory. S o m e say that the only t h i n g that q u a n t u m theory has
g o i n g for it, in fact, is t h a t it is unquestionably correct.
However, t h e r e is a third way of dealing with this p a r a d o x , called the
many-worlds theory. This theory (like the a n t h r o p i c principle) fell o u t of
favor in the past decades, b u t is b e i n g revived again by Hawking's wave
function of t h e universe.
9
Many Worlds
In 1957, physicist H u g h Everett raised the possibility that d u r i n g the
evolution of t h e universe, it continually " s p l i t " in half, like a fork in a
road. I n o n e universe, t h e u r a n i u m a t o m did n o t disintegrate a n d the
cat was n o t shot. In t h e o t h e r , t h e u r a n i u m a t o m did disintegrate a n d
the cat was shot. If Everett is correct, t h e r e are an infinite n u m b e r of
universes. Each universe is linked to every o t h e r t h r o u g h the network of
forks in t h e road. O r , as the A r g e n t i n i a n writer J o r g e Luis Borges wrote
in The Garden of Forking Paths, " t i m e forks perpetually toward i n n u m e r able f u t u r e s . "
Physicist Bryce DeWitt, o n e of the p r o p o n e n t s of the many-worlds
theory, describes t h e lasting impact it m a d e on him: "Every q u a n t u m
transition taking place on every star, in every galaxy, in every r e m o t e
c o r n e r of t h e universe is splitting o u r local world on earth into myriads
of copies of itself. I still recall vividly the shock I e x p e r i e n c e d on first
e n c o u n t e r i n g this multiworld c o n c e p t . " T h e many-worlds theory postulates t h a t all possible q u a n t u m worlds exist. In some worlds, h u m a n s
exist as t h e d o m i n a n t life form on e a r t h . In o t h e r worlds, subatomic
events took place that p r e v e n t e d h u m a n s from ever evolving on this
planet.
As physicist Frank Wilczek n o t e d ,
10
It is said that the history of the world w o u l d be entirely different if H e l e n
o f T r o y h a d h a d a w a r t a t t h e t i p o f h e r n o s e . W e l l , warts c a n a r i s e f r o m
m u t a t i o n s i n s i n g l e c e l l s , o f t e n t r i g g e r e d b y e x p o s u r e t o t h e u l t r a v i o l e t rays
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o f the sun. C o n c l u s i o n : there are many, m a n y worlds i n w h i c h H e l e n o f
T r o y did h a v e a wart a t t h e tip o f h e r n o s e . "
Actually, t h e idea that t h e r e may be multiple universes is an old o n e .
T h e p h i l o s o p h e r St. Albertus M a g n u s o n c e wrote, " D o t h e r e exist m a n y
worlds, or is t h e r e b u t a single world? This is o n e of the most n o b l e a n d
exalted questions in t h e study of N a t u r e . " However, t h e new twist on
this ancient idea is that these m a n y worlds resolve t h e S c h r o d i n g e r cat
paradox. In o n e universe, the cat may be d e a d ; in a n o t h e r , the cat is
alive.
As strange as Everett's many-worlds theory seems, o n e can show that
it is mathematically equivalent to the usual i n t e r p r e t a t i o n s of q u a n t u m
theory. But traditionally, Everett's many-worlds theory has n o t b e e n p o p ular a m o n g physicists. A l t h o u g h it c a n n o t be r u l e d out, the idea of an
infinite n u m b e r of equally valid universes, each fissioning in half at every
instant in time, poses a philosophical n i g h t m a r e for physicists, w h o love
simplicity. T h e r e is a principle of physics called O c c a m ' s razor, which
states that we should always take t h e simplest possible p a t h a n d i g n o r e
m o r e clumsy alternatives, especially if t h e alternatives can never be measured. (Thus O c c a m ' s razor dismisses t h e old " a e t h e r " theory, which
stated that a mysterious gas o n c e p e r v a d e d t h e entire universe. T h e
a e t h e r theory provided a c o n v e n i e n t answer to an e m b a r r a s s i n g question: If light is a wave, a n d light can travel in a v a c u u m , t h e n what is
waving? T h e answer was that a e t h e r , like a fluid, was vibrating even in a
vacuum. Einstein showed that t h e a e t h e r was unnecessary. However, he
never said that the a e t h e r d i d n ' t exist. He merely said it was irrelevant.
T h u s by O c c a m ' s razor, physicists d o n ' t refer to the a e t h e r anymore.)
O n e can show that c o m m u n i c a t i o n between Everett's many worlds
is n o t possible. T h e r e f o r e , each universe is unaware of t h e existence of
the others. If e x p e r i m e n t s c a n n o t test for t h e existence of these worlds,
we should, by O c c a m ' s razor, eliminate t h e m .
Somewhat in the same vein, physicists do n o t say categorically that
angels a n d miracles c a n n o t exist. P e r h a p s they d o . But miracles, almost
by definition, are n o t r e p e a t a b l e a n d therefore n o t m e a s u r a b l e by experiment. T h e r e f o r e , by O c c a m ' s razor, we must dismiss t h e m (unless, of
course, we can find a r e p r o d u c i b l e , m e a s u r a b l e miracle or a n g e l ) . O n e
of the developers of the many-worlds theory, Everett's m e n t o r J o h n
Wheeler, reluctantly rejected it because " i t r e q u i r e d too m u c h metaphysical baggage to carry a r o u n d . "
T h e u n p o p u l a r i t y of t h e many-worlds theory, however, may subside
as Hawking's wave function of t h e universe gains popularity. Everett's
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theory was based on single particles, with no possibility of communication between different universes as they fissioned. However, Hawking's
theory, a l t h o u g h related, goes m u c h further: It is based on an infinite
n u m b e r of self-contained universes ( a n d n o t j u s t particles) a n d postulates t h e possibility of t u n n e l i n g (via wormholes) between t h e m .
Hawking has even u n d e r t a k e n the d a u n t i n g task of calculating the
solution to t h e wave function of t h e universe. He is confident that his
a p p r o a c h is c o r r e c t partly b e c a u s e t h e theory is well defined (if, as we
m e n t i o n e d , the theory is ultimately defined in ten d i m e n s i o n s ) . His goal
is to show that t h e wave function of t h e universe assumes a large value
n e a r a universe that looks like ours. T h u s o u r universe is the most likely
universe, b u t certainly n o t t h e only o n e .
By now, t h e r e have b e e n a n u m b e r of international conferences on
t h e wave function of the universe. However, as before, the mathematics
involved in t h e wave function of the universe is b e y o n d the calculational
ability of any h u m a n on this planet, a n d we may have to wait years before
any e n t e r p r i s i n g individual can find a rigorous solution to Hawking's
equations.
Parallel Worlds
A major difference between Everett's many-worlds theory a n d Hawking's
wave function of t h e universe is that Hawking's theory places wormholes
t h a t c o n n e c t these parallel universes at the c e n t e r of his theory. However, t h e r e is no n e e d to w o n d e r w h e t h e r you will someday walk h o m e
from work, o p e n t h e d o o r , e n t e r a parallel universe, a n d discover that
your family never h e a r d of you. Instead of r u s h i n g to m e e t you after a
h a r d day's work, your family is thrown into a panic, scream a b o u t an
i n t r u d e r , a n d have you thrown in jail for illegal entry. This kind of scen a r i o h a p p e n s only on television or in t h e movies. In Hawking's
a p p r o a c h , t h e w o r m h o l e s d o , in fact, constantly c o n n e c t o u r universe
with billions u p o n billions of parallel universes, b u t t h e size of these
w o r m h o l e s , on t h e average, is extremely small, a b o u t the size of the
Planck l e n g t h ( a b o u t a 100 billion billion times smaller t h a n a p r o t o n ,
too small for h u m a n travel). F u r t h e r m o r e , since large q u a n t u m transitions between these universes are infrequent, we may have to wait a long
time, l o n g e r t h a n t h e lifetime of the universe, before such an event takes
place.
T h u s it is perfectly consistent with t h e laws of physics (although highly
unlikely) that s o m e o n e may e n t e r a twin universe that is precisely like
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o u r universe except for o n e small crucial difference, created at s o m e
p o i n t in time w h e n t h e two universes split apart.
This type of parallel world was e x p l o r e d by J o h n W y n d h a m in t h e
story " R a n d o m Q u e s t . " Colin Trafford, a British n u c l e a r physicist, is
almost killed in 1954 w h e n a n u c l e a r e x p e r i m e n t blows u p . Instead of
winding up in the hospital, he wakes u p , a l o n e a n d u n h u r t , in a r e m o t e
part of L o n d o n . He is relieved that everything a p p e a r s n o r m a l , b u t soon
discovers that s o m e t h i n g is very wrong. T h e n e w s p a p e r headlines are all
impossible. World War II never took place. T h e atomic b o m b was never
discovered.
World history has b e e n twisted. F u r t h e r m o r e , he glances at a store
shelf a n d notices his own n a m e , with his picture, as t h e a u t h o r of a bestselling book. He is shocked. An exact c o u n t e r p a r t of himself exists in
this parallel world as an a u t h o r instead of a n u c l e a r physicist!
Is he d r e a m i n g all this? Years ago, he o n c e t h o u g h t of b e c o m i n g a
writer, b u t instead he chose to b e c o m e a n u c l e a r physicist. Apparently
in this parallel universe, different choices were m a d e in the past.
Trafford scans the L o n d o n t e l e p h o n e book a n d finds his n a m e listed,
b u t the address is wrong. Shaking, he decides to visit " h i s " h o m e .
E n t e r i n g " h i s " a p a r t m e n t , he is shocked to m e e t " h i s " wife—someo n e he has never seen before—a beautiful w o m a n w h o is bitter a n d
angry over " h i s " n u m e r o u s affairs with o t h e r w o m e n . She berates " h i m "
for his extramarital indiscretions, b u t she notices that h e r h u s b a n d
seems confused. His c o u n t e r p a r t , Trafford finds out, is a cad a n d a womanizer. However, he finds it difficult to a r g u e with a beautiful stranger
he has never seen before, even if she h a p p e n s to be " h i s " wife. Apparently, he a n d his c o u n t e r p a r t have switched universes.
He gradually finds himself falling in love with " h i s " own wife. He
c a n n o t u n d e r s t a n d how his c o u n t e r p a r t could ever have t r e a t e d his
lovely wife in such a despicable m a n n e r . T h e n e x t few weeks s p e n t
t o g e t h e r are the best of their lives. He decides to u n d o all the h a r m his
c o u n t e r p a r t inflicted on his wife over the years. T h e n , j u s t as the two are
rediscovering each o t h e r , he is suddenly w r e n c h e d back into his own
universe, leaving " h i s " love b e h i n d . T h r o w n back into his own universe
against his will, he begins a frantic quest to find " h i s " wife. He has
discovered that most, b u t n o t all, p e o p l e in his universe have a c o u n t e r p a r t in the other. Surely, he reasons, " h i s " wife m u s t have a c o u n t e r p a r t
in his own world.
He b e c o m e s obsessed, tracking down all t h e clues that he r e m e m b e r s
from the twin universe. Using all his knowledge of history a n d physics,
he concludes that two worlds diverged from each o t h e r because of s o m e
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pivotal event in 1926 or 1927. A single event, he reasons, must have split
the two universes apart.
He t h e n meticulously traces t h e birth a n d d e a t h records of several
families. He s p e n d s his r e m a i n i n g savings interviewing scores of p e o p l e
until he locates " h i s " wife's family tree. Eventually, he succeeds in tracking down " h i s " wife in his own universe. In t h e e n d , he marries her.
Attack of the Giant Wormholes
O n e Harvard physicist w h o has j u m p e d into the fray c o n c e r n i n g wormholes is Sidney C o l e m a n . Resembling a cross between Woody Allen a n d
Albert Einstein, he shuffles t h r o u g h t h e corridors of Jefferson Hall, trying to convince the skeptics of his latest theory of wormholes. With his
C h a p l i n e s q u e m o u s t a c h e , his hair swept back like Einstein's, a n d his
oversize sweatshirt, C o l e m a n stands o u t in any crowd. Now he claims to
have solved t h e celebrated cosmological constant p r o b l e m , which has
puzzled physicists for the past 80 years.
His work even m a d e t h e cover of Discover Magazine, with an article
entitled "Parallel Universes: T h e New Reality—From Harvard's Wildest
Physicist." He is also wild a b o u t science fiction; a serious science-fiction
fan, he even co-founded Advent Publishers, which published books on
science-fiction criticism.
At present, C o l e m a n vigorously engages t h e critics who say that scientists w o n ' t be able to verify w o r m h o l e theories within o u r lifetime. If
we believe in T h o m e ' s w o r m h o l e s , t h e n we have to wait until s o m e o n e
discovers exotic m a t t e r or masters t h e Casimir effect. Until t h e n , o u r
time m a c h i n e s have no " e n g i n e " capable of s h o o t i n g us into the past.
Similarly, if we believe in Hawking's w o r m h o l e s , t h e n we have to travel
in " i m a g i n a r y t i m e " in o r d e r to travel between wormholes. Either way,
it a very sad state of affairs for the average theoretical physicist, who feels
frustrated by t h e i n a d e q u a t e , feeble technology of the twentieth century
a n d w h o can only d r e a m of harnessing the Planck energy.
This is w h e r e C o l e m a n ' s work comes in. He recently m a d e the claim
that the w o r m h o l e s m i g h t yield a very tangible, very measurable result
in t h e present, a n d n o t in some distant, unforeseeable future. As we
p o i n t e d o u t earlier, Einstein's e q u a t i o n s state that the m a t t e r - e n e r g y
c o n t e n t of an object d e t e r m i n e s the curvature of s p a c e - t i m e surroundi n g it. Einstein w o n d e r e d w h e t h e r the p u r e v a c u u m of empty space could
contain energy. Is p u r e emptiness devoid of energy? This vacuum energy
is m e a s u r e d by s o m e t h i n g called t h e cosmological constant; in principle,
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t h e r e is n o t h i n g to prevent a cosmological c o n s t a n t from a p p e a r i n g in
the equations. Einstein t h o u g h t this t e r m was aesthetically ugly, b u t he
could n o t rule it o u t on physical or m a t h e m a t i c a l g r o u n d s .
In the 1920s, w h e n Einstein tried to solve his e q u a t i o n s for the universe, he found, m u c h to his chagrin, that t h e universe was e x p a n d i n g .
Back t h e n , t h e prevailing wisdom was that the universe was static a n d
u n c h a n g i n g . In o r d e r to " f u d g e " his e q u a t i o n s to p r e v e n t the e x p a n s i o n
of the universe, Einstein inserted a tiny cosmological c o n s t a n t into this
solution, chosen so it would j u s t balance o u t the expansion, yielding a
static universe by fiat. In 1929, w h e n H u b b l e conclusively p r o v e d that
t h e universe is i n d e e d e x p a n d i n g , Einstein b a n i s h e d the cosmological
constant a n d said it was t h e " g r e a t e s t b l u n d e r of my life."
Today, we know that the cosmological c o n s t a n t is very close to zero.
If t h e r e were a small negative cosmological constant, t h e n gravity would
be powerfully attractive a n d the e n t i r e universe m i g h t b e , say, j u s t a few
feet across. (By r e a c h i n g o u t with your h a n d , you s h o u l d be able to g r a b
the p e r s o n in front of you, who h a p p e n s to be yourself.) If t h e r e were a
small positive cosmological constant, t h e n gravity would be repulsive a n d
everything would be flying away from you so fast that their light would
never reach you. Since n e i t h e r n i g h t m a r i s h scenario occurs, we are confident that t h e cosmological constant is extremely tiny or even zero.
But this p r o b l e m resurfaced in t h e 1970s, w h e n symmetry b r e a k i n g
was b e i n g intensively studied in t h e S t a n d a r d Model a n d G U T theory.
Whenever a symmetry is b r o k e n , a large a m o u n t of energy is d u m p e d
into the vacuum. In fact, the a m o u n t of energy flooding t h e v a c u u m is
10
times larger t h a n the experimentally observed a m o u n t . In all of
physics, this discrepancy of 1 0 is unquestionably t h e largest. N o w h e r e
in physics do we see such a large divergence between theory (which
predicts a large v a c u u m energy whenever a symmetry is b r o k e n ) a n d
e x p e r i m e n t (which measures zero cosmological c o n s t a n t in t h e universe). This is where C o l e m a n ' s w o r m h o l e s comes in; they're n e e d e d to
cancel the u n w a n t e d contributions to t h e cosmological constant.
According to Hawking, t h e r e may be an infinite n u m b e r of alternative universes coexisting with ours, all of which are c o n n e c t e d by an
infinite web of interlocking wormholes. C o l e m a n tried to a d d up t h e
contribution from this infinite series. After t h e s u m was p e r f o r m e d , he
found a surprising result: T h e wave function of the universe prefers to
have zero cosmological constant, as desired. If t h e cosmological c o n s t a n t
was zero, the wave function b e c a m e exceptionally large, m e a n i n g that
there was a high probability of finding a universe with zero cosmological
constant. Moreover, t h e wave function of the universe quickly vanished
1 0 0
1 0 0
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if t h e cosmological constant b e c a m e n o n z e r o , m e a n i n g that t h e r e was
zero probability for that u n w a n t e d universe. This was exactly what was
n e e d e d to cancel the cosmological constant. In o t h e r words, t h e cosmological c o n s t a n t was zero because t h a t was t h e most p r o b a b l e outc o m e . T h e only effect of having billions u p o n billions of parallel universes was to k e e p the cosmological c o n s t a n t zero in o u r universe.
Because this was such an i m p o r t a n t result, physicists immediately
b e g a n to leap into the field. " W h e n Sidney c a m e o u t with this work,
everyone j u m p e d , " recalls Stanford physicist L e o n a r d Susskind. In his
typical puckish way, C o l e m a n p u b l i s h e d this potentially i m p o r t a n t result
with a bit of h u m o r . " I t is always possible that u n k n o w n to myself I am
up to my neck in quicksand a n d sinking fast," he w r o t e .
C o l e m a n likes to impress a u d i e n c e s vividly with the i m p o r t a n c e of
this p r o b l e m , that t h e chances of canceling o u t a cosmological constant
to o n e p a r t in 10
is fantastically small. " I m a g i n e that over a ten-year
p e r i o d you s p e n d millions of dollars without looking at your salary, a n d
w h e n you finally c o m p a r e what you e a r n with what you spent, they bala n c e o u t to t h e p e n n y , " he n o t e s . T h u s his calculation, which shows
t h a t you can cancel t h e cosmological constant to o n e p a r t in 1 0 , is a
highly nontrivial result. To a d d frosting to t h e cake, C o l e m a n e m p h a sizes that these w o r m h o l e s also solve a n o t h e r p r o b l e m : They h e l p to
d e t e r m i n e the values of the f u n d a m e n t a l constants of t h e universe. Colem a n adds, " I t was a completely different m e c h a n i s m from any that h a d
been considered. It was B a t m a n swinging in on his r o p e . "
13
14
l00
15
100
1 6
But criticisms also b e g a n to surface; t h e most persistent criticism was
that he assumed that t h e wormholes were small, on t h e o r d e r of the
Planck length, a n d that he forgot to sum over large wormholes. According to the critics, large w o r m h o l e s should also be i n c l u d e d in his sum.
But since we d o n ' t see large, visible w o r m h o l e s anywhere, it seems that
his calculation has a fatal flaw.
Unfazed by this criticism, C o l e m a n shot back in his usual way: choosing o u t r a g e o u s titles for his p a p e r s . To prove that large wormholes can
be neglected in his calculation, he wrote a rebuttal to his critics with the
title " E s c a p e from the M e n a c e of t h e Giant W o r m h o l e s . " W h e n asked
a b o u t his titles, he replied, "If Nobel Prizes were given for titles, I'd have
already collected m i n e . "
If C o l e m a n ' s purely m a t h e m a t i c a l a r g u m e n t s are correct, they would
give h a r d e x p e r i m e n t a l evidence that w o r m h o l e s are an essential feature
of all physical processes, a n d n o t j u s t s o m e p i p e d r e a m . It would m e a n
that w o r m h o l e s c o n n e c t i n g o u r universe with an infinite n u m b e r of dead
universes a r e essential to p r e v e n t o u r universe from wrapping itself up
1 7
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into a tight, tiny ball, or from e x p l o d i n g outward at fantastic rates. It
would m e a n that w o r m h o l e s are t h e essential feature m a k i n g o u r universe relatively stable.
But as with most d e v e l o p m e n t s t h a t occur at t h e Planck l e n g t h , t h e
final solution to these w o r m h o l e e q u a t i o n s will have to wait until we
have a b e t t e r grasp of q u a n t u m gravity. Many of C o l e m a n ' s e q u a t i o n s
r e q u i r e a m e a n s of eliminating t h e infinities c o m m o n to all q u a n t u m
theories of gravity, a n d this m e a n s using superstring theory. In particular, we may have to wait until we can confidently calculate finite q u a n t u m
corrections to his theory. Many of these strange predictions will have to
wait until we can s h a r p e n o u r calculational tools.
As we have emphasized, t h e p r o b l e m is mainly theoretical. We simply
do n o t have the m a t h e m a t i c a l b r a i n p o w e r to b r e a k o p e n these welldefined p r o b l e m s . T h e equations stare at us from t h e blackboard, b u t
we are helpless to find rigorous, finite solutions to t h e m at present. O n c e
physicists have a b e t t e r grasp of the physics at the Planck energy, t h e n
a whole new universe of possibilities o p e n s u p . Anyone, or any civilization, that truly masters t h e energy f o u n d at t h e Planck l e n g t h will
b e c o m e the master of all f u n d a m e n t a l forces. T h a t is t h e n e x t topic to
which we will t u r n . W h e n can we expect to b e c o m e masters of hyperspace?
PART IV
Masters of
Hyperspace
13
Beyond the Future
W h a t d o e s it m e a n for a civilization to be a m i l l i o n years old?
We have h a d radio t e l e s c o p e s a n d spaceships for a few decades; o u r technical civilization is a few h u n d r e d years o l d .. .
an
a d v a n c e d civilization m i l l i o n s of years o l d is as
much
b e y o n d us as we are b e y o n d a b u s h baby or a m a c a q u e .
Carl Sagan
P
HYSICIST Paul Davies o n c e c o m m e n t e d on what to expect o n c e
we have solved the mysteries of t h e unification of all forces into a
single superforce. He wrote that
w e c o u l d c h a n g e t h e s t r u c t u r e o f s p a c e a n d t i m e , tie o u r o w n k n o t s i n
nothingness, and build matter to order. Controlling the superforce would
e n a b l e u s t o c o n s t r u c t a n d t r a n s m u t e p a r t i c l e s a t will, t h u s g e n e r a t i n g
exotic forms of matter. We might even be able to manipulate the d i m e n s i o n a l i t y o f s p a c e itself, c r e a t i n g b i z a r r e artificial w o r l d s w i t h u n i m a g i n a b l e
properties. Truly we s h o u l d be lords of the universe.'
W h e n can we expect to h a r n e s s t h e power of hyperspace? Experim e n t a l verification of t h e hyperspace theory, at least indirectly, may
c o m e in the twenty-first century. However, the energy scale necessary to
m a n i p u l a t e (and n o t j u s t verify) ten-dimensional s p a c e - t i m e , to b e c o m e
"lords of the universe," is many centuries b e y o n d today's technology.
As we have seen, e n o r m o u s a m o u n t s of m a t t e r - e n e r g y are necessary to
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p e r f o r m near-miraculous feats, such as creating wormholes a n d altering
t h e direction of time.
To be masters of the t e n t h d i m e n s i o n , e i t h e r we e n c o u n t e r intellig e n t life within t h e galaxy that has already harnessed these astronomical
energy levels, or we struggle for several t h o u s a n d years before we attain
this ability ourselves. For e x a m p l e , o u r c u r r e n t a t o m smashers or particle
accelerators can boost t h e energy of a particle to over 1 trillion electron
volts (the energy created if an electron were accelerated by 1 trillion
volts). T h e largest accelerator is currently located in Geneva, Switzerland, a n d o p e r a t e d by a c o n s o r t i u m of 14 E u r o p e a n nations. But this
energy pales before the energy necessary to p r o b e hyperspace: 1 0 billion e l e c t r o n volts, or a quadrillion times larger t h a n the energy that
m i g h t have b e e n p r o d u c e d by t h e SSC.
19
A quadrillion (1 with 15 zeros after it) may seem like an impossibly
large n u m b e r . T h e technology necessary to p r o b e this incredible energy
may r e q u i r e a t o m smashers billions of miles long, or an entirely new
technology altogether. Even if we were to liquidate the entire gross
n a t i o n a l p r o d u c t of t h e world a n d build a super-powerful a t o m smasher,
we would n o t be able to c o m e close to this energy. At first, it seems an
impossible task to h a r n e s s this level of energy.
However, this n u m b e r d o e s n o t seem so ridiculously large if we
u n d e r s t a n d t h a t technology e x p a n d s exponentially, which is difficult for
o u r m i n d s t o c o m p r e h e n d . T o u n d e r s t a n d how fast exponential
growth is, imagine a b a c t e r i u m that splits in half every 30 minutes. If its
growth is u n i m p e d e d , t h e n within a few weeks this single bacterium
will p r o d u c e a colony that will weigh as m u c h as the entire p l a n e t
earth.
A l t h o u g h h u m a n s have existed on this p l a n e t for p e r h a p s 2 million
years, t h e rapid climb to m o d e r n civilization within t h e last 200 years
was possible d u e to t h e fact t h a t t h e growth of scientific knowledge is
e x p o n e n t i a l ; that is, its rate of e x p a n s i o n is p r o p o r t i o n a l to how m u c h
is already known. T h e m o r e we know, the faster we can know m o r e . For
e x a m p l e , we have amassed m o r e knowledge since World War II t h a n all
t h e knowledge amassed in o u r 2-million-year evolution on this planet.
In fact, the a m o u n t of knowledge that o u r scientists gain doubles approximately every 10 to 20 years.
T h u s it b e c o m e s i m p o r t a n t to analyze o u r own d e v e l o p m e n t historically. To a p p r e c i a t e how technology can grow exponentially, let us analyze o u r own evolution, focusing strictly on the energy available to the
average h u m a n . This will h e l p p u t t h e energy necessary to exploit the
ten-dimensional theory i n t o p r o p e r historical perspective.
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The Exponential Rise of Civilization
Today, we may think n o t h i n g a b o u t taking a Sunday drive in the country
in a car with a 200-horsepower e n g i n e . But t h e energy available to t h e
average h u m a n d u r i n g most of o u r evolution on this p l a n e t was considerably less.
D u r i n g this period, the basic energy s o u r c e was t h e power of o u r own
h a n d s , a b o u t one-eighth of a horsepower. H u m a n s r o a m e d the e a r t h in
small bands, h u n t i n g a n d foraging for food in packs m u c h like animals,
using only t h e energy of their own muscles. F r o m an energy p o i n t of
view, this c h a n g e d only within t h e last 100,000 years. With t h e invention
of h a n d tools, h u m a n s could e x t e n d the power of their limbs. Spears
e x t e n d e d the power of their arms, clubs t h e power of their fists, a n d
knives the power of their jaws. In this p e r i o d , their energy o u t p u t doubled, to a b o u t o n e - q u a r t e r of a horsepower.
Within the past 10,000 or so years, t h e energy o u t p u t of a h u m a n
d o u b l e d o n c e again. T h e m a i n reason for this c h a n g e was probably t h e
e n d of the Ice Age, which h a d r e t a r d e d h u m a n d e v e l o p m e n t for
t h o u s a n d s of years.
H u m a n society, which consisted of small b a n d s of h u n t e r s a n d gatherers for h u n d r e d s of t h o u s a n d s of years, c h a n g e d with the discovery of
agriculture soon after t h e ice m e l t e d . Roving b a n d s of h u m a n s , n o t having to follow g a m e across t h e plains a n d forests, settled in stable villages
where crops could be harvested a r o u n d the year. Also, with the m e l t i n g
of the ice sheet came t h e domestication of animals such as horses a n d
oxen; the energy available to a h u m a n rose to approximately 1 horsepower.
With the b e g i n n i n g of a stratified, agrarian life c a m e the division of
labor, until society u n d e r w e n t an i m p o r t a n t c h a n g e : t h e transition to a
slave society. This m e a n t that o n e p e r s o n , t h e slave owner, could comm a n d the energy of h u n d r e d s of slaves. This s u d d e n increase in energy
m a d e possible i n h u m a n brutality; it also m a d e possible t h e first true
cities, where kings could c o m m a n d their slaves to use large cranes, levers, a n d pulleys to erect fortresses a n d m o n u m e n t s to themselves.
Because of this increase in energy, o u t of the deserts a n d forests rose
temples, towers, pyramids, a n d cities.
F r o m an energy p o i n t of view, for a b o u t 99.99% of the existence of
h u m a n i t y on this planet, the technological level of o u r species was only
o n e step above that of animals. It has only b e e n within t h e past few
h u n d r e d years that h u m a n s have h a d m o r e t h a n 1 h o r s e p o w e r available
to t h e m .
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A decisive c h a n g e c a m e with t h e Industrial Revolution. Newton's discovery of t h e universal law of gravity a n d m o t i o n m a d e it possible to
r e d u c e m e c h a n i c s to a set of well-defined equations. T h u s Newton's classical theory of t h e gravitational force, in s o m e sense, paved the way for
t h e m o d e r n theory of m a c h i n e s . This h e l p e d to m a k e possible the widespread use of steam-powered e n g i n e s in t h e n i n e t e e n t h century; with
steam, t h e average h u m a n could c o m m a n d tens to h u n d r e d s of horsepowers. For e x a m p l e , the railroads o p e n e d up entire c o n t i n e n t s to develo p m e n t , a n d steamships o p e n e d u p m o d e r n international trade. Both
were e n e r g i z e d by t h e power of steam, h e a t e d by coal.
It took over 10,000 years for h u m a n i t y to create m o d e r n civilization
over t h e face of E u r o p e . With steam-driven a n d later oil-fired machines,
t h e U n i t e d States was industrialized within a century. T h u s the mastery
of j u s t a single f u n d a m e n t a l force of n a t u r e vastly increased the energy
available to a h u m a n b e i n g a n d irrevocably c h a n g e d society.
By t h e late n i n e t e e n t h century, Maxwell's mastery of the electromagnetic force o n c e again set off a revolution in energy. T h e electromagnetic force m a d e possible the electrification of o u r cities a n d o u r h o m e s ,
exponentially increasing t h e versatility a n d power of o u r machines.
Steam e n g i n e s were now b e i n g replaced by powerful dynamos.
Within t h e past 50 years, t h e discovery of t h e nuclear force has
increased t h e power available to a single h u m a n by a factor of a million.
Because t h e energy of chemical reactions is m e a s u r e d in electron volts,
while t h e energy of fission a n d fusion is m e a s u r e d in millions of electron
volts, we have a millionfold increase in the power available to us.
T h e lesson from analyzing t h e historical energy n e e d s of humanity
shows graphically how for only 0 . 0 1 % of o u r existence we have manipulated energy levels b e y o n d that of animals. Yet within j u s t a few centuries, we have u n l e a s h e d vast a m o u n t s of energy via the electromagnetic
a n d n u c l e a r forces. Let us now leave t h e past a n d begin a discussion of
the future, using the same m e t h o d o l o g y , to u n d e r s t a n d the p o i n t at
which we may harness t h e superforce.
Type I, II, and III Civilizations
Futurology, or t h e prediction of the future from reasonable scientific
j u d g m e n t s , is a risky science. S o m e would n o t even call it a science at
all, b u t s o m e t h i n g that m o r e resembles h o c u s p o c u s or witchcraft. Futurology has deservedly e a r n e d this unsavory r e p u t a t i o n because every
"scientific" poll c o n d u c t e d by futurologists a b o u t t h e n e x t d e c a d e has
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proved to be wildly off t h e mark. W h a t makes futurology such a primitive
science is that o u r brains t h i n k linearly, while knowledge progresses
exponentially. For e x a m p l e , polls of futurologists have shown t h a t they
take known technology a n d simply d o u b l e or triple it to p r e d i c t t h e
future. Polls taken in the 1920s showed that futurologists p r e d i c t e d t h a t
we would have, within a few decades, h u g e fleets of blimps taking passengers across the Atlantic.
But science also develops in u n e x p e c t e d ways. In t h e s h o r t r u n , w h e n
extrapolating within a few years, it is a safe b e t that science will progress
t h r o u g h steady, quantitative i m p r o v e m e n t s on existing technology. However, w h e n extrapolating over a few decades, we find that qualitative
b r e a k t h r o u g h s in new areas b e c o m e the d o m i n a n t factor, w h e r e new
industries o p e n up in u n e x p e c t e d places.
P e r h a p s the most famous e x a m p l e of futurology g o n e w r o n g is t h e
predictions m a d e by J o h n von N e u m a n n , t h e father of the m o d e r n electronic c o m p u t e r a n d o n e of the great m a t h e m a t i c i a n s of t h e century.
After the war, he m a d e two predictions: first, t h a t in t h e future c o m p u t ers would b e c o m e so m o n s t r o u s a n d costly that only large g o v e r n m e n t s
would be able to afford t h e m , a n d second, that c o m p u t e r s would be able
to predict the weather accurately.
In reality, the growth of c o m p u t e r s went in precisely t h e opposite
direction: We are flooded with inexpensive, m i n i a t u r e c o m p u t e r s that
can fit in the palm of o u r h a n d s . C o m p u t e r chips have b e c o m e so c h e a p
a n d plentiful that they a r e an integral p a r t of s o m e m o d e r n appliances.
Already, we have the " s m a r t " typewriter (the word p r o c e s s o r ) , a n d eventually we will have the " s m a r t " v a c u u m cleaner, t h e " s m a r t " kitchen,
the " s m a r t " television, a n d the like. Also, c o m p u t e r s , no m a t t e r how
powerful, have failed to p r e d i c t t h e weather. A l t h o u g h t h e classical
motion of individual molecules can, in principle, be p r e d i c t e d , t h e
weather is so c o m p l e x that even s o m e o n e sneezing can create distortions
that will ripple a n d be magnified across t h o u s a n d s of miles, eventually,
p e r h a p s , unleashing a h u r r i c a n e .
With all these i m p o r t a n t caveats, let us d e t e r m i n e w h e n a civilization
(either o u r own or o n e in o u t e r space) may attain t h e ability to master
the tenth dimension. A s t r o n o m e r Nikolai Kardashev of t h e f o r m e r
Soviet U n i o n o n c e categorized future civilizations in t h e following way.
A Type I civilization is o n e that controls t h e energy resources of an
entire planet. This civilization can control t h e weather, p r e v e n t earthquakes, m i n e d e e p in t h e e a r t h ' s crust, a n d harvest t h e oceans. This
civilization has already c o m p l e t e d the exploration of its solar system.
A Type II civilization is o n e t h a t controls t h e power of the sun itself.
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This does n o t m e a n passively h a r n e s s i n g solar energy; this civilization
m i n e s t h e sun. T h e energy n e e d s of this civilization are so large that it
directly c o n s u m e s the power of the sun to drive its m a c h i n e s . This civilization will begin t h e colonization of local star systems.
A Type III civilization is o n e that controls t h e power of an entire
galaxy. For a power source, it harnesses t h e power of billions of star
systems. It has probably m a s t e r e d Einstein's equations a n d can manipulate s p a c e - t i m e at will.
T h e basis of this classification is r a t h e r simple: Each level is categorized on t h e basis of the power source that energizes the civilization.
Type I civilizations use t h e power of an entire planet. Type II civilizations
use t h e power of an e n t i r e star. Type III civilizations use the power of
an e n t i r e galaxy. This classification ignores any predictions c o n c e r n i n g
the detailed n a t u r e of future civilizations (which are b o u n d to be wrong)
a n d instead focuses on aspects that can be reasonably u n d e r s t o o d by the
laws of physics, such as energy supply.
O u r civilization, by contrast, can be categorized as a Type 0 civilization, o n e that is j u s t b e g i n n i n g to tap planetary resources, b u t does n o t
have t h e technology a n d resources to control t h e m . A Type 0 civilization
like o u r s derives its e n e r g y from fossil fuels like oil a n d coal a n d , in m u c h
of t h e T h i r d World, from raw h u m a n labor. O u r largest c o m p u t e r s cann o t even p r e d i c t t h e weather, let a l o n e control it. Viewed from this larger
perspective, we as a civilization are like a n e w b o r n infant.
A l t h o u g h o n e m i g h t guess t h a t t h e slow m a r c h from a Type 0 civilization to a Type III civilization m i g h t take millions of years, the extraordinary fact a b o u t this classification s c h e m e is that this climb is an exponential o n e a n d h e n c e p r o c e e d s m u c h faster t h a n anything we can
readily conceive.
With all these qualifications, we can still m a k e e d u c a t e d guesses
a b o u t w h e n o u r civilization will r e a c h these milestones. Given the rate
at which o u r civilization is growing, we m i g h t expect to reach Type I
status within a few centuries.
For e x a m p l e , the largest energy source available to o u r Type 0 civilization is the h y d r o g e n b o m b . O u r technology is so primitive that we
can u n l e a s h t h e power of h y d r o g e n fusion only by d e t o n a t i n g a b o m b ,
r a t h e r t h a n c o n t r o l l i n g it in a power g e n e r a t o r . However, a simple hurr i c a n e g e n e r a t e s t h e power of h u n d r e d s of h y d r o g e n b o m b s . T h u s
w e a t h e r c o n t r o l , which is o n e feature of Type I civilizations, is at least a
century away from today's technology.
Similarly, a Type I civilization has already colonized most of its solar
system. By contrast, milestones in today's d e v e l o p m e n t of space travel
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are painfully m e a s u r e d on the scale of decades, a n d therefore qualitative
leaps such as space colonization m u s t be m e a s u r e d in centuries. For
example, the earliest d a t e for NASA's m a n n e d l a n d i n g on t h e p l a n e t
Mars is 2020. T h e r e f o r e , the colonization of Mars may take place 40 to
50 years after that, a n d the colonization of t h e solar system within a
century.
By contrast, the transition from a Type I to a Type II civilization may
take only 1,000 years. Given the e x p o n e n t i a l growth of civilization, we
may expect that within 1,000 years t h e energy n e e d s of a civilization will
b e c o m e so large that it m u s t begin to m i n e the sun to energize its
machines.
A typical e x a m p l e of a Type II civilization is t h e F e d e r a t i o n of Planets
portrayed in the "Star T r e k " series. This civilization has j u s t b e g u n to
master the gravitational force—that is, t h e art of warping s p a c e - t i m e via
w o r m h o l e s — a n d h e n c e , for the first time, has the capability of r e a c h i n g
nearby stars. It has evaded t h e limit placed by t h e s p e e d of light by
mastering Einstein's theory of general relativity. Small colonies have
b e e n established on some of these systems, which t h e starship Enterprise
is sworn to protect. T h e civilization's starships are p o w e r e d by the collision of matter a n d antimatter. T h e ability to create large c o n c e n t r a t i o n s
of a n t i m a t t e r suitable for space travel places t h a t civilization many centuries to a m i l l e n n i u m away from ours.
Advancing to a Type III civilization may take several t h o u s a n d years
or m o r e . This is, in fact, t h e time scale p r e d i c t e d by Isaac Asimov in his
classic F o u n d a t i o n Series, which describes t h e rise, fall, a n d re-emergence of a galactic civilization. T h e time scale involved in each of these
transitions involves t h o u s a n d s of years. This civilization has h a r n e s s e d
the energy source c o n t a i n e d within t h e galaxy itself. To it, warp drive,
instead of b e i n g an exotic form of travel to t h e nearby stars, is t h e stand a r d m e a n s of trade a n d c o m m e r c e between sectors of t h e galaxy. T h u s
although it took 2 million years for o u r species to leave t h e safety of t h e
forests a n d build a m o d e r n civilization, it may take only t h o u s a n d s of
years to leave the safety of o u r solar system a n d build a galactic civilization.
O n e o p t i o n o p e n to a Type III civilization is h a r n e s s i n g t h e power
of supernovae or black holes. Its starships may even be able to p r o b e t h e
galactic nucleus, which is p e r h a p s t h e most mysterious of all energy
sources. Astrophysicists have theorized that because of the e n o r m o u s
size of the galactic nucleus, the c e n t e r of o u r galaxy may c o n t a i n millions
of black holes. If true, this would provide virtually u n l i m i t e d a m o u n t s of
energy.
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At this p o i n t , m a n i p u l a t i n g energies a million billion times larger
t h a n present-day energies should be possible. T h u s for a Type III civilization, with t h e energy o u t p u t of u n c o u n t a b l e star systems a n d p e r h a p s
the galactic n u c l e u s at its disposal, the mastery of the t e n t h dimension
b e c o m e s a real possibility.
Astrochicken
I o n c e h a d l u n c h with physicist F r e e m a n Dyson of the Institute for
Advanced Study. Dyson is a senior figure in t h e world of physics who has
tackled s o m e of t h e most intellectually challenging a n d intriguing questions facing h u m a n i t y , such as new directions in space exploration, the
n a t u r e of extraterrestrial life, a n d t h e future of civilization.
Unlike o t h e r physicists, w h o dwell excessively in narrow, well-defined
areas of specialization, Dyson's fertile imagination has r o a m e d across
the galaxy. "I c a n n o t , as B o h r a n d F e y n m a n did, sit for years with my
whole m i n d c o n c e n t r a t e d u p o n o n e d e e p question. I am interested in
too m a n y different d i r e c t i o n s , " he confessed. T h i n , remarkably spry,
with t h e owlish expression of an Oxford d o n , a n d speaking with a trace
of his British accent, he e n g a g e d in a long, wide-ranging l u n c h conversation with m e , t o u c h i n g on many of t h e ideas that have fascinated h i m
over t h e years.
Viewing t h e transition of o u r civilization to Type I status, Dyson finds
that o u r primitive space p r o g r a m is h e a d e d in the w r o n g direction. T h e
c u r r e n t t r e n d is toward heavier payloads a n d g r e a t e r lag time between
space shots, which is severely r e t a r d i n g t h e exploration of space. In his
writings, he has p r o p o s e d a radical d e p a r t u r e from this t r e n d , based on
what he calls t h e Astrochicken.
Small, lightweight, a n d intelligent, Astrochicken is a versatile space
p r o b e t h a t has a clear advantage over t h e bulky, exorbitantly expensive
space missions of t h e past, which have b e e n a b o t t l e n e c k to space exploration. "Astrochicken will weight a kilogram instead of Voyager's t o n , "
he claims. "Astrochicken will n o t be built, it will be g r o w n , " he adds.
"Astrochicken could be as agile as a h u m m i n g b i r d with a brain weighing
no more than a g r a m . "
It will be p a r t m a c h i n e a n d p a r t animal, using t h e most advanced
d e v e l o p m e n t s in b i o e n g i n e e r i n g . It will be small b u t powerful e n o u g h
to e x p l o r e t h e o u t e r planets, such as U r a n u s a n d N e p t u n e . It will n o t
n e e d h u g e quantities of r o c k e t fuel; it will be b r e d a n d p r o g r a m m e d to
" e a t " ice a n d h y d r o c a r b o n s f o u n d i n t h e rings s u r r o u n d i n g the o u t e r
2
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planet. Its genetically e n g i n e e r e d s t o m a c h will t h e n digest these materials into chemical fuel. O n c e its a p p e t i t e has b e e n satisfied, it will t h e n
rocket to the n e x t m o o n or planet.
Astrochicken d e p e n d s on technological b r e a k t h r o u g h s in genetic
e n g i n e e r i n g , artificial intelligence, a n d solar-electric p r o p u l s i o n . Given
the remarkable progress in these ares, Dyson expects that t h e various
technologies for Astrochicken may be available by t h e year 2016.
Taking the larger view of t h e d e v e l o p m e n t of civilization, Dyson also
believes that, at the c u r r e n t rate of d e v e l o p m e n t , we may attain Type I
status within a few centuries. He d o e s n o t believe that m a k i n g t h e transition between the various types of civilizations will be very difficult. He
estimates that the difference in size a n d power separating t h e various
types of civilizations is roughly a factor of 10 billion. A l t h o u g h this may
seem like a large n u m b e r , a civilization growing at the sluggish rate of
1 p e r c e n t p e r year can expect to m a k e the transition between t h e various
civilizations within 2,500 years. T h u s it is almost g u a r a n t e e d that a civilization can steadily progress toward Type III status.
Dyson has written, "A society which h a p p e n s to possess a s t r o n g
expansionist drive will e x p a n d its habitat from a single p l a n e t (Type I)
to a b i o s p h e r e exploiting an e n t i r e star (Type II) within a few t h o u s a n d
years, a n d from a single star to an e n t i r e galaxy (Type III) within a few
million years. A species which has o n c e passed b e y o n d Type II status is
invulnerable to extinction by even the worst imaginable n a t u r a l or artificial catastrophe."
However, t h e r e is o n e p r o b l e m . Dyson has c o n c l u d e d that t h e transition from a Type II to a Type III civilization may pose formidable physical difficulties, d u e mainly to t h e limitation i m p o s e d by t h e speed of
light. T h e expansion of a Type II civilization will necessarily p r o c e e d at
less t h a n the speed of light, which he feel places a severe restriction on
its development.
Will a Type II civilization break the light b a r r i e r a n d t h e b o n d s of
special relativity by e x p l o r i n g the power of hyperspace? Dyson is n o t
sure. N o t h i n g can b e ruled out, b u t the Planck l e n g t h , h e r e m i n d e d m e ,
is a fantastically small distance, a n d t h e energies r e q u i r e d to p r o b e down
to that distance are u n i m a g i n a b l e . P e r h a p s , he m u s e d , t h e Planck l e n g t h
is a natural barrier facing all civilizations.
4
Type III Civilizations in Outer Space
If the long j o u r n e y to r e a c h Type III status seems r e m o t e for o u r own
civilization, p e r h a p s o n e day we will m e e t an extraterrestrial civilization
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that has already h a r n e s s e d hyperspace for its n e e d s a n d is willing to share
its technology with us. T h e puzzle facing us, however, is that we do n o t
see signs of any a d v a n c e d civilization in the heavens, at least n o t in o u r
solar system or even in o u r small sector of t h e galaxy. O u r space probes,
especially the Viking l a n d i n g on Mars in the 1970s a n d the Voyager missions to J u p i t e r , Saturn, U r a n u s , a n d N e p t u n e in the 1980s, have sent
back discouraging information c o n c e r n i n g the bleak, lifeless n a t u r e of
o u r solar system.
T h e two most p r o m i s i n g planets, Venus a n d Mars, have t u r n e d up
no signs of life, let a l o n e advanced civilizations. Venus, n a m e d after the
goddess of love, was o n c e envisioned by a s t r o n o m e r s as well as romantics
to be a lush, tropical planet. Instead, o u r space p r o b e s have found a
h a r s h , b a r r e n planet, with a suffocating a t m o s p h e r e of c a r b o n dioxide,
blistering t e m p e r a t u r e s e x c e e d i n g 800°F, a n d toxic rains of sulfuric acid.
Mars, t h e focus of speculation even before O r s o n Welles caused
p a n i c in the country in 1938 d u r i n g t h e Depression with his fictional
broadcast a b o u t an invasion from that planet, has b e e n equally disapp o i n t i n g . We know it to be a desolate, desert p l a n e t without traces of
surface water. A n c i e n t riverbeds a n d long-vanished oceans have left their
distinctive m a r k on t h e surface of Mars, b u t we see no ruins or any
indications of civilization.
G o i n g b e y o n d o u r solar system, scientists have analyzed the radio
emissions from nearby stars with equally fruitless results. Dyson has
stressed t h a t any advanced civilization, by the Second Law of T h e r m o dynamics, must necessarily g e n e r a t e large quantities of waste heat. Its
energy c o n s u m p t i o n s h o u l d be e n o r m o u s , a n d a small fraction of that
waste h e a t s h o u l d be easily d e t e c t e d by o u r instruments. T h u s , Dyson
claims, by s c a n n i n g the nearby stars, o u r i n s t r u m e n t s should be able find
the telltale fingerprint of waste h e a t b e i n g g e n e r a t e d by an advanced
civilization. But no m a t t e r w h e r e we scan t h e heavens, we see no traces
of waste h e a t or r a d i o c o m m u n i c a t i o n s from Type I, II, or III civilizations. On o u r own earth, for e x a m p l e , we have m a s t e r e d the art of radio
a n d television within t h e past half-century. T h u s an e x p a n d i n g s p h e r e
of radio waves, a b o u t 50 light-years in radius, s u r r o u n d s o u r planet. Any
star within 50 light-years of earth, if it contains intelligent life, should be
able to d e t e c t o u r p r e s e n c e . Likewise, any Type II or III civilization
s h o u l d be b r o a d c a s t i n g copious quantities of electromagnetic radiation
continuously for the past several t h o u s a n d years, so that any intelligent
life within several t h o u s a n d light-years of t h e civilization's p l a n e t should
be able to d e t e c t its p r e s e n c e .
In 1978, a s t r o n o m e r Paul Horowitz s c a n n e d all sunlike star systems
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(185 in all) within 80 light-years of o u r solar system, a n d f o u n d no traces
of radio emissions from intelligent life. A s t r o n o m e r s D o n a l d Goldsmith
a n d Tobius Owen r e p o r t e d in 1979 a search of m o r e t h a n 600 star systems, also with negative results. This search, called SETI (search for
extraterrestrial intelligence), has m e t with consistent failure. (Encouragingly, in a rare display of scientific generosity, in 1992 Congress a p p r o priated $100 million to be s p e n t over a 10-year p e r i o d for t h e H i g h
Resolution Microwave Survey, which will scan t h e nearby stars for intelligent life. T h e s e funds will m a k e it possible for t h e gigantic 305-meter
fixed r a d i o dish at Arecibo, P u e r t o Rico, to scan select stars systematically
within 100 light-years of t h e e a r t h . This will be c o m p l e m e n t e d by t h e 34m e t e r movable radio a n t e n n a at Goldstone, California, which will sweep
b r o a d p o r t i o n s of the n i g h t sky. After years of negative results, astronom e r Frank Drake of t h e University of California at Santa Cruz is cautiously optimistic that they will find s o m e positive signs of intelligent life.
H e remarks, " M a n y h u m a n societies developed science i n d e p e n d e n t l y
t h r o u g h a c o m b i n a t i o n of curiosity a n d trying to create a b e t t e r life, a n d
I think those same motivations would exist in o t h e r c r e a t u r e s . " )
T h e puzzle d e e p e n s w h e n we realize t h a t the probability of intellig e n t life e m e r g i n g within o u r galaxy is surprisingly large. Drake even
derived a simple e q u a t i o n to calculate t h e n u m b e r of planets with intelligent life forms in t h e galaxy.
O u r galaxy, for e x a m p l e , contains a b o u t 200 billion stars. To get a
ballpark figure for the n u m b e r of stars with intelligent life forms, we can
make the following very c r u d e estimate. We can be conservative a n d say
that 10% of these stars are yellow stars m u c h like the sun, that 10% of
those have planets orbiting t h e m , that 10% of those have earthlike planets, that 10% of those have earthlike planets with a t m o s p h e r e s compatible with life, that 10% have earthlike a t m o s p h e r e s with life forms growing in t h e m , a n d that 10% of those have s o m e form of intelligent life.
This m e a n s that one-millionth of t h e 200 billion stars in the galaxy will
probably have some intelligent life form. This implies that a staggering
200,000 stars will have planets h a r b o r i n g some form of intelligent life.
A slightly m o r e optimistic set of values for Drake's e q u a t i o n shows t h a t
intelligent life m i g h t b e , on t h e average, as close as 15 light-years from
o u r sun.
With r e c e n t advanced c o m p u t e r t e c h n i q u e s , scientists have b e e n able
to refine Drake's original back-of-the-envelope calculation. G e o r g e W.
Wetherill of the Carnegie Institution of W a s h i n g t o n , for e x a m p l e , has
r u n c o m p u t e r simulations of t h e early evolution of o u r solar system,
b e g i n n i n g with a large, swirling disk of gas a n d dust a r o u n d t h e sun. He
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lets the c o m p u t e r evolve the disk until small, rocky masses begin to coalesce out of the dust. Much to his pleasant surprise, he found that planets
of approximately the size of the earth were easy to evolve o u t of these
rocky cores. Most of the time, in fact, earth-size planets spontaneously
coalesced with masses between 80% a n d 130% of the earth's distance
from the sun. (Curiously, he also found that the formation of Jupitersize planets far from the sun was i m p o r t a n t for the evolution of the earthsize planets. T h e Jupiter-size planets were essential to sweep o u t swarms
of comets a n d debris that would eventually strike the earthlike planet,
extinguishing any primitive life forms on it. Wetherill's c o m p u t e r simulations show that without a Jupiter-like planet to clean out these comets
with its gigantic gravitational pull, these comets would hit the earthlike
planet a b o u t 1,000 times m o r e frequently than they do in reality, making
a life-destroying impact every 100,000 years or so.)
T h u s it is a compelling (but certainly n o t rigorous) conclusion that
the laws of probability favor the presence of other intelligence within
the galaxy. T h e fact that o u r galaxy is p e r h a p s 10 billion years old means
that there has b e e n ample time for scores of intelligent life forms to have
flourished within it. Type II a n d III civilizations, broadcasting for several
h u n d r e d to several thousand years, should be sending out an easily
detectable sphere of electromagnetic radiation measuring several h u n d r e d to several thousand light-years in diameter. Yet we see no signs of
intelligent life forms in the heavens.
Why?
Several speculative theories have been advanced to explain why we
have b e e n unable to detect signs of intelligent life o u t to 100 light-years
of o u r planet. N o n e of t h e m is particularly satisfying, a n d the final truth
may be a combination of all of t h e m .
O n e theory holds that Drake's equation may give us r o u g h probabilities of how many planets contain intelligent life, b u t tells us n o t h i n g
a b o u t when these planets attain this level of development. Given the
astronomical time scales involved, p e r h a p s Drake's equation predicts
intelligent life forms that existed millions of years before us, or will exist
millions of years after us.
For example, o u r solar system is approximately 4.5 billion years old.
Life started on the earth about 3 to 4 billion years ago, b u t only within
the past million years has intelligent life developed on the planet (and
only within the past few decades has this civilization built radio stations
capable of sending signals into o u t e r space). However, 1 million years,
on the time scale of billions of years, is b u t an instant of time. It is
reasonable to assume that thousands of advanced civilizations existed
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before o u r distant ancestors even left the forest a n d have since perished,
or that thousands m o r e civilizations will develop long after ours has died.
Either way, we would n o t be able to detect t h e m via o u r instruments.
T h e second theory holds that the galaxy is, in fact, teeming with
advanced forms of civilizations, b u t they are advanced e n o u g h to conceal
their existence from o u r prying instruments. We would m e a n n o t h i n g
to t h e m because they are so many millions of years a h e a d of us. For
example, if we stumble on an a n t colony while walking in a field, o u r
first impulse is certainly not to make contact with the ants, ask to see
their leader, wave trinkets before their eyes, a n d offer t h e m unparalleled
prosperity a n d the fruits of o u r advanced technology. More likely, o u r
first temptation is to ignore t h e m (or p e r h a p s even step on a few of
them).
Puzzled by these long-standing questions, I asked Dyson if he t h o u g h t
we would soon be making contact with extraterrestrial life forms. His
answer r a t h e r surprised me. He said, "I h o p e n o t . " I t h o u g h t it was
strange that s o m e o n e who h a d spent decades speculating a b o u t intellig e n t civilizations in o u t e r space should have reservations a b o u t actually
m e e t i n g them. Knowing British history, however, he must have h a d good
reasons for n o t rushing in to e m b r a c e o t h e r civilizations. British civilization was probably only several h u n d r e d years m o r e advanced t h a n
many of the civilizations, such as the Indian a n d the African, c o n q u e r e d
by the British army a n d navy.
Although most science-fiction writers bewail the limitations on space
exploration placed by the speed of light, Dyson takes the u n o r t h o d o x
view that p e r h a p s this is a good thing. Viewing the often bloody history
of colonialism t h r o u g h o u t o u r own world history, p e r h a p s it is a blessing
in disguise, he muses, that various Type II civilizations will be separated
by large distances a n d that the Planck energy is inaccessible. Looking at
the bright side, he q u i p p e d , "At least, o n e can evade the tax collector."
Unfortunately, the m e e t i n g of two u n e q u a l civilizations has often h a d
catastrophic implications for the weaker o n e . For example, the Aztec
civilization h a d risen over thousands of years to great p r o m i n e n c e in
central Mexico. In some areas, its mastery of science, art, a n d technology
rivaled the achievements of E u r o p e . However, in the area of g u n p o w d e r
a n d warships, the Aztecs were p e r h a p s several centuries b e h i n d the
Spanish. T h e s u d d e n clash between a small, ragged b a n d of 400 conquistadors a n d the advanced civilizations of the Aztecs e n d e d in tragedy
in 1521. Within a brief period of time, the Aztec people, with a population n u m b e r i n g in the millions, were systematically crushed a n d
enslaved to work in the mines. Their treasuries were looted, their history
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was erased, a n d even the faintest memory of the great Aztec civilization
was obliterated by waves of missionaries.
W h e n we think of how we might react to visitors from outer space,
it is sobering to read how the Aztecs reacted to the visitors from Spain:
"They seized u p o n the gold as if they were monkeys, their faces gleaming. For clearly their thirst for gold was insatiable; they starved for it;
they lusted for it; they wanted to stuff themselves with it as if they were
pigs. So they went about fingering, taking up the streamers of gold,
moving t h e m back and forth, grabbing t h e m to themselves, babbling,
talking gibberish a m o n g themselves."*
5
On a cosmic scale, the s u d d e n interactions between civilizations
could be even m o r e dramatic. Because we are talking about astronomical
time scales, it is likely that a civilization that is a million years ahead of
us will find us totally uninteresting. F u r t h e r m o r e , t h e r e is probably little
that our planet can offer these aliens in terms of natural resources that
isn't simultaneously available in n u m e r o u s other star systems.
In the "Star T r e k " series, however, the Federation of Planets encounters other hostile civilizations, the Klingons and Romulans, which are
precisely at the same stage of technological development as the Federation. This may increase the d r a m a and tension of t h e series, b u t the odds
of this h a p p e n i n g are truly astronomical. More likely, as we venture off
into the galaxy in starships, we will e n c o u n t e r civilizations at vastly different levels of technological development, some perhaps millions of
years ahead of us.
The Rise and Fall of Civilizations
In addition to the possibilities that we may have missed other civilizations
by millions of years and that o t h e r civilizations may n o t consider us worthy of notice, a third theory, which is m o r e interesting, holds that
thousands of intelligent life forms did arise from the swamp, b u t they
were unable to negotiate a series of catastrophes, both natural a n d self-
*So perhaps we shouldn't be so enthusiastic about making contact with intelligent
extraterrestrials. Scientists point out that on the earth, there are two types of animals:
predators like cats, dogs, and tigers (which have eyes to the front of their face, so they can
stereoscopically zero in on their target) and prey like rabbits and deer (which have eyes
to the side of their face in order to look around 360 degrees for the predators). Typically,
predators are more intelligent then prey. Tests show that cats are more intelligent than
mice, and foxes are more intelligent than rabbits. Humans, with eyes to the front, are also
predators. In our search for intelligent life in the heavens, we should keep in mind that
the aliens we meet will probably also have evolved from predators.
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inflicted. If this theory is correct, t h e n p e r h a p s someday o u r starships
will find t h e ruins of ancient civilizations on far-off planets, or, m o r e
likely, o u r own civilization will be faced with these catastrophes. Instead
of b e c o m i n g "lords of the universe," we may follow the r o a d to selfdestruction. T h u s the question we ask is: What is the fate of advanced
civilizations? Will we (they) survive l o n g e n o u g h to master the physics
of the t e n t h dimension?
T h e rise of civilizations is not m a r k e d by a steady a n d sure growth in
technology a n d knowledge. History shows us that civilizations rise,
m a t u r e , a n d then disappear, sometimes without a trace. In the future,
p e r h a p s h u m a n i t y will unleash a P a n d o r a ' s box of technological h o r r o r s
that t h r e a t e n o u r very existence, from atomic b o m b s to carbon dioxide.
Far from t r u m p e t i n g the c o m i n g of t h e Age of Aquarius, some futurologists predict that we may be facing technological a n d ecological collapse. For the future, they conjure up the frightening image of humanity
r e d u c e d to a pathetic, terrified Scrooge in Charles Dickens's fable, groveling on the g r o u n d of his own grave a n d p l e a d i n g for a second c h a n c e .
Unfortunately, the bulk of humanity is largely uncaring, or unaware,
of the potential disasters facing us. Some scientists have argued that
p e r h a p s humanity, considered as a single entity, can be c o m p a r e d to a
teenager c a r e e n i n g o u t of control. For example, psychologists tell us that
teenagers act as if they are invulnerable. T h e i r driving, drinking, a n d
d r u g habits are graphic proof, they say, of the devil-may-care recklessness
that pervades their life-style a n d outlook. T h e main cause of d e a t h
a m o n g teenagers in this country is no longer disease, b u t accidents,
probably caused by the fact that they think they will live forever.
If that is true, t h e n we are abusing technology a n d the e n v i r o n m e n t
as if we will live forever, unaware of the catastrophes that lie in the future.
Society as a whole may have a " P e t e r Pan c o m p l e x , " never wanting to
grow up a n d face the consequences of its own irresponsibility.
To concretize o u r discussion, using the knowledge at o u r disposal,
we can identify several i m p o r t a n t hurdles that m u s t be crossed over during t h e next several aeons before we can b e c o m e masters of the t e n t h
dimension: the u r a n i u m barrier, ecological collapse, a new ice age, astronomical close e n c o u n t e r s , Nemesis a n d extinction, a n d t h e d e a t h of t h e
sun a n d t h e Milky Way galaxy.
The Uranium Barrier
J o n a t h a n Schell, in his watershed b o o k The Fate of the Earth, points o u t
how perilously close we have c o m e to m u t u a l annihilation. Although t h e
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r e c e n t collapse of the Soviet U n i o n has m a d e possible sweeping arms
cuts, there are still close to 50,000 nuclear weapons, b o t h tactical a n d
strategic, in the world today, a n d with deadly accurate rockets to deliver
t h e m . Humanity has finally mastered the possibility of total annihilation.
If the missiles do n o t destroy everyone in the o p e n i n g shots of a
nuclear war, we can still look forward to the agonizing death caused by
nuclear winter, d u r i n g which the soot and ash from b u r n i n g cities slowly
chokes off all the life-giving sunlight. C o m p u t e r studies have shown that
as few as 100 megatons of explosives may generate e n o u g h fire storms
in the cities to cloud the a t m o s p h e r e significantly. As temperatures
p l u m m e t , crops fail, a n d cities freeze over, the last vestiges of civilization
will be snuffed out like a candle.
Finally, there is the increasing d a n g e r of nuclear proliferation.
United States intelligence estimates that India, which d e t o n a t e d its first
b o m b in 1974, now has a stockpile of a b o u t 20 atomic bombs. Arche n e m y Pakistan, these sources claim, has built four atomic bombs, o n e
of which weighs no m o r e than 400 p o u n d s , at its secret Kahuta nuclear
facility. An atomic worker at Israel's D i m o n a nuclear installation in the
Negev desert claimed that he saw e n o u g h material to build 200 atomic
b o m b s there. And South Africa admitted that it had m a d e seven atomic
b o m b s a n d apparently tested two atomic b o m b s in the late 1970s off its
coast. T h e U.S. spy satellite Vela picked up the " f i n g e r p r i n t " of the
atomic b o m b , a characteristic, unmistakable double-flash, on two occasions off the coast of South Africa in the presence of Israeli warships.
Nations like North Korea, South Korea, a n d Taiwan are poised at the
brink of going nuclear. It's highly probable, given r e c e n t U.S. intelligence disclosures, that 20 nations will possess the b o m b by the year 2000.
T h e b o m b will have proliferated into the hottest spots a r o u n d the world,
including the Middle East.
This situation is highly unstable, and will continue to b e c o m e m o r e
so as nations c o m p e t e for diminishing resources a n d spheres of influence. Not j u s t o u r society, but every intelligent civilization in the galaxy
building an industrial society, will discover e l e m e n t 92 (uranium) a n d
with it the ability for mass destruction. E l e m e n t 92 has the curious property of sustaining a chain reaction and releasing the vast a m o u n t of
energy stored within its nucleus. With the ability to master e l e m e n t 92
comes the ability either to liberate our species from want, ignorance,
a n d h u n g e r , or to c o n s u m e the planet in nuclear fire. T h e power of
e l e m e n t 92, however, can be unleashed only when an intelligent species
reaches a certain point of development as a Type 0 civilization. It
d e p e n d s on the size of its cohesive social unit a n d its state of industrial
development.
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Fire, for example, can be harnessed by isolated groups of intelligent
individuals (such as a tribe). Smelting a n d primitive metallurgy, necessary for the manufacture of weapons, requires a larger social unit, perhaps n u m b e r i n g in the thousands (such as a small village). T h e develo p m e n t of the internal-combustion e n g i n e (for example, a car engine)
requires the development of a complex chemical a n d industrial base,
which can be accomplished by only a cohesive social unit n u m b e r i n g in
the millions (for example, a nation-state).
T h e discovery of e l e m e n t 92 upsets this balance between the slow,
steady rise of the cohesive social unit a n d its technological development.
T h e releasing of nuclear energy dwarfs chemical explosives by a factor
of a million, b u t the same nation-state that can harness the internalcombustion e n g i n e can also refine e l e m e n t 92. T h u s a severe mismatch
occurs, especially when the social development of this hypothetical civilization is still locked in the form of hostile nation-states. T h e technology for mayhem a n d destruction abruptly outpaces the slow developm e n t of social relations with the discovery of e l e m e n t 92.
It is natural to conclude, therefore, that Type 0 civilizations arose on
n u m e r o u s occasions within the past 5- to 10-billion-year history of o u r
galaxy, but that they all eventually discovered e l e m e n t 92. If a civilization's technological capability outraced its social development, then,
with the rise of hostile nation-states, there was a large c h a n c e that the
civilization destroyed itself long ago in an atomic war. Regrettably, if we
live long e n o u g h to reach nearby stars in o u r sector of the galaxy, we
may see the ashes of n u m e r o u s , dead civilizations that settled national
passions, personal jealousies, a n d racial hatreds with nuclear bombs.
6
As Heinz Pagels has said,
The challenge to our civilization which has come from our knowledge of
the cosmic energies that fuel the stars, the movement of light and electrons
through matter, the intricate molecular order which is the biological basis
of life, must be met by the creation of a moral and political order which
will accommodate these forces or we shall be destroyed. It will try our
deepest resources of reason and compassion.
7
It seems likely, therefore, that advanced civilizations sprang up on
n u m e r o u s occasions within o u r galaxy, but that few of t h e m negotiated
the u r a n i u m barrier, especially if their technology outpaced their social
development.
If we plot, for example, the rise of radio technology on a graph, we
see that o u r planet evolved for 5 billion years before an intelligent species discovered how to manipulate the electromagnetic a n d nuclear
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forces. However, if we annihilate ourselves in a nuclear war, then this
curve will b e c o m e a spike and return to zero. T h u s in o r d e r to communicate with an advanced civilization, we must scan at precisely the
right era, to an accuracy of a few decades, before the civilization blows
itself up. T h e r e is a vanishingly small "window" t h r o u g h which we may
make contact with a n o t h e r living civilization, before it destroys itself. In
Figure 13.1, we see the rise of alien civilizations t h r o u g h o u t the galaxy
represented as a series of peaks, each representing the rapid rise of a
civilization a n d the even m o r e rapid fall d u e to nuclear war. Scanning
the heavens for intelligent life, therefore, may be a difficult task. Perhaps
there have b e e n many thousands of peaks within the past few billion
years, with thousands of planets briefly mastering radio technology
before blowing themselves u p . Each brief peak, unfortunately, takes
place at different cosmic times.
Ecological Collapse
Assuming that a Type 0 civilization can master u r a n i u m without destroying itself in a nuclear war, the next barrier is the possibility of ecological
collapse.
We recall t h e earlier example of a single bacterium, which divides so
frequently that it eventually outweighs the planet earth. However, in
reality we do n o t see gigantic masses of bacteria on the earth—in fact,
bacterial colonies usually do n o t even grow to the size of a penny. Laboratory bacteria placed in a dish filled with nutrients will indeed grow
exponentially, but eventually die because they p r o d u c e too m u c h waste
a n d exhaust the food supply. These bacterial colonies essentially suffocate in their own waste products.
Like bacterial colonies, we may also be exhausting our resources
while drowning in the waste products that we relentlessly p r o d u c e . O u r
oceans a n d the atmosphere are n o t limitless, but ultrathin films on the
surface of the earth. T h e population of a Type 0 civilization, before it
reaches Type I status, may soar to the billions, creating a strain on
resources and exacerbating the problems of pollution. O n e of the most
immediate dangers is the poisoning of the atmosphere, in the form of
carbon dioxide, which traps sunlight and raises the average world temperature, possibly initiating a runaway greenhouse effect.
Since 1958, carbon dioxide concentrations in the air have increased
2 5 % , mostly from oil a n d coal b u r n i n g (45% of carbon dioxide comes
from the United States a n d the former Soviet U n i o n ) . This, in turn, may
have accelerated the m e a n t e m p e r a t u r e rise of the earth. It took almost
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Radio
telescope
technology
of l i f e
in t h e
galaxy
Billions of y e a r s
Figure 13.1. Why don't we see other intelligent life in the galaxy ? Perhaps intelligent life forms that could build radio telescopes flourished millions of years in
the past, but perished in a nuclear war. Our galaxy could have been teeming with
intelligent life, but perhaps most are dead now. Will our civilization be any different?
a century, from 1880, to raise the m e a n world t e m p e r a t u r e 1°F. However,
t h e m e a n t e m p e r a t u r e is now rising at almost 0.6°F p e r decade. By the
year 2050, this translates into a rise of coastal waters by 1 to 4 feet, which
could swamp nations like Bangladesh a n d flood areas like Los Angeles
and Manhattan. Even m o r e serious would be a devastation of t h e
nation's food basket in t h e Midwest, the acceleration of the spread of
deserts, a n d destruction of tropical rain forests, which in t u r n accelerates
the g r e e n h o u s e effect. Famine a n d economic ruin could spread on a
global scale.
T h e fault lies in an u n c o o r d i n a t e d planetary policy. Pollution takes
place in millions of individual factories all over t h e planet, b u t the power
to c u r b this u n b r i d l e d pollution resides with a planetary policy, which is
difficult, if not impossible, to enforce if the d o m i n a n t cohesive social
unit is the nation-state, n u m b e r i n g only in the h u n d r e d s of millions. In
the short term, this may m e a n emergency policies a n d t h e sharp curtailment of the internal-combustion engine a n d coal a n d oil b u r n i n g .
T h e standard of living could also d r o p . It means additional hardships in
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developing nations, which n e e d access to cheap sources of energy. In
the long term, however, our society may be forced to resort to o n e of
three possible solutions that do n o t give off carbon dioxide and are
essentially inexhaustible: solar energy, fusion plants, and b r e e d e r reactors. Of these, solar and fusion hold the most promise. Fusion power
(which fuses the hydrogen atoms found in sea water) a n d solar energy
are still several decades away, b u t should provide ample energy supplies
into the next few centuries, until society makes the transition to a Type
I civilization.
T h e fault once again lies in the fact that the technology has outpaced
social development. As long as pollution is p r o d u c e d by individual
nation-states, while the measures necessary to correct this are planetary,
there will be a fatal mismatch that invites disaster. T h e u r a n i u m barrier
and ecological collapse will exist as life-threatening disasters for Type 0
civilizations until this mismatch is bridged.
O n c e a civilization passes Type 0 status, however, there is m u c h m o r e
r o o m for optimism. To reach Type I status requires a remarkable degree
of social cooperation on a planetary scale. Aggregates on the o r d e r of
tens to h u n d r e d s of millions of individuals are necessary to exploit the
resources of u r a n i u m , internal combustion, and chemicals. However,
aggregates on the o r d e r of billions are probably necessary truly to harness planetary resources. T h u s the social organization of a Type I civilization must be very complex and very advanced, or else the technology
c a n n o t be developed.
By definition, a Type I civilization requires a cohesive social unit that
is the entire planet's population. A Type I civilization by its very n a t u r e
must be a planetary civilization. It c a n n o t function on a smaller scale.
This can, in some sense, be c o m p a r e d to childbirth. T h e most dangerous period for a child is the first few m o n t h s of life, when the transition to an external, potentially hostile environment places e n o r m o u s
biological strains on the baby. After the first year of life, the death rate
plunges dramatically. Similarly, the most dangerous period for a civilization is the first few centuries after it has reached nuclear capability. It
may turn out that once a civilization has achieved a planetary political
system, the worst is over.
A New Ice Age
No o n e knows what causes an ice age, which has a duration measured
in tens to h u n d r e d s of thousands of years. O n e theory is that it is caused
by m i n u t e variations in the earth's rotation, which are too small to be
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noticed even over a period of centuries. These tiny effects, over h u n dreds of thousands of years, apparently accumulate to cause slight
changes in the j e t stream over the poles. Eventually, the j e t streams are
diverted, sending freezing polar air masses farther a n d farther south,
causing temperatures to p l u m m e t a r o u n d the globe, until an ice age
begins. T h e ice ages did considerable d a m a g e to the ecology of the
earth, wiping o u t scores of m a m m a l i a n life forms a n d p e r h a p s even isolating b a n d s of h u m a n s on different continents, p e r h a p s even giving rise
to the various races, which is a relatively recent p h e n o m e n o n .
Unfortunately, o u r c o m p u t e r s a r e too primitive even to predict
tomorrow's weather, let alone when the next ice age will strike. For
example, c o m p u t e r s are now e n t e r i n g their fifth generation. We sometimes forget that no matter how large or complex a fourth-generation
c o m p u t e r is, it can only a d d two n u m b e r s at a time. This is an e n o r m o u s
bottleneck that is j u s t b e g i n n i n g to be solved with fifth-generation computers, which have parallel processors that can perform several operations simultaneously.
It is highly likely that o u r civilization (if it successfully negotiates t h e
u r a n i u m barrier a n d ecological collapse) will attain Type I status, a n d
with it the ability to control the weather, within a few h u n d r e d years. If
humanity reaches Type I status or h i g h e r before the n e x t ice age occurs,
then t h e r e is a m p l e reason to believe that an ice age will not destroy
humanity. H u m a n s either will c h a n g e the weather a n d prevent the ice
age or will leave the earth.
Astronomical Close Encounters
On a time scale of several t h o u s a n d to several million years, Types 0 a n d
I civilizations have to worry about asteroid collisions a n d nearby supernovas.
Only within this century, with refined astronomical m e a s u r e m e n t s ,
has it b e c o m e a p p a r e n t that the earth's orbit cuts across the orbits of
many asteroids, making t h e possibility of n e a r misses uncomfortably
large. ( O n e way for a Type 0 or I civilization to prevent a direct collision
is to send rockets with hydrogen b o m b s to intercept a n d deflect t h e
asteroid while it is still tens of millions of miles away from t h e earth. This
m e t h o d has, in fact, b e e n p r o p o s e d by international bodies of scientists.)
These n e a r misses are m o r e frequent than most p e o p l e realize. T h e
last o n e took place on J a n u a r y 3, 1993, a n d was actually p h o t o g r a p h e d
using r a d a r by NASA astronomers. Photos of t h e asteroid Toutatis show
that it consists of two rocky cores, each 2 miles in diameter. It c a m e
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within 2.2 million miles of the planet earth. On March 23, 1989, an
asteroid a b o u t half a mile across drifted even closer to the earth, about
0.7 million miles (roughly three times the distance from the earth to the
moon).
In fact, it was also a n n o u n c e d in late 1992 that a gigantic comet would
hit the earth on exactly August 14, 2126, perhaps e n d i n g all life on the
planet. Astronomer Brian Marsden of the Harvard-Smithsonian Center
for Astrophysics estimated the chances of a direct hit as 1 in 10,000. T h e
Swift-Tuttle comet ( n a m e d after the two American astronomers who
first spotted it d u r i n g the Civil War) was soon d u b b e d the Doomsday
Rock by the media. Soon-to-be-unemployed nuclear weapons physicists
argued, p e r h a p s in a self-serving way, that they should be allowed to
build massive hydrogen bombs to blow it to smithereens when the time
comes.
Bits a n d pieces of the Swift-Tuttle comet have already impacted on
the earth. Making a complete revolution a r o u n d the sun every 130 years,
it sheds a considerable a m o u n t of debris, creating a river of meteors and
particles in outer space. W h e n the earth crosses this river, we have the
annual Perseid meteor shower, which rarely fails to light up the sky with
celestial fireworks. (We should also point out that predicting n e a r misses
of comets is a risky business. Because the heat of the sun's radiation
causes the comet's icy surface to vaporize irregularly and sputter like
thousands of small firecrackers, there are slight b u t i m p o r t a n t distortions in its trajectory. Not surprisingly, Marsden retracted his prediction
a few weeks later as being incorrect. " W e ' r e safe for the next millenn i u m , " admitted Marsden.)
A NASA panel in January 1991 estimated that there are about 1,000
to 4,000 asteroids that cross the earth's orbit and are bigger than a halfmile across, sufficient to pose a threat to h u m a n civilization. However,
only about 150 of these large asteroids have b e e n adequately tracked by
radar. F u r t h e r m o r e , there are estimated to be about 300,000 asteroids
that cross the earth's orbit that are at least 300 feet across. Unfortunately,
scientists hardly know the orbits of any of these smaller asteroids.
My own personal close e n c o u n t e r with an extraterrestrial object came
when I was a senior at Harvard in the winter of 1967. A close friend of
m i n e in my dormitory, who had a part-time j o b at the university observatory, told me a closely held secret: T h e astronomers there h a d detected
a gigantic asteroid, several miles across, h e a d i n g directly for the planet
earth. F u r t h e r m o r e , although it was too early to tell, he informed me
that their computers calculated it might strike the earth in J u n e 1968,
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t h e time of o u r graduation. An object that size would crack the e a r t h ' s
crust, spew o p e n billions of tons of m o l t e n m a g m a , a n d send h u g e earthquakes a n d tidal waves a r o u n d the world. As the m o n t h s went by, I would
get periodic updates on t h e course of the Doomsday asteroid. T h e
astronomers at t h e observatory were obviously b e i n g careful not to cause
any u n d u e panic with this information.
Twenty years later, I h a d forgotten all a b o u t the asteroid, until I was
browsing t h r o u g h an article on asteroid n e a r misses. Sure e n o u g h , t h e
article m a d e reference to t h e asteroid of 1968. Apparently, the asteroid
came within about 1 million miles of a direct impact with the earth.
More rare, b u t m o r e spectacular than asteroid collisions are supernova bursts in t h e vicinity of the earth. A supernova releases e n o r m o u s
quantities of energy, greater than the o u t p u t of h u n d r e d s of billions of
stars, until eventually it outshines the entire galaxy itself. It creates a
burst of x-rays, which would be sufficient to cause severe disturbances in
any nearby star system. At t h e very m i n i m u m , a nearby supernova would
create a gigantic EMP (electromagnetic pulse), similar to the o n e that
would be unleashed by a hydrogen b o m b d e t o n a t e d in o u t e r space. T h e
x-ray burst would eventually hit o u r a t m o s p h e r e , smashing electrons o u t
of atoms; t h e electrons would t h e n spiral t h r o u g h the earth's magnetic
field, creating e n o r m o u s electric fields. These fields are sufficient to
black out all electrical a n d c o m m u n i c a t i o n devices for h u n d r e d s of
miles, creating confusion a n d panic. In a large-scale nuclear war, the
EMP would be sufficient to wipe out or d a m a g e any form of electronics
over a wide area of the earth's population. At worst, in fact, a supernova
burst in the vicinity of a star system might be sufficient to destroy all life.
A s t r o n o m e r Carl Sagan speculates that such an event may have wiped
o u t the dinosaurs:
I f t h e r e w e r e b y c h a n c e a s u p e r n o v a w i t h i n t e n o r t w e n t y light-years o f t h e
s o l a r s y s t e m s o m e sixty-five m i l l i o n y e a r s a g o , i t w o u l d h a v e s p r a y e d a n
i n t e n s e f l u x o f c o s m i c rays i n t o s p a c e , a n d s o m e o f t h e s e , e n t e r i n g t h e
E a r t h ' s e n v e l o p e o f air, w o u l d h a v e b u r n e d t h e a t m o s p h e r i c n i t r o g e n . T h e
o x i d e s o f n i t r o g e n t h u s g e n e r a t e d w o u l d h a v e r e m o v e d t h e p r o t e c t i v e layer
o f o z o n e f r o m t h e a t m o s p h e r e , i n c r e a s i n g t h e f l u x o f s o l a r u l t r a v i o l e t radia t i o n a t t h e s u r f a c e a n d frying a n d m u t a t i n g t h e m a n y o r g a n i s m s i m p e r fectly p r o t e c t e d a g a i n s t i n t e n s e u l t r a v i o l e t l i g h t .
Unfortunately, the supernova would give little warning of its explosion. A supernova e r u p t i o n takes place quite rapidly, a n d its radiation
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travels at the speed of light, so a Type I civilization would have to make
a speedy escape into outer space. T h e only precaution that a civilization
can take is to m o n i t o r carefully those nearby stars that are on the verge
of going supernova.
The Nemesis Extinction Factor
In 1980, the late Luis Alvarez, his son Walter, a n d Frank Asaro a n d Helen
Michel of the University of California at Berkeley proposed that a comet
or an asteroid hit the earth 65 million years ago, thereby initiating vast
atmospheric disturbances that led to the s u d d e n extinction of the dinosaurs. By examining the rocky strata laid down by river beds 65 million
years ago, they were able to d e t e r m i n e the presence of unusually high
a m o u n t s of iridium, which is rarely found on earth b u t commonly found
in extraterrestrial objects, like meteors. T h e theory is quite plausible,
since a comet 5 miles in diameter hitting the earth at about 20 miles p e r
second (ten times faster than a speeding bullet) would have the force
of 100 million megatons of T N T (or 10,000 times the world's total
nuclear arsenal). It would create a crater 60 miles across a n d 20 miles
d e e p , sending up e n o u g h debris to cut off all sunlight for an extended
period of time. As temperatures fall dramatically, the vast majority of
the species on this planet would be either killed off or seriously
depleted.
In fact, it was a n n o u n c e d in 1992 that a strong candidate for the
dinosaur-killing comet or asteroid had b e e n identified. It was already
known that there is a large impact crater, measuring 110 miles across,
in Mexico, in the Yucatan, near the village of Chicxulub Puerto. In 1981,
geophysicists with the Mexican national p e t r o l e u m company, Pemex,
told geologists that they h a d picked up gravitational a n d magnetic anomalies that were circular in shape at the site. However, only after Alvarez's
theory became popular did geologists actively analyze the r e m n a n t s of
that cataclysmic impact. Radioactive-dating m e t h o d s using argon-39
have shown that the Yucatan crater is 64.98 ± 0.05 million years old.
More impressively, it was shown that Mexico, Haiti, a n d even Florida are
littered with small, glassy debris called tektites, which were probably silicates that were glassified by the impact of this large asteroid or comet.
These glassy tektites can be found in sediment that was laid
down between the Tertiary a n d Cretaceous periods. Analyses of five
different tektite samples show an average age of 65.07 ± 0.10 million
years. Given the accuracy of these i n d e p e n d e n t measurements,
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geologists now have the " s m o k i n g g u n " for the dinosaur-killing
asteroid or comet.
But o n e of t h e astonishing features of life on earth is that the extinction of the dinosaurs is but o n e of several well-documented mass extinctions. O t h e r mass extinctions were m u c h worse than t h e o n e that e n d e d
the Cretaceous period 65 million years ago. T h e mass extinction that
e n d e d the Permian period, for example, destroyed fully 9 6 % of all plant
a n d animal species 250 million years ago. T h e trilobites, which ruled the
oceans as o n e of earth's d o m i n a n t life forms, mysteriously a n d abruptly
perished d u r i n g this great mass extinction. In fact, there have b e e n five
mass extinctions of animal a n d plant life. If o n e includes mass extinctions that are less well d o c u m e n t e d , a pattern becomes evident: Every
26 million years or so, t h e r e is a mass extinction. Paleontologists David
R a u p a n d J o h n Sepkoski have shown that if we plot the n u m b e r of known
species on the earth at any given time, then the chart shows a sharp d r o p
in t h e n u m b e r of life forms on the earth every 26 million years, like
clockwork. This can be shown to extend over ten cycles going back 260
million years (excluding two cycles).
In o n e extinction cycle, at the e n d of the Cretaceous period, 65 million years ago, most of the dinosaurs were killed off. In a n o t h e r extinction cycle, at the e n d of the Eocene period, 35 million years ago, many
species of land m a m m a l s were extinguished. But the central puzzle to
this is: What in heaven's n a m e has a cycle time of 26 million years? A
search t h r o u g h biological, geological, or even astronomical data suggests
that n o t h i n g has a cycle time of 26 million years.
Richard Muller of Berkeley has theorized that our sun is actually part
of a double-star system, a n d that o u r sister star (called Nemesis or the
Death Star) is responsible for periodic extinctions of life on the earth.
T h e conjecture is that our sun has a massive unseen p a r t n e r that circles
it every 26 million years. As it passes t h r o u g h t h e O o r t cloud (a cloud
of comets that supposedly exists beyond the orbit of Pluto), it brings
with it an unwelcome avalanche of comets, some of which strike t h e
earth, causing e n o u g h debris that the sunlight is blocked from reaching
the earth's surface.
Experimental evidence for this unusual theory comes from the fact
that t h e geological layers from the past, c o r r e s p o n d i n g to the e n d of
each extinction cycle, contain unusually large quantities of the e l e m e n t
iridium. Since iridium is naturally found in extraterrestrial meteors, it is
possible that these traces of iridium are r e m n a n t s of the comets sent
down by Nemesis. At present, we are half-way between extinction cycles,
m e a n i n g that Nemesis, if it exists, is at its farthest p o i n t in its orbit (prob-
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ably several light-years away). This would give us over 10 million years
or so until its next arrival.*
Fortunately, by the time comets from the O o r t cloud streak t h r o u g h
the solar system again, we will have reached Type III status, m e a n i n g
that we will have c o n q u e r e d n o t just the nearby stars, b u t travel through
space-time.
The Death of the Sun
Scientists sometimes wonder what will eventually h a p p e n to the atoms
of o u r bodies long after we are dead. T h e most likely possibility is that
o u r molecules will eventually r e t u r n to the sun.
O u r sun is a middle-aged star. It is approximately 5 billion years old,
and will probably remain a yellow star for a n o t h e r 5 billion years. W h e n
our sun exhausts its supply of hydrogen fuel, however, it will b u r n helium
a n d b e c o m e vastly inflated—a red giant. Its atmosphere will expand rapidly, eventually extending out to the orbit of Mars, a n d the earth's orbit
will be entirely within the sun's atmosphere, so that the earth will be
fried by the sun's e n o r m o u s temperatures. T h e molecules making up
o u r bodies, a n d in fact the earth itself, will be consumed by the solar
atmosphere.
Sagan paints the following picture:
B i l l i o n s of years f r o m n o w , t h e r e will be a last p e r f e c t day on E a r t h . . . .
T h e Arctic a n d A n t a r c t i c i c e c a p s will m e l t , f l o o d i n g t h e c o a s t s o f t h e w o r l d .
T h e h i g h o c e a n i c t e m p e r a t u r e s will r e l e a s e m o r e water v a p o r i n t o t h e air,
i n c r e a s i n g c l o u d i n e s s , s h i e l d i n g t h e Earth f r o m s u n l i g h t a n d d e l a y i n g t h e
e n d a little. B u t s o l a r e v o l u t i o n i s i n e x o r a b l e . E v e n t u a l l y t h e o c e a n s will
b o i l , t h e a t m o s p h e r e will e v a p o r a t e away t o s p a c e a n d a c a t a s t r o p h e o f t h e
m o s t i m m e n s e p r o p o r t i o n s i m a g i n a b l e will o v e r t a k e o u r p l a n e t .
8
Thus, for those who wish to know whether the earth will be c o n s u m e d
in ice or fire, physics actually gives a definite answer. It will be consumed
in fire. However, it is highly likely that h u m a n s , if we have survived that
* A n o t h e r t h e o r y t h a t m i g h t e x p l a i n p e r i o d i c e x t i n c t i o n s o n t h i s vast t i m e scale i s t h e
o r b i t o f o u r s o l a r system a r o u n d t h e M i l k y W a y galaxy. T h e solar System a c t u a l l y d i p s b e l o w
a n d a b o v e t h e g a l a c t i c p l a n e i n its o r b i t a r o u n d t h e g a l a x y , m u c h l i k e c a r o u s e l h o r s e s m o v e
up a n d d o w n as a m e r r y - g o - r o u n d turns. As it dips periodically t h r o u g h the galactic plane,
t h e s o l a r system m a y e n c o u n t e r l a r g e q u a n t i t i e s o f d u s t t h a t d i s t u r b t h e O o r t c l o u d , b r i n g i n g d o w n a hail of comets.
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long, will have long d e p a r t e d from t h e solar system. Unlike a supernova,
there is ample warning of the demise of o u r sun.
The Death of the Galaxy
On a time scale of several billions of years, we must confront the fact
that the Milky Way galaxy in which we live, will die. More precisely, we
live on the O r i o n spiral a r m of the Milky Way. W h e n we gaze at t h e night
sky a n d feel dwarfed by t h e immensity of the celestial lights dotting the
heavens, we are actually looking at a tiny p o r t i o n of t h e stars located on
the O r i o n arm. T h e millions of stars that have inspired b o t h lovers a n d
poets for generations occupy only a tiny part of the O r i o n arm. T h e rest
of t h e 200 billion stars within the Milky Way a r e so distant that they can
barely be seen as a hazy ribbon that cuts across the night sky.
About 2 million light-years from t h e Milky Way is o u r nearest galactic
neighbor, the great A n d r o m e d a galaxy, which is two to t h r e e times larger
than o u r own galaxy. T h e two galaxies are hurtling toward each o t h e r
at 125 kilometers p e r second, a n d should collide within 5 to 10 billion
years. As a s t r o n o m e r Lars Hernquist at the University of California at
Santa Cruz has said, this collision will be " a n a l o g o u s to a hostile takeover. O u r galaxy will be c o n s u m e d a n d d e s t r o y e d . "
9
As seen from o u t e r space, the A n d r o m e d a galaxy will a p p e a r to collide with a n d t h e n slowly absorb the Milky Way galaxy. C o m p u t e r simulations of colliding galaxies show that the gravitational pull of the larger
galaxy will slowly overwhelm the gravity of the smaller galaxy, a n d after
several rotations the smaller galaxy will be eaten u p . But because the
stars within the Milky Way galaxy are so widely separated by the vacuum
of space, the n u m b e r of collisions between stars will be quite low, on the
o r d e r of several collisions p e r century. So o u r sun may avoid a direct
collision for an e x t e n d e d period of time.
Ultimately, on this time scale of billions of years, we have a m u c h
m o r e deadly fate, t h e d e a t h of the universe itself. Clever forms of intelligent life may find ways to build space arks to avoid most natural catastrophes, b u t how can we avoid the d e a t h of t h e universe, when space
itself is o u r worst enemy?
T h e Aztecs believed that the e n d of the world would c o m e w h e n the
sun o n e day falls from the sky. They foretold that this would c o m e ' ' w h e n
the Earth has b e c o m e tired . . . , w h e n the seed of Earth has e n d e d . "
T h e stars would be shaken from the heavens.
P e r h a p s they were close to the truth.
O n e can h o p e that by t h e time o u r sun begins to flicker out, h u m a n -
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ity will have long since left the solar system and r e a c h e d for the stars.
(In fact, in Asimov's Foundation series, the location of our original star
system has b e e n lost for thousands of years.) However, inevitably, all the
stars in the heavens will flicker o u t as their nuclear fuel is exhausted.
On a scale of tens to h u n d r e d s of billions of years, we are facing the
d e a t h of the universe itself. Either the universe is o p e n , in which case it
will e x p a n d forever until temperatures gradually reach n e a r absolute
zero, or the universe is closed, in which case the expansion will be
reversed and the universe will die in a fiery Big Crunch. Even for a Type
III civilization, this is a d a u n t i n g threat to its existence. Can mastery of
hyperspace save civilization from its ultimate catastrophe, the death of
the universe?
14
The Fate of the Universe
S o m e say t h e w o r l d will e n d i n fire.
S o m e say in ice.
F r o m w h a t I've t a s t e d o f d e s i r e
I h o l d w i t h t h o s e w h o favor fire.
Robert Frost
It a i n ' t o v e r 'til it's o v e r .
Yogi Berra
W
H E T H E R a civilization, either on e a r t h or in o u t e r space, can
reach a p o i n t in its technological d e v e l o p m e n t to harness the
power of hyperspace d e p e n d s partly, as we have seen, on negotiating a
series of disasters typical of Type 0 civilizations. T h e d a n g e r period is
the first several h u n d r e d years after t h e dawn of the nuclear age, when
a civilization's technological d e v e l o p m e n t has far o u t p a c e d its social a n d
political maturity in h a n d l i n g regional conflicts.
By t h e time a civilization has attained Type III status, it will have
achieved a planetary social structure advanced e n o u g h to avoid self-annihilation a n d a technology powerful e n o u g h to avoid an ecological or a
natural disaster, such as an ice age or solar collapse. However, even a
Type III civilization will have difficulty avoiding t h e ultimate catastrophe:
the d e a t h of the universe itself. Even the mightiest a n d most sophisticated of t h e Type III civilization's starships will be u n a b l e to escape the
final destiny of the universe.
T h a t t h e universe itself must die was known to nineteenth-century
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scientists. Charles Darwin, in his Autobiography, wrote of his anguish when
he realized this profound but depressing fact: "Believing as I do that
m a n in the distant future will be a far m o r e perfect creature than he
now is, it is an intolerable t h o u g h t that he and all other sentient beings
are d o o m e d to complete annihilation after such long-continued slow
progress."
1
T h e mathematician and philosopher Bertrand Russell wrote that the
ultimate extinction of humanity is a cause of "unyielding despair." In
what must be o n e of the most depressing passages ever written by a
scientist, Russell noted:
That m a n is the product of causes which h a d no prevision of the e n d they
w e r e a c h i e v i n g ; that h i s o r i g i n , h i s g r o w t h , h i s h o p e s a n d fears, h i s l o v e s
a n d h i s b e l i e f s , are b u t t h e o u t c o m e o f a c c i d e n t a l c o l l o c a t i o n s o f a t o m s ;
that n o fire, n o h e r o i s m , n o i n t e n s i t y o f t h o u g h t o r f e e l i n g , c a n p r e s e r v e
a life b e y o n d t h e grave; that all t h e l a b o r s of t h e a g e s , all t h e d e v o t i o n , all
t h e i n s p i r a t i o n , all t h e n o o n d a y b r i g h t n e s s o f h u m a n g e n i u s , are d e s t i n e d
t o e x t i n c t i o n i n t h e vast d e a t h o f t h e s o l a r system; a n d t h e w h o l e t e m p l e
o f M a n ' s a c h i e v e m e n t m u s t inevitably b e b u r i e d b e n e a t h t h e d e b r i s o f a
u n i v e r s e i n r u i n s — a l l t h e s e t h i n g s , i f n o t q u i t e b e y o n d d i s p u t e , are yet s o
n e a r l y c e r t a i n , t h a t n o p h i l o s o p h y w h i c h rejects t h e m c a n h o p e t o s t a n d .
O n l y w i t h i n t h e s c a f f o l d i n g o f t h e s e t r u t h s , o n l y o n t h e firm f o u n d a t i o n
o f u n y i e l d i n g d e s p a i r , c a n t h e s o u l ' s h a b i t a t i o n b e safely b u i l t .
2
Russell wrote this passage in 1923, decades before the advent of space
travel. T h e death of the solar system loomed large in his mind, a rigorous
conclusion of the laws of physics. Within the confines of the limited
technology of his time, this depressing conclusion seemed inescapable.
Since that time, we have learned e n o u g h about stellar evolution to know
that our sun will eventually b e c o m e a red giant a n d consume the earth
in nuclear fire. However, we also u n d e r s t a n d the basics of space travel.
In Russell's time, the very t h o u g h t of large ships capable of placing
h u m a n s on the m o o n or the planets was universally considered to be
the thinking of a m a d m a n . However, with the exponential growth of
technology, the prospect of the death of the solar system is not such a
fearsome event for humanity, as we have seen. By the time o u r sun turns
into a red giant, humanity either will have long perished into nuclear
dust or, hopefully, will have found its rightful place a m o n g the stars.
Still, it is a simple matter to generalize Russell's "unyielding despair"
from the death of our solar system to the death of the entire universe.
In that event, it appears that no space ark can transport humanity out
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of h a r m ' s way. T h e conclusion seems irrefutable; physics predicts that
all intelligent life forms, no matter how advanced, will eventually perish
w h e n the universe itself dies.
According to Einstein's general theory of relativity, the universe
either will c o n t i n u e to e x p a n d forever in a Cosmic W h i m p e r , in which
case the universe reaches n e a r absolute zero t e m p e r a t u r e s , or will contract into a fiery collapse, the Big C r u n c h . T h e universe will die either
in " i c e , " with an o p e n universe, or in " f i r e , " with a closed universe.
Either way, a Type III civilization is d o o m e d because t e m p e r a t u r e s will
a p p r o a c h either absolute zero or infinity.
To tell which fate awaits us, cosmologists use Einstein's equations to
calculate t h e total a m o u n t of m a t t e r - e n e r g y in t h e universe. Because the
matter in Einstein's e q u a t i o n d e t e r m i n e s the a m o u n t of s p a c e - t i m e curvature, we must know the average matter density of t h e universe in o r d e r
to d e t e r m i n e if t h e r e is e n o u g h m a t t e r a n d energy for gravitation to
reverse t h e cosmic expansion of the original Big Bang.
A critical value for t h e average matter density d e t e r m i n e s the ultimate fate of the universe a n d all intelligent life within it. If the average
density of the universe is less than 1 0
g r a m p e r cubic centimeter,
which a m o u n t s to 10 milligrams of matter spread over the volume of the
earth, t h e n the universe will c o n t i n u e to e x p a n d forever, until it b e c o m e s
a uniformly cold, lifeless space. However, if the average density is larger
than this value, t h e n there is e n o u g h matter for the gravitational force
of t h e universe to reverse the Big Bang, a n d suffer the fiery t e m p e r a t u r e s
of the Big C r u n c h .
- 2 9
At present, t h e e x p e r i m e n t a l situation is confused. Astronomers have
several ways of measuring the mass of a galaxy, and h e n c e the mass of
the universe. T h e first is to c o u n t t h e n u m b e r of stars in a galaxy, a n d
multiply that n u m b e r by t h e average weight of each star. Calculations
p e r f o r m e d in this tedious fashion show that t h e average density is less
than the critical a m o u n t , a n d that t h e universe will c o n t i n u e to e x p a n d
forever. T h e p r o b l e m with this calculation is that it omits matter that is
n o t l u m i n o u s (for example, dust clouds, black holes, cold dwarf stars).
T h e r e is also a second way to perform this calculation, which is to
use Newton's laws. By calculating t h e time it takes for stars to move
a r o u n d a galaxy, astronomers can use Newton's laws to estimate t h e total
mass of t h e galaxy, in t h e same way that Newton used t h e time it took
for the m o o n to orbit the earth to estimate t h e mass of the m o o n a n d
earth.
T h e p r o b l e m is t h e mismatch between these two calculations. In fact,
astronomers know that up to 9 0 % of the mass of a galaxy is in t h e form
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of h i d d e n , undetectable "missing mass" or " d a r k m a t t e r , " which is n o t
luminous b u t has weight. Even if we include an approximate value for
the mass of n o n l u m i n o u s interstellar gas, Newton's laws predict that the
galaxy is far heavier than the value calculated by c o u n t i n g stars.
Until astronomers resolve the question of this missing mass or dark
matter, we c a n n o t resolve the question of whether the universe will contract and collapse into a fiery ball or will expand forever.
Entropy Death
Assume, for the m o m e n t , that the average density of the universe is less
than the critical value. Since the m a t t e r - e n e r g y content determines the
curvature of space-time, we find that there is n o t e n o u g h m a t t e r - e n e r g y
to make the universe recollapse. It will then e x p a n d limitlessly until its
t e m p e r a t u r e reaches almost absolute zero. This increases entropy (which
measures the total a m o u n t of chaos or r a n d o m n e s s in the universe).
Eventually, the universe dies in an entropy death.
T h e English physicist and astronomer Sir J a m e s J e a n s wrote about
the ultimate death of the universe, which he called the " h e a t d e a t h , " as
early as the turn of the century: " T h e second law of thermodynamics
predicts that there can be but o n e e n d to the universe—a 'heat death'
in which [the] t e m p e r a t u r e is so low as to make life impossible."
3
To u n d e r s t a n d how entropy d e a t h occurs, it is i m p o r t a n t to understand the three laws of thermodynamics, which govern all chemical a n d
nuclear processes on the earth a n d in the stars. T h e British scientist a n d
a u t h o r C. P. Snow had an elegant way of r e m e m b e r i n g the three laws:
1. You cannot win (that is, you c a n n o t get something for nothing,
because matter a n d energy are conserved).
2. You cannot break even (you c a n n o t return to the same energy
state, because there is always an increase in disorder; entropy
always increases).
3. You cannot get out of the game (because absolute zero is unattainable).
For the death of the universe, the most i m p o r t a n t is the Second Law,
which states that any process creates a n e t increase in the a m o u n t of
disorder (entropy) in the universe. T h e Second Law is actually an integral part of our everyday lives. For example, consider p o u r i n g cream
into a c u p of coffee. O r d e r (separate cups of cream a n d coffee) has
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naturally c h a n g e d into disorder (a r a n d o m mixture of cream a n d coffee). However, reversing entropy, extracting o r d e r from disorder, is
exceedingly difficult. " U n m i x i n g " t h e liquid back into separate cups of
c r e a m a n d coffee is impossible without an elaborate chemistry laboratory. Also, a lighted cigarette can fill an empty r o o m with wisps of smoke,
increasing entropy in that r o o m . O r d e r (tobacco a n d p a p e r ) has again
t u r n e d into disorder (smoke a n d charcoal). Reversing e n t r o p y — t h a t is,
forcing the smoke back into the cigarette a n d t u r n i n g t h e charcoal back
into u n b u r n e d tobacco—is impossible even with t h e finest chemistry
laboratory on t h e planet.
Similarly, everyone knows that it's easier to destroy t h a n to build. It
may take a year to construct a house, but only an h o u r or so to destroy
it in a fire. It took almost 5,000 years to transform roving b a n d s of hunters into t h e great Aztec civilization, which flourished over Mexico a n d
Central America a n d built towering m o n u m e n t s to its gods. However, it
only took a few m o n t h s for Cortez a n d the conquistadors to demolish
that civilization.
Entropy is relentlessly increasing in the stars as well as on o u r planet.
Eventually, this m e a n s that the stars will exhaust their nuclear fuel a n d
die, t u r n i n g into d e a d masses of nuclear matter. T h e universe will
d a r k e n as t h e stars, o n e by o n e , cease to twinkle.
Given o u r u n d e r s t a n d i n g of stellar evolution, we can paint a r a t h e r
dismal picture of how the universe will die. All stars will b e c o m e black
holes, n e u t r o n stars, or cold dwarf stars ( d e p e n d i n g on their mass)
within 1 0 years as their nuclear furnaces shut down. Entropy increases
as stars slide down the curve of b i n d i n g energy, until no m o r e energy
can be extracted by fusing their nuclear fuel. Within 1 0 years, all p r o tons a n d n e u t r o n s in the universe will probably decay. According to t h e
GUTs, the p r o t o n s a n d n e u t r o n s are unstable over that vast time scale.
This m e a n s that eventually all matter as we know it, including the earth
and t h e solar system, will dissolve into smaller particles, such as electrons
a n d neutrinos. T h u s intelligent beings will have to face the unpleasant
possibility that t h e protons a n d n e u t r o n s in their bodies will disintegrate.
T h e bodies of intelligent organisms will no longer be m a d e of t h e familiar 100 chemical elements, which are unstable over that i m m e n s e period
of time. Intelligent life will have to find ways of creating new bodies m a d e
of energy, electrons, a n d neutrinos.
24
32
100
After a fantastic 10
(a googol) years, the universe's t e m p e r a t u r e
will reach n e a r absolute zero. Intelligent life in this dismal future will
face t h e p r o s p e c t of extinction. U n a b l e to h u d d l e n e x t to stars, they will
freeze to d e a t h . But even in a desolate, cold universe at t e m p e r a t u r e s
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n e a r absolute zero, there is o n e last remaining flickering source of
energy: black holes. According to cosmologist Stephen Hawking, black
holes are n o t completely black, b u t slowly leak energy into outer space
over an e x t e n d e d period of time.
In this distant future, black holes may b e c o m e "life preservers"
because they slowly evaporate energy. Intelligent life would necessarily
congregate n e x t to these black holes and extract energy from t h e m to
k e e p their machines functioning. Intelligent civilizations, like shivering
homeless people h u d d l e d next to a fading fire, would be r e d u c e d to
pathetic outposts of misery clinging to a black h o l e .
4
1 0 0
But what, we may ask, h a p p e n s after 1 0 years, w h e n the evaporating
black holes will have exhausted most of their own energy? Astronomers
J o h n D. Barrow of the University of Sussex and J o s e p h Silk of the University of California at Berkeley caution that this question may ultimately
have no answer with present-day knowledge. On that time scale, quant u m theory, for example, leaves o p e n the possibility that o u r universe
may " t u n n e l " into a n o t h e r universe.
T h e probabilities for these kinds of events are exceedingly small; o n e
would have to wait a time interval larger than the lifetime of o u r present
universe, so we n e e d n o t worry that reality will suddenly collapse in our
lifetime, bringing with it a new set of physical laws. However, on the scale
of 10 years, these kinds of rare cosmic q u a n t u m events can no longer
be ruled out.
l00
Barrow a n d Silk add, " W h e r e there is q u a n t u m theory there is h o p e .
We can never be completely sure this cosmic h e a t death will occur
because we can never predict the future of a q u a n t u m mechanical universe with complete certainty; for in an infinite q u a n t u m future anything
that can h a p p e n , eventually will."
5
Escape Through a Higher Dimension
T h e Cosmic W h i m p e r is i n d e e d a dismal fate awaiting us if the average
density of the universe is too low. Now assume that the average density
is larger than the critical value. This means that t h e expansion process
will contract within tens of billions of years, a n d the universe will e n d in
fire, n o t ice.
In this scenario, there is e n o u g h matter and h e n c e a strong e n o u g h
gravitational pull in the universe to halt the expansion, a n d then the
universe will begin to slowly recollapse, bringing the distant galaxies
together again. Starlight will b e c o m e " b l u e shifted," instead of red
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shifted, indicating that the stars are rapidly a p p r o a c h i n g o n e a n o t h e r .
T h e t e m p e r a t u r e s o n c e again will rise to astronomical limits. Eventually,
the h e a t will b e c o m e sufficiently great to vaporize all m a t t e r into a gas.
Intelligent beings will find that their planets' oceans have boiled away
a n d that their a t m o s p h e r e s have t u r n e d into a searing furnace. As their
planets begin to disintegrate, they will be forced to flee into o u t e r space
in giant rockets.
Even t h e sanctuary of o u t e r space may prove to be inhospitable, however. T e m p e r a t u r e s will eventually rise past the p o i n t where atoms are
stable, a n d electrons will be ripped off their nuclei, creating a plasma
(like that found in o u r s u n ) . At this point, intelligent life may have to
build gigantic shields a r o u n d their ships a n d use their entire energy
o u t p u t to k e e p their shields from disintegrating from the intense heat.
As t e m p e r a t u r e s c o n t i n u e to rise, the p r o t o n s a n d n e u t r o n s in t h e
nucleus will be ripped apart. Eventually, the p r o t o n s a n d n e u t r o n s themselves will be torn apart into quarks. As in a black hole, t h e Big C r u n c h
devours everything. N o t h i n g survives it. T h u s it seems impossible that
ordinary matter, let alone intelligent life, can survive t h e violent disruption.
However, t h e r e is o n e possible escape. If all of s p a c e - t i m e is collapsing into a fiery cataclysm, t h e n the only way to escape the Big C r u n c h is
to leave space a n d time—escape via hyperspace. This may n o t be as farfetched as it sounds. C o m p u t e r calculations p e r f o r m e d with KaluzaKlein a n d superstring theories have shown that m o m e n t s after Creation,
the four-dimensional universe e x p a n d e d at t h e expense of t h e sixdimensional universe. T h u s the ultimate fate of the four- and the sixdimensional universes are linked.
Assuming that this basic picture is correct, o u r six-dimensional twin
universe may gradually expand, as o u r own four-dimensional universe
collapses. M o m e n t s before o u r universe shrinks to n o t h i n g , intelligent
life may realize that t h e six-dimensional universe is o p e n i n g u p , a n d find
a m e a n s to exploit that fact.
Interdimensional travel is impossible today because o u r sister universe has s h r u n k down to t h e Planck scale. However, in the final stages
of a collapse, the sister universe may o p e n u p , making dimensional travel
possible once again. If the sister universe e x p a n d s e n o u g h , then m a t t e r
a n d energy may escape into it, making an escape hatch possible for any
intelligent beings smart e n o u g h to calculate the dynamics of space-time.
T h e late Columbia University physicist Gerald Feinberg speculated
on this long shot of escaping the ultimate compression of the universe
t h r o u g h extra dimensions:
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At present, this is no more than a science fiction plot. However, if there
are more dimensions than those we know, or four-dimensional spacetimes in addition to the one we inhabit, then I think it very likely that there
are physical phenomena that provide connections between them. It seems
plausible that if intelligence persists in the universe, it will, in much less
time than the many billions of years before the Big Crunch, find out
whether there is anything to this speculation, and if so how to take advantage of it.
6
Colonizing the Universe
Almost all scientists who have investigated the death of the universe,
from Bertrand Russell to current cosmologists, have assumed that intelligent life will be almost helpless in the face of the inevitable, final death
throes of the universe. Even the theory that intelligent beings can tunnel
through hyperspace and avoid the Big Crunch assumes that these beings
are passive victims until the final m o m e n t s of the collapse.
However, physicists J o h n D. Barrow of the University of Sussex and
Frank J. Tipler of T u l a n e University, in their book The Anthropic Cosmological Principle, have departed from conventional wisdom and concluded
just the opposite: that intelligent life, over billions of years of evolution,
will play an active role in the final m o m e n t s of o u r universe. They take
the rather u n o r t h o d o x view that technology will continue to rise exponentially over billions of years, constantly accelerating in p r o p o r t i o n to
existing technology. T h e m o r e star systems that intelligent beings have
colonized, the m o r e star systems they can colonize. Barrow a n d Tipler
argue that over several billion years, intelligent beings will have completely colonized vast portions of the visible universe. But they are conservative; they do n o t assume that intelligent life will have mastered the
art of hyperspace travel. They assume only that their rockets will travel
at near-light velocities.
This scenario should be taken seriously for several reasons. First,
rockets traveling at near-light velocities (propelled, say, by p h o t o n
engines using the power of large laser beams) may take h u n d r e d s of
years to reach distant star systems. But Barrow a n d Tipler believe that
intelligent beings will thrive for billions of years, which is sufficient time
to colonize their own a n d neighboring galaxies even with sub-light-speed
rockets.
Without assuming hyperspace travel, Barrow a n d Tipler argue that
intelligent beings will send millions of small "von N e u m a n n p r o b e s "
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into t h e galaxy at near-light speeds to find suitable star systems for colonization. J o h n von N e u m a n n , the mathematical genius w h o developed
t h e first electronic c o m p u t e r at Princeton University d u r i n g World W a r
II, proved rigorously that robots or a u t o m a t o n s could be built with the
ability to p r o g r a m themselves, repair themselves, a n d even create c a r b o n
copies of themselves. T h u s Barrow a n d Tipler suggest that the von Neum a n n p r o b e s will function largely i n d e p e n d e n t l y of their creators. These
small p r o b e s will be vastly different from the c u r r e n t generation of Viking
a n d Pioneer probes, which are little m o r e than passive, p r e p r o g r a m m e d
machines obeying orders from their h u m a n masters. T h e von N e u m a n n
p r o b e s will be similar to Dyson's Astrochicken, except vastly m o r e powerful a n d intelligent. They will e n t e r new star systems, l a n d on planets,
a n d m i n e t h e rock for suitable chemicals a n d metals. They will t h e n
create a small industrial c o m p l e x capable of m a n u f a c t u r i n g n u m e r o u s
robotic copies of themselves. From these bases, m o r e von N e u m a n n
p r o b e s will be l a u n c h e d to explore even m o r e star systems.
Being self-programming a u t o m a t o n s , these p r o b e s will n o t n e e d
instructions from their m o t h e r planet; they will explore millions of star
systems entirely on their own, pausing only to periodically radio back
their findings. With millions of these von N e u m a n n p r o b e s scattered
t h r o u g h o u t the galaxy, creating millions of copies of themselves as they
" e a t " a n d " d i g e s t " t h e chemicals on each planet, an intelligent civilization will be able to cut down the time wasted exploring uninteresting
star systems. (Barrow a n d Tipler even consider t h e possibility that von
N e u m a n n p r o b e s from distant civilizations have already e n t e r e d o u r own
solar system. Perhaps the monolith featured so mysteriously in 2001: A
Space Odyssey was a von N e u m a n n probe.)
In the "Star T r e k " series, for example, the exploration of o t h e r star
systems by the Federation is rather primitive. T h e exploration process
d e p e n d s totally on t h e skills of h u m a n s a b o a r d a small n u m b e r of starships. Although this scenario may make for intriguing human-interest
dramas, it is a highly inefficient m e t h o d of stellar exploration, given t h e
large n u m b e r of planetary systems that are probably unsuitable for life.
Von N e u m a n n probes, a l t h o u g h they may n o t have t h e interesting
adventures of Captain Kirk or Captain Picard a n d their crews, would be
m o r e suitable for galactic exploration.
Barrow a n d Tipler make a second assumption that is crucial to their
a r g u m e n t : T h e expansion of the universe will eventually slow down a n d
reverse itself over tens of billions of years. D u r i n g t h e contraction p h a s e
of the universe, the distance between galaxies will decrease, making it
vastly easier for intelligent beings to c o n t i n u e the colonization of the
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galaxies. As the contraction of the universe accelerates, the rate of colonization of n e i g h b o r i n g galaxies will also accelerate, until the entire
universe is eventually colonized.
Even t h o u g h Barrow a n d Tipler assume that intelligent life will populate the entire universe, they are still at a loss to explain how any life
form will be able to withstand the unbelievably large temperatures and
pressures created by the final collapse of the universe. They concede
that the heat created by the contraction phase will be great e n o u g h to
vaporize any living being, b u t p e r h a p s the robots that they have created
will be sufficiently heat resistant to withstand the final m o m e n t s of the
collapse.
Re-Creating the Big Bang
Along these lines, Isaac Asimov has conjectured how intelligent beings
might react to the final death of the universe. In " T h e Last Q u e s t i o n , "
Asimov asks the ancient question of whether the universe must inevitably
die, and what will h a p p e n to all intelligent life when we reach Doomsday.
Asimov, however, assumes that the universe will die in ice, rather than
in fire, as the stars cease to b u r n hydrogen a n d temperatures p l u m m e t
to absolute zero.
T h e story begins in the year 2061, when a colossal c o m p u t e r has
solved the earth's energy problems by designing a massive solar satellite
in space that can b e a m the sun's energy back to earth. T h e AC (analog
c o m p u t e r ) is so large a n d advanced that its technicians have only the
vaguest idea of how it operates. On a $5 bet, two d r u n k e n technicians
ask the c o m p u t e r whether the sun's eventual death can be avoided or,
for that matter, whether the universe must inevitably die. After quietly
mulling over this question, the AC responds: INSUFFICIENT DATA FOR A
MEANINGFUL ANSWER.
Centuries into the future, the AC has solved the p r o b l e m of hyperspace travel, a n d h u m a n s begin colonizing thousands of star systems.
T h e AC is so large that it occupies several h u n d r e d square miles on each
planet and so complex that it maintains a n d services itself. A young
family is rocketing t h r o u g h hyperspace, unerringly guided by the AC, in
search of a new star system to colonize. W h e n the father casually mentions that the stars must eventually die, the children become hysterical.
" D o n ' t let t h e stars d i e , " plead the children. To calm t h e children, he
asks the AC if entropy can be reversed. " S e e , " reassures the father, reading the AC's response, the AC can solve everything. He comforts t h e m
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by saying, " I t will take care of everything when t h e time comes, so d o n ' t
worry." He never tells the children that the AC actually prints out: INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.
T h o u s a n d s of years into t h e future, t h e Galaxy itself has b e e n colonized. T h e AC has solved t h e p r o b l e m of immortality a n d harnesses t h e
energy of the Galaxy, but must find new galaxies for colonization. T h e
AC is so complex that it is long past t h e p o i n t where anyone u n d e r s t a n d s
how it works. It continually redesigns a n d improves its own circuits. Two
m e m b e r s of t h e Galactic Council, each h u n d r e d s of years old, d e b a t e
t h e u r g e n t question of finding new galactic energy sources, a n d w o n d e r
if the universe itself is r u n n i n g down. Can entropy be reversed? they ask.
T h e AC r e s p o n d s : INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.
Millions of years into t h e future, humanity has spread across the
u n c o u n t a b l e galaxies of t h e universe. T h e AC has solved the p r o b l e m
of releasing the m i n d from the body, a n d h u m a n minds are free to
explore t h e vastness of millions of galaxies, with their bodies safely stored
on some long forgotten planet. Two m i n d s accidentally m e e t each o t h e r
in outer space, a n d casually wonder where a m o n g the u n c o u n t a b l e galaxies h u m a n s originated. T h e AC, which is now so large that most of it
has to be h o u s e d in hyperspace, responds by instantly transporting t h e m
to an obscure galaxy. They are disappointed. T h e galaxy is so ordinary,
like millions of o t h e r galaxies, a n d t h e original star has l o n g since died.
T h e two minds b e c o m e anxious because billions of stars in the heavens
are slowly m e e t i n g the same fate. T h e two minds ask, can the d e a t h of
the universe itself be avoided? From hyperspace, the AC responds: INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.
Billions of years into the future, humanity consists of a trillion, trillion, trillion immortal bodies, each cared for by a u t o m a t o n s . Humanity's
collective mind, which is free to r o a m anywhere in the universe at will,
eventually fuses into a single mind, which in turn fuses with the AC itself.
It no longer makes sense to ask what the AC is m a d e of, or where in
hyperspace it really is. " T h e universe is dying," thinks Man, collectively.
O n e by o n e , as the stars a n d galaxies cease to g e n e r a t e energy, temperatures t h r o u g h o u t the universe a p p r o a c h absolute zero. Man desperately
asks if the cold a n d darkness slowly engulfing t h e galaxies m e a n its eventual death. From hyperspace, the AC answers: INSUFFICIENT DATA FOR A
MEANINGFUL ANSWER.
W h e n Man asks t h e AC to collect the necessary data, it responds: I
WILL DO SO. I HAVE BEEN DOING SO FOR A HUNDRED BILLION YEARS. MY PREDECESSORS HAVE BEEN ASKED THIS QUESTION MANY TIMES. ALL THE DATA I
HAVE REMAINS INSUFFICIENT.
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A timeless interval passes, a n d the universe has finally reached its
ultimate death. From hyperspace, the AC spends an eternity collecting
data and contemplating the final question. At last, the AC discovers the
solution, even though there is no longer anyone to give the answer. T h e
AC carefully formulates a p r o g r a m , a n d t h e n begins the process of
reversing Chaos. It collects cold, interstellar gas, brings together the
d e a d stars, until a gigantic ball is created.
T h e n , when its labors are d o n e , from hyperspace the AC thunders:
LET THERE BE LIGHT!
And there was light—
And on the seventh day, He rested.
15
Conclusion
T h e k n o w n is finite, the u n k n o w n infinite; intellectually we
s t a n d o n a n islet i n t h e m i d s t o f a n i l l i m i t a b l e o c e a n o f i n e x p l icability. O u r b u s i n e s s i n every g e n e r a t i o n i s t o r e c l a i m a little
more land.
Thomas H. Huxley
P
ERHAPS t h e most p r o f o u n d discovery of t h e past century in physics
has b e e n t h e realization that n a t u r e , at its most fundamental level,
is simpler t h a n a n y o n e t h o u g h t . Although t h e mathematical complexity
of the ten-dimensional theory has soared to dizzying heights, o p e n i n g
up new areas of mathematics in the process, t h e basic concepts driving
unification forward, such as higher-dimensional space a n d strings, are
basically simple a n d geometric.
Although it is too early to tell, future historians of science, w h e n
looking back at the t u m u l t u o u s twentieth century, may view o n e of the
great conceptual revolutions to be t h e introduction of higher-dimensional s p a c e - t i m e theories, such as superstring and Kaluza-Klein-type
theories. As Copernicus simplified t h e solar system with his series of
concentric circles a n d d e t h r o n e d t h e central role of t h e earth in the
heavens, t h e ten-dimensional theory promises to vastly simplify the laws
of n a t u r e a n d d e t h r o n e t h e familiar world of three dimensions. As we
have seen, the crucial realization is that a three-dimensional description
of the world, such as the Standard Model, is " t o o small" to unite all the
fundamental forces of n a t u r e into o n e comprehensive theory. J a m m i n g
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the four fundamental forces into a three-dimensional theory creates an
ugly, contrived, and ultimately incorrect description of n a t u r e .
Thus the main current dominating theoretical physics in the past
decade has been the realization that the fundamental laws of physics
appear simpler in higher dimensions, a n d that all physical laws appear
to be unified in ten dimensions. These theories allow us to r e d u c e an
e n o r m o u s a m o u n t of information into a concise, elegant fashion that
unites the two greatest theories of the twentieth century: q u a n t u m theory
a n d general relativity. Perhaps it is time to explore some of the many
implications that the ten-dimensional theory has for the future of physics
and science, the debate between reductionism a n d holism in nature,
and the aesthetic relation a m o n g physics, mathematics, religion, and
philosophy.
Ten Dimensions and Experiment
W h e n caught up in the excitement and turmoil accompanying the birth
of any great theory, there is a tendency to forget that ultimately all theories must be tested against the bedrock of experiment. No matter how
elegant or beautiful a theory may appear, it is d o o m e d if it disagrees
with reality.
Goethe once wrote, "Gray is the dogma, b u t green is the tree of life."
History has repeatedly b o r n e out the correctness of his p u n g e n t observation. T h e r e are many examples of old, incorrect theories that stubbornly persisted for years, sustained only by the prestige of foolish b u t
well-connected scientists. At times, it even became politically risky to
oppose the power of ossified, senior scientists. Many of these theories
have b e e n killed off only when some decisive e x p e r i m e n t exposed their
incorrectness.
For example, because of H e r m a n n von Helmholtz's fame and considerable influence in nineteenth-century Germany, his theory of electromagnetism was m u c h m o r e popular a m o n g scientists than Maxwell's
relatively obscure theory. But no matter how well known Helmholtz was,
ultimately e x p e r i m e n t confirmed the theory of Maxwell a n d relegated
Helmholtz's theory to obscurity. Similarly, when Einstein proposed his
theory of relativity, many politically powerful scientists in Nazi Germany,
like Nobel laureate Philip Lenard, h o u n d e d him until he was driven out
of Berlin in 1933. T h u s the yeoman's work in any science, a n d especially
physics, is d o n e by the experimentalist, who must keep the theoreticians
honest.
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Victor Weisskopf, a theoretical physicist at MIT, o n c e summarized
the relationship between theoretical a n d experimental science w h e n he
observed that t h e r e are t h r e e kinds of physicists: the m a c h i n e builders
(who build the a t o m smashers that m a k e the e x p e r i m e n t possible), the
experimentalists (who plan a n d execute the e x p e r i m e n t ) , a n d the theoreticians (who devise the theory to explain t h e e x p e r i m e n t ) . He t h e n
c o m p a r e d these three classes to Columbus's voyage to America. He
observed that
the m a c h i n e builders correspond to the captains and ship builders w h o
really d e v e l o p e d t h e t e c h n i q u e s a t t h a t t i m e . T h e e x p e r i m e n t a l i s t s w e r e
t h o s e fellows on t h e ships that sailed to t h e o t h e r side of t h e world a n d
t h e n j u m p e d u p o n t h e n e w i s l a n d s a n d j u s t w r o t e d o w n w h a t t h e y saw.
T h e theoretical physicists are those fellows w h o stayed back in M a d r i d a n d
told C o l u m b u s that he was g o i n g to l a n d in India.'
If, however, the laws of physics b e c o m e united in ten dimensions
only at energies far beyond anything available with o u r p r e s e n t technology, t h e n the future of experimental physics is in jeopardy. In the
past, every new generation of atom smashers has b r o u g h t forth a new
generation of theories. This period may be c o m i n g to a close.
Although everyone expected new surprises if the SSC b e c a m e operational by a b o u t the year 2000, some were betting that it would simply
reconfirm the correctness of o u r present-day Standard Model. Most
likely, the decisive experiments that will prove or disprove the correctness of t h e ten-dimensional theory c a n n o t be p e r f o r m e d anytime in t h e
n e a r future. We may be e n t e r i n g a l o n g dry spell where research in tendimensional theories will b e c o m e an exercise in p u r e mathematics. All
theories derive their power a n d strength from e x p e r i m e n t , which is like
fertile soil that can nourish a n d sustain a field of flowering plants o n c e
they take root. If the soil b e c o m e s b a r r e n a n d dry, t h e n t h e plants will
wither along with it.
David Gross, o n e of t h e originators of the heterotic string theory, has
c o m p a r e d the d e v e l o p m e n t of physics to the relationship between two
m o u n t a i n climbers:
It used to be that as we were climbing the m o u n t a i n of nature, the experi m e n t a l i s t s w o u l d l e a d t h e way. W e lazy t h e o r i s t s w o u l d l a g b e h i n d . Every
o n c e i n a w h i l e t h e y w o u l d kick d o w n a n e x p e r i m e n t a l s t o n e w h i c h w o u l d
b o u n c e off o u r heads. Eventually we w o u l d get the idea a n d we w o u l d
f o l l o w t h e p a t h t h a t was b r o k e n b y t h e e x p e r i m e n t a l i s t s . . . . B u t n o w w e
theorists m i g h t have to take the lead. This is a m u c h m o r e lonely enter-
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prise. In the past we always knew where the experimentalists were and thus
what we should aim for. Now we have no idea how large the mountain is,
nor where the summit is.
Although experimentalists have traditionally taken the lead in breaking o p e n new territory, the next era in physics may be an exceptionally
difficult o n e , forcing theoreticians to assume t h e lead, as Gross
notes.
T h e SSC probably would have found new particles. T h e Higgs particles may have been discovered, or " s u p e r " partners of the quarks may
have shown u p , or maybe a sublayer b e n e a t h the quarks may have b e e n
revealed. However, the basic forces b i n d i n g these particles will, if the
theory holds u p , be the same. We may have seen m o r e complex Y a n g Mills fields a n d gluons coming forth from the SSC, b u t these fields may
represent only larger a n d larger symmetry groups, representing fragments of the even larger E(8) X E(8) symmetry coming from string
theory.
In some sense, the origin of this uneasy relation between theory a n d
e x p e r i m e n t is d u e to the fact that this theory represents, as Witten has
noted, "21st century physics that fell accidentally into the 20th century." Because the natural dialectic between theory a n d e x p e r i m e n t was
disrupted by the fortuitous accidental discovery of the theory in 1968,
p e r h a p s we must wait until the twenty-first century, when we expect the
arrival of new technologies that will hopefully o p e n up a new generation
of atom smashers, cosmic-ray counters, a n d d e e p space probes. Perhaps
this is the price we must pay for having a forbidden "sneak preview"
into the physics of the next century. Perhaps by then, t h r o u g h indirect
means, we may experimentally see the glimmer of the tenth dimension
in o u r laboratories.
2
Ten Dimensions and Philosophy: Reductionism versus Holism
Any great theory has equally great repercussions on technology and the
foundations of philosophy. T h e birth of general relativity o p e n e d up
new areas of research in astronomy a n d practically created the science
of cosmology. T h e philosophical implications of the Big Bang have sent
reverberations t h r o u g h o u t the philosophical a n d theological communities. A few years ago, this even led to leading cosmologists having a
special audience with the p o p e at the Vatican to discuss the implications
of the Big Bang theory on the Bible and Genesis.
Similarly, q u a n t u m theory gave birth to the science of subatomic
particles a n d h e l p e d fuel the c u r r e n t revolution in electronics. T h e tran-
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sistor—the linchpin of m o d e r n technological society—is a purely q u a n tum-mechanical device. Equally p r o f o u n d was t h e impact that the Heis e n b e r g Uncertainty Principle has h a d on the debate over free will a n d
determinism, affecting religious d o g m a on the role of sin a n d r e d e m p tion for the c h u r c h . Both the Catholic C h u r c h a n d the Presbyterian
C h u r c h , with a large ideological stake in the o u t c o m e of this controversy
over predestination, have b e e n affected by this debate over q u a n t u m
mechanics. Although the implications of the ten-dimensional theory are
still unclear, we ultimately expect that the revolution now g e r m i n a t i n g
in the world of physics will have a similar far-reaching impact o n c e t h e
theory becomes accessible to the average person.
In general, however, most physicists feel uncomfortable talking a b o u t
philosophy. They are s u p r e m e pragmatists. They stumble across physical
laws n o t by design or ideology, b u t largely t h r o u g h trial a n d e r r o r a n d
shrewd guesses. T h e younger physicists, who do the lion's share of
research, are too busy discovering new theories to waste time philosophizing. Younger physicists, in fact, look askance at older physicists if
they s p e n d too m u c h time sitting on distinguished policy committees or
pontificating on the philosophy of science.
Most physicists feel that, outside of vague notions of " t r u t h " a n d
" b e a u t y , " philosophy has no business i n t r u d i n g on their private d o m a i n .
In general, they argue, reality has always proved to be m u c h m o r e sophisticated a n d subtle than any preconceived philosophy. They r e m i n d us
of some well-known figures in science who, in their waning years, took
up embarrassingly eccentric philosophical ideas that led down blind
alleys.
W h e n confronted with sticky philosophical questions, such as the
role of "consciousness" in performing a q u a n t u m m e a s u r e m e n t , most
physicists s h r u g their shoulders. As l o n g as they can calculate the outc o m e of an experiment, they really d o n ' t care a b o u t its philosophical
implications. In fact, Richard Feynman almost m a d e a career trying to
expose the p o m p o u s pretenses of certain philosophers. T h e greater
their puffed-up rhetoric a n d erudite vocabulary, he t h o u g h t , the weaker
the scientific foundation of their arguments. (When debating the relative merits of physics a n d philosophy, I am sometimes r e m i n d e d of the
n o t e written by an a n o n y m o u s university president who analyzed the
differences between t h e m . He wrote, "Why is it that you physicists always
require so m u c h expensive e q u i p m e n t ? Now the D e p a r t m e n t of Mathematics requires n o t h i n g b u t m o n e y for paper, pencils, a n d waste p a p e r
baskets a n d the D e p a r t m e n t of Philosophy is still better. It d o e s n ' t even
ask for waste p a p e r baskets." )
Nevertheless, although the average physicist is n o t b o t h e r e d by philo3
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sophical questions, the greatest of t h e m were. Einstein, Heisenberg, a n d
Bohr spent long hours in heated discussions, wrestling late into the night
with the m e a n i n g of m e a s u r e m e n t , the problems of consciousness, a n d
the m e a n i n g of probability in their work. Thus it is legitimate to ask how
higher-dimensional theories reflect on this philosophical conflict, especially regarding the debate between " r e d u c t i o n i s m " a n d " h o l i s m . "
Heinz Pagels once said, "We are passionate about our experience of
reality, and most of us project o u r hopes a n d fears o n t o the universe."
T h u s it is inevitable that philosophical, even personal questions will
intrude into the discussion on higher-dimensional theories. Inevitably,
the revival of higher dimensions in physics will rekindle the debate
between " r e d u c t i o n i s m " a n d " h o l i s m " that has flared, on a n d off, for
the past decade.
4
Webster's Collegiate Dictionary defines reductionism as a " p r o c e d u r e or
theory that reduces complex data or p h e n o m e n a to simple t e r m s . " This
has b e e n o n e of the guiding philosophies of subatomic physics—to
reduce atoms and nuclei to their basic c o m p o n e n t s . T h e p h e n o m e n a l
experimental success, for example, of the Standard Model in explaining
the properties of h u n d r e d s of subatomic particles shows that there is
merit in looking for the basic building blocks of matter.
Webster's Collegiate Dictionary defines holism as the "theory that the
d e t e r m i n i n g factors esp. in living nature are irreducible wholes." This
philosophy maintains that the Western philosophy of breaking things
down into their c o m p o n e n t s is overly simplistic, that o n e misses the
larger picture, which may contain vitally important information. For
example, think of an ant colony containing thousands of ants that obeys
complex, dynamic rules of social behavior. T h e question is: What is the
best way to u n d e r s t a n d the behavior of an ant colony? T h e reductionist
would break the ants into their constituents: organic molecules. However, o n e may spend h u n d r e d s of years dissecting ants a n d analyzing
their molecular m a k e u p without finding the simplest clues as to how an
a n t colony behaves. T h e obvious way is to analyze the behavior of an a n t
colony as an integral whole, without breaking it down.
Similarly, this debate has sparked considerable controversy within
the area of brain research a n d artificial intelligence. T h e reductionist
approach is to reduce the brain to its ultimate units, the brain cells, a n d
try to reassemble the brain from them. A whole school of research in
artificial intelligence held that by creating elemental digital circuits we
could build up increasingly complex circuits, until we created artificial
intelligence. Although this school of t h o u g h t h a d initial success in the
1950s by modeling "intelligence" along the lines of m o d e r n digital com-
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puters, it proved disappointing because it could n o t mimic even the
simplest of brain functions, such as recognizing patterns in a p h o t o graph.
T h e second school of t h o u g h t has tried to take a m o r e holistic
a p p r o a c h to t h e brain. It attempts to define t h e functions of t h e brain
a n d create models that treat the brain as a whole. Although this has
proved m o r e difficult to initiate, it holds great promise because certain
brain functions that we take for g r a n t e d (for example, tolerance of error,
weighing of uncertainty, a n d m a k i n g creative associations between different objects) a r e built i n t o t h e system from t h e start. Neural network
theory, for example, uses aspects of this organic a p p r o a c h .
Each side of this reductionist-holistic d e b a t e takes a dim view of the
other. In their strenuous attempts to d e b u n k each other, they sometimes
only diminish themselves. They often talk past each o t h e r , n o t addressing each o t h e r ' s m a i n points.
T h e latest twist in the d e b a t e is that the reductionists have, for the
past few years, declared victory over holism. Recently, t h e r e has b e e n a
flurry of claims in t h e p o p u l a r press by t h e reductionists that the successes of the Standard Model a n d t h e G U T theory are vindications of
r e d u c i n g n a t u r e to smaller a n d m o r e basic constituents. By p r o b i n g
down to t h e elemental quarks, leptons, a n d Yang-Mills fields, physicists
have finally isolated the basic constituents of all matter. For e x a m p l e ,
physicist J a m e s S. Trefil of the University of Virginia takes a swipe at
holism when he writes a b o u t the " T r i u m p h of Reductionism":
During the 1960s a n d 1970s, w h e n the complexity of the particle world
w a s b e i n g m a d e m a n i f e s t i n o n e e x p e r i m e n t after a n o t h e r , s o m e p h y s i c i s t s
b r o k e faith with t h e r e d u c t i o n i s t p h i l o s o p h y a n d b e g a n t o l o o k o u t s i d e o f
t h e W e s t e r n t r a d i t i o n f o r g u i d a n c e . In h i s b o o k
The Tao of Physics, f o r
e x a m p l e , Fritjhof Capra a r g u e d that t h e p h i l o s o p h y o f r e d u c t i o n i s m h a d
failed a n d that it was t i m e to take a m o r e holistic, mystical view of
nature.. .. [ T ] h e 1970s [however] can be t h o u g h t of as the period in
w h i c h t h e great traditions of Western scientific t h o u g h t , s e e m i n g l y i m p e r iled by the advances of twentieth-century science, have b e e n thoroughly
v i n d i c a t e d . P r e s u m a b l y , i t will t a k e a w h i l e f o r t h i s r e a l i z a t i o n t o p e r c o l a t e
away f r o m a s m a l l g r o u p o f t h e o r e t i c a l p h y s i c i s t s a n d b e c o m e i n c o r p o r a t e d
into our general world view.
5
T h e disciples of holism, however, t u r n this debate a r o u n d . They
claim that the idea of unification, p e r h a p s the greatest t h e m e in all of
physics, is holistic, n o t reductionist. They p o i n t to how reductionists
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would sometimes snicker b e h i n d Einstein's back in the last years of his
life, saying that he was getting senile trying to unite all the forces of the
world. T h e discovery of unifying patterns in physics was an idea pion e e r e d by Einstein, n o t the reductionists. F u r t h e r m o r e , the inability of
the reductionists to offer a convincing resolution of the Schrodinger's
cat paradox shows that they have simply chosen to ignore the deeper,
philosophical questions. T h e reductionists may have had great success
with q u a n t u m field theory and the Standard Model, b u t ultimately that
success is based on sand, because q u a n t u m theory, in the final analysis,
is an incomplete theory.
Both sides, of course, have merit. Each side is merely addressing
different aspects of a difficult problem. However, taken to extremes, this
debate sometimes degenerates into a battle between what I call bellige r e n t science versus know-nothing science.
Belligerent science clubs the opposition with a heavy, rigid view of
science that alienates rather than persuades. Belligerent science seeks
to win points in a debate, rather than win over the audience. Instead of
appealing to the finer instincts of the lay audience by presenting itself
as the defender of enlightened reason a n d sound experiment, it comes
off as a new Spanish Inquisition. Belligerent science is science with a
chip on its shoulder. Its scientists accuse t h e holists of being soft-headed,
of getting their physics confused, of throwing pseudoscientific gibberish
to cover their ignorance. Thus belligerent science may be winning the
individual battles, b u t is ultimately losing the war. In every one-on-one
skirmish, belligerent science may trounce the opposition by parading
out mountains of data a n d learned Ph.D.s. However, in the long run,
arrogance a n d conceit may eventually backfire by alienating the very
audience that it is trying to persuade.
Know-nothing science goes to the opposite extreme, rejecting experi m e n t and embracing whatever faddish philosophy h a p p e n s to come
along. Know-nothing science sees unpleasant facts as m e r e details, and
the overall philosophy as everything. If t h e facts do n o t seem to fit the
philosophy, t h e n obviously something is wrong with the facts. Known o t h i n g science comes in with a preformed agenda, based on personal
fulfillment rather than objective observation, and tries to fit in the science as an afterthought.
This split between these two factions first a p p e a r e d d u r i n g the Vietn a m War, when the flower generation was appalled by the massive, excessive use of deadly technology against a peasant nation. But p e r h a p s the
area in which this legitimate debate has flared up most recently is personal health. For example, well-paid lobbyists for the powerful agri-busi-
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ness a n d food industry in the 1950s a n d 1960s exerted considerable
influence on Congress a n d the medical establishment, preventing a
t h o r o u g h examination of the harmful effects of cholesterol, tobacco,
animal fats, pesticides, a n d certain food additives on h e a r t disease a n d
cancer, which have now b e e n thoroughly d o c u m e n t e d .
A r e c e n t example is the scandal that s u r r o u n d e d t h e u p r o a r over the
pesticide Alar in apples. W h e n the environmentalists at the National
Resources Defense Council a n n o u n c e d that c u r r e n t levels of pesticides
in apples could kill upward of 5,000 children, they sparked c o n c e r n
a m o n g consumers a n d indignation within the food industry, which
d e n o u n c e d t h e m as alarmists. T h e n it was revealed that t h e r e p o r t used
figures a n d data from the federal g o v e r n m e n t to arrive at these conclusions. This, in turn, implied that the Food a n d D r u g Administration was
sacrificing 5,000 children in the interests of "acceptable risk."
In addition, the revelations a b o u t the widespread possible contamination of o u r drinking water by lead, which can cause serious neurological problems in children, only served to lower the prestige of science
in the minds of most Americans. T h e medical profession, the food industry, a n d t h e chemical industry have b e g u n to earn t h e distrust of wide
portions of society. These a n d o t h e r scandals have also c o n t r i b u t e d to
the national flareup of faddish health diets, most of which are well intentioned, b u t some of which are n o t scientifically sound.
Higher Synthesis in Higher Dimensions
These two philosophical viewpoints, apparently irreconcilable, must be
viewed from t h e larger perspective. They are antagonistic only when
viewed in their e x t r e m e form.
Perhaps a h i g h e r synthesis of b o t h viewpoints lies in higher dimensions. Geometry, almost by definition, c a n n o t fit the usual reductionist
m o d e . By studying a tiny strand of fiber, we c a n n o t possibly u n d e r s t a n d
an entire tapestry. Similarly, by isolating a microscopic region of a surface, we c a n n o t d e t e r m i n e the overall structure of the surface. H i g h e r
dimensions, by definition, imply that we must take t h e larger, global
viewpoint.
Similarly, geometry is n o t purely holistic, either. Simply observing
that a higher-dimensional surface is spherical does n o t provide the information necessary to calculate the properties of the quarks c o n t a i n e d
within it. T h e precise way in which a dimension curls up into a ball
d e t e r m i n e s the n a t u r e of t h e symmetries of the quarks a n d gluons living
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on that surface. Thus holism by itself does n o t give us the data necessary
to turn the ten-dimensional theory into a physically relevant theory.
T h e geometry of higher dimensions, in some sense, forces us to realize the unity between the holistic a n d reductionist approaches. They are
simply two ways of a p p r o a c h i n g the same thing: geometry. They are two
sides of the same coin. From the vantage p o i n t of geometry, it makes no
difference whether we a p p r o a c h it from the reductionist point of view
(assembling quarks and gluons in a Kaluza-Klein space) or the holistic
a p p r o a c h (taking a Kaluza-Klein surface a n d discovering the symmetries
of the quarks a n d gluons).
We may prefer o n e a p p r o a c h over the other, b u t this is only for
historical or pedagogical purposes. For historical reasons, we may stress
the reductionist roots of subatomic physics, emphasizing how particle
physicists over a period of 40 years pieced together three of the fundamental forces by smashing atoms, or we may take a m o r e holistic
a p p r o a c h a n d claim that the final unification of q u a n t u m forces with
gravity implies a d e e p u n d e r s t a n d i n g of geometry. This leads us to
a p p r o a c h particle physics t h r o u g h Kaluza-Klein and string theories and
to view the Standard Model as a consequence of curling up higherdimensional space.
T h e two approaches are equally valid. In o u r book Beyond Einstein:
The Cosmic Quest for the Theory of the Universe, Jennifer Trainer a n d I took
a m o r e reductionist a p p r o a c h a n d described how the discoveries of phen o m e n a in the visible universe eventually led to a geometric description
of matter. In this book, we took the opposite approach, b e g i n n i n g with
the invisible universe and taking the concept of how the laws of nature
simplify in higher dimensions as our basic t h e m e . However, both
approaches yield the same result.
By analogy, we can discuss the controversy over the "left" brain a n d
" r i g h t " brain. T h e neurologists who originally m a d e the experimental
discovery that the left and right hemispheres of our brain perform distinctly different functions became distressed that their data were grossly
misrepresented in the p o p u l a r press. Experimentally, they found that
when s o m e o n e is shown a picture, the left eye (or right brain) pays m o r e
attention to particular details, while the right eye (or left brain) m o r e
easily grasps the entire p h o t o . However, they became disturbed when
popularizers began to say that the left brain was the "holistic b r a i n " a n d
the right brain was the "reductionist b r a i n . " This took the distinction
between the two brains o u t of context, resulting in many bizarre interpretations of how o n e should organize o n e ' s thoughts in daily life.
A m o r e correct a p p r o a c h to brain function, they found, was that the
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brain necessarily uses b o t h halves in synchrony, that the dialectic
between b o t h halves of t h e brain is m o r e i m p o r t a n t t h a n t h e specific
function of each half individually. T h e truly interesting dynamics take
place w h e n b o t h halves of the brain interact in harmony.
Similarly, anyone who sees the victory of o n e philosophy over the
o t h e r in recent advances in physics is p e r h a p s reading too m u c h into
the experimental data. Perhaps the safest conclusion that we can reach
is that science benefits most from the intense interaction between these
two philosophies.
Let us see concretely how this takes place, analyzing how the theory
of h i g h e r dimensions gives us a resolution between diametrically
o p p o s e d philosophies, using two examples, Schrodinger's cat a n d t h e S
matrix theory.
Schrodinger's Cat
T h e disciples of holism sometimes attack reductionism by hitting quantum theory where it is weakest, on t h e question of Schrodinger's cat.
T h e reductionists c a n n o t give a reasonable explanation of the paradoxes
of q u a n t u m mechanics.
T h e most embarrassing feature of q u a n t u m theory, we recall, is that
an observer is necessary to make a m e a s u r e m e n t . T h u s before the observation is m a d e , cats can be either d e a d or alive a n d the m o o n may or
may n o t be in t h e sky. Usually, this would be considered crazy, b u t quantum mechanics has b e e n verified repeatedly in the laboratory. Since the
process of making an observation requires an observer, a n d since an
observer requires consciousness, then the disciples of holism claim that
a cosmic consciousness must exist in o r d e r to explain t h e existence of
any object.
Higher-dimensional theories do n o t resolve this difficult question
completely, b u t they certainly p u t it in a new light. T h e p r o b l e m lies in
the distinction between t h e observer a n d t h e observed. However, in
q u a n t u m gravity we write down the wave function of t h e entire universe.
T h e r e is no m o r e distinction between the observer a n d t h e observed;
q u a n t u m gravity allows for the existence of only the wave function of
everything.
In the past, such statements were meaningless because q u a n t u m gravity did n o t really exist as a theory. Divergences would c r o p up every time
s o m e o n e wanted to do a physically relevant calculation. So t h e c o n c e p t
of a wave function for the entire universe, a l t h o u g h appealing, was m e a n -
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ingless. However, with the coming of the ten-dimensional theory, the
m e a n i n g of the wave function of the entire universe becomes a relevant
concept once again. Calculations with the wave function of the universe
can appeal to the fact that the theory is ultimately a ten-dimensional
theory, a n d is h e n c e renormalizable.
This partial solution to the question of observation once again takes
the best of b o t h philosophies. On the o n e hand, this picture is reductionist because it adheres closely to the standard quantum-mechanical
explanation of reality, without recourse to consciousness. On the other
h a n d , it is also holistic because it begins with the wave function of the
entire universe, which is the ultimate holistic expression! This picture
does not make the distinction between t h e observer and the observed.
In this picture, everything, including all objects a n d their observers, is
included in the wave function.
This is still only a partial solution because the cosmic wave function
itself, which describes the entire universe, does n o t live in any definite
state, but is actually a composite of all possible universes. T h u s the problem of indeterminacy, first discovered by Heisenberg, is now extended
to the entire universe.
T h e smallest unit that o n e can manipulate in these theories is the
universe itself, and the smallest unit that o n e can quantize is the space
of all possible universes, which includes both dead cats and live cats.
T h u s in o n e universe, the cat is i n d e e d dead; but in another, the cat is
alive. However, both universes reside in the same h o m e : the wave function of the universe.
A Child of 5-Matrix Theory
Ironically, in the 1960s, the reductionist approach looked like a failure;
the q u a n t u m theory of fields was hopelessly riddled with divergences
found in the perturbation expansion. With q u a n t u m physics in disarray,
a b r a n c h of physics called S-matrix (scattering matrix) theory broke off
from the mainstream a n d began to germinate. Originally founded by
Heisenberg, it was further developed by Geoffrey Chew at the University
of California at Berkeley. S-matrix theory, unlike reductionism, tried to
look at the scattering of particles as an inseparable, irreducible whole.
In principle, if we know the S matrix, we know everything about
particle interactions and how they scatter. In this approach, how particles b u m p into o n e a n o t h e r is everything; the individual particle is nothing. S-matrix theory said that the self-consistency of t h e scattering matrix,
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a n d self-consistency alone, was sufficient to d e t e r m i n e the S matrix. T h u s
fundamental particles a n d fields were banished forever from t h e E d e n
of S-matrix theory. In the final analysis, only t h e S matrix h a d any physical m e a n i n g .
As an analogy, let us say that we are given a complex, strange-looking
m a c h i n e a n d are asked to explain what it does. T h e reductionist will
immediately get a screw driver a n d take the m a c h i n e apart. By breaking
down the m a c h i n e to thousands of tiny pieces, the reductionist h o p e s
to find out how the m a c h i n e functions. However, if the m a c h i n e is too
complicated, taking it apart only makes matters worse.
T h e holists, however, do n o t want to take the m a c h i n e a p a r t for
several reasons. First, analyzing thousands of gears a n d screws may n o t
give us the slightest h i n t of what the overall m a c h i n e does. Second, trying
to explain how each tiny gear works may send us on a wild-goose chase.
T h e correct way, they feel, is to look at the m a c h i n e as a whole. They
turn t h e m a c h i n e on a n d ask how the parts move a n d interact with o n e
a n o t h e r . In m o d e r n language, this m a c h i n e is the S matrix, a n d this
philosophy b e c a m e t h e S-matrix theory.
In 1971, however, the tide shifted dramatically in favor of reductionism with Gerard 't Hooft's discovery that the Yang-Mills field can provide
a self-consistent theory of subatomic forces. Suddenly, each of the particle interactions c a m e tumbling down like h u g e trees in a forest. T h e
Yang-Mills field gave u n c a n n y a g r e e m e n t with the experimental data
from a t o m smashers, leading to the establishment of the Standard
Model, while S-matrix theory b e c a m e entangled in m o r e a n d m o r e
obscure mathematics. By t h e late 1970s, it s e e m e d like a total, irreversible victory of reductionism over holism a n d t h e S-matrix theory. T h e
reductionists began to declare victory over t h e prostrate body of the
holists a n d the S matrix.
T h e tide, however, shifted o n c e again in the 1980s. With the failure
of the GUTs to yield any insight into gravitation or yield any experimentally verifiable results, physicists b e g a n to look for new avenues of
research. This d e p a r t u r e from GUTs began with a new theory, which
owed its existence to the S-matrix theory.
In 1968, w h e n S-matrix theory was in its heyday, Veneziano a n d
Suzuki were deeply influenced by t h e philosophy of d e t e r m i n i n g t h e S
matrix in its entirety. They hit on the Euler beta function because they
were searching for a mathematical representation of the entire S matrix.
If they h a d looked for reductionist Feynman diagrams, they never would
have stumbled on o n e of t h e great discoveries of t h e past several decades.
Twenty years later, we see t h e flowering of the seed p l a n t e d by the
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S-matrix theory. T h e Veneziano-Suzuki theory gave birth to string theory, which in turn has b e e n reinterpreted via Kaluza-Klein as a tendimensional theory of the universe.
T h u s we see that the ten-dimensional theory straddles both
traditions. It was b o r n as a child of a holistic Smatrix theory, b u t contains
the reductionist Yang-Mills and quark theories. In essence, it has
m a t u r e d e n o u g h to absorb both philosophies.
Ten Dimensions and Mathematics
O n e of the intriguing features of superstring theory is the level to which
the mathematics has soared. No o t h e r theory known to science uses such
powerful mathematics at such a fundamental level. In hindsight, this is
necessarily so, because any unified field theory first must absorb the
Riemannian geometry of Einstein's theory a n d the Lie groups coming
from q u a n t u m field theory, a n d then must incorporate an even higher
mathematics to make t h e m compatible. This new mathematics, which is
responsible for the merger of these two theories, is topology, and it is
responsible for accomplishing the seemingly impossible task of abolishing the infinities of a q u a n t u m theory of gravity.
T h e a b r u p t introduction of advanced mathematics into physics via
string theory has caught many physicists off guard. More than o n e physicists has secretly g o n e to the library to check out h u g e volumes of mathematical literature to u n d e r s t a n d the ten-dimensional theory. CERN
physicist J o h n Ellis admits, "I find myself touring t h r o u g h the bookshops
trying to find encyclopedias of mathematics so that I can m u g up on all
these mathematical concepts like homology and h o m o t o p y a n d all this
sort of stuff which I never b o t h e r e d to learn b e f o r e ! " To those who
have worried about the ever-widening split between mathematics a n d
physics in this century, this is a gratifying, historic event in itself.
6
Traditionally, mathematics a n d physics have b e e n inseparable since
the time of the Greeks. Newton a n d his contemporaries never m a d e a
sharp distinction between mathematics a n d physics; they called themselves natural philosophers, a n d felt at h o m e in the disparate worlds of
mathematics, physics, a n d philosophy.
Gauss, Riemann, a n d Poincare all considered physics to be of the
utmost importance as a source of new mathematics. T h r o u g h o u t the
eighteenth a n d n i n e t e e n t h centuries, there was extensive cross-pollination between mathematics a n d physics. But after Einstein and Poincare,
the development of mathematics and physics took a sharp turn. For the
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past 70 years, t h e r e has b e e n little, if any, real c o m m u n i c a t i o n between
mathematicians a n d physicists. Mathematicians explored the topology
of N-dimensional space, developing new disciplines such as algebraic
topology. F u r t h e r i n g t h e work of Gauss, R i e m a n n , a n d Poincare, mathematicians in t h e past century developed an arsenal of abstract t h e o r e m s
a n d corollaries that have no c o n n e c t i o n to t h e weak or strong forces.
Physics, however, b e g a n to p r o b e t h e realm of t h e n u c l e a r force, using
three-dimensional mathematics known in t h e n i n e t e e n t h century.
All this c h a n g e d with the introduction of t h e t e n t h d i m e n s i o n .
Rather abruptly, t h e arsenal of the past century of mathematics is being
i n c o r p o r a t e d into t h e world of physics. Enormously powerful t h e o r e m s
in mathematics, l o n g cherished only by mathematicians, now take on
physical significance. At last, it seems as t h o u g h the diverging gap
between mathematics a n d physics will be closed. In fact, even the mathematicians have b e e n startled at the flood of new mathematics that the
theory has i n t r o d u c e d . Some distinguished mathematicians, such as Isad o r e A. Singer of MIT, have stated that p e r h a p s superstring theory
should be treated as a b r a n c h of mathematics, i n d e p e n d e n t of w h e t h e r
it is physically relevant.
No o n e has t h e slightest inkling why mathematics a n d physics a r e so
intertwined. T h e physicist Paul A. M. Dirac, o n e of t h e founders of quantum theory, stated that " m a t h e m a t i c s can lead us in a direction we would
n o t take if we only followed up physical ideas by themselves."
7
Alfred N o r t h Whitehead, o n e of the greatest mathematicians of the
past century, o n c e said that mathematics, at t h e deepest level, is inseparable from physics at the deepest level. However, the precise reason for
the miraculous convergence seems totally obscure. No o n e has even a
reasonable theory to explain why the two disciplines should share concepts.
It is often said that " m a t h e m a t i c s is the language of physics." For
example, Galileo o n c e said, " N o o n e will be able to r e a d t h e great b o o k
of t h e Universe if he does n o t u n d e r s t a n d its language, which is that of
m a t h e m a t i c s . " But this begs t h e question of why. F u r t h e r m o r e , mathematicians would be insulted to think that their entire discipline is b e i n g
r e d u c e d to m e r e semantics.
8
Einstein, n o t i n g this relationship, r e m a r k e d that p u r e mathematics
might be o n e avenue to solve the mysteries of physics: " I t is my conviction that p u r e mathematical construction enables us to discover t h e concepts a n d the laws c o n n e c t i n g t h e m , which gives us t h e key to the u n d e r standing of n a t u r e . . . . In a certain sense, therefore, I hold it true that
p u r e t h o u g h t can grasp reality, as the ancients d r e a m e d . " Heisenberg
9
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e c h o e d this belief: "If nature leads us to mathematical forms of great
simplicity a n d beauty . . . that no o n e has previously e n c o u n t e r e d , we
c a n n o t help thinking that they are 'true,' that they reveal a g e n u i n e
feature of n a t u r e . "
Nobel laureate Eugene Wigner once even p e n n e d an essay with the
candid title " T h e Unreasonable Effectiveness of Mathematics in the Natural Sciences."
Physical Principles versus Logical Structures
Over the years, I have observed that mathematics a n d physics have
obeyed a certain dialectical relationship. Physics is n o t just an aimless,
r a n d o m sequence of Feynman diagrams and symmetries, a n d mathematics is n o t just a set of messy equations, b u t rather physics a n d mathematics obey a definite symbiotic relationship.
Physics, I believe, is ultimately based on a small set of physical principles. These principles can usually be expressed in plain English without
reference to mathematics. From the Copernican theory, to Newton's
laws of motion, and even Einstein's relativity, the basic physical principles can be expressed in just a few sentences, largely i n d e p e n d e n t of any
mathematics. Remarkably, only a handful of fundamental physical principles are sufficient to summarize most of m o d e r n physics.
Mathematics, by contrast, is the set of all possible self-consistent structures, and there are vastly many m o r e logical structures than physical
principles. T h e hallmark of any mathematical system (for example, arithmetic, algebra, or geometry) is that its axioms and t h e o r e m s are consistent with o n e another. Mathematicians are mainly c o n c e r n e d that these
systems never result in a contradiction, a n d are less interested in discussing the relative merits of o n e system over another. Any self-consistent
structure, of which there are many, is worthy of study. As a result, mathematicians are m u c h m o r e fragmented than physicists; mathematicians
in o n e area usually work in isolation from mathematicians in o t h e r areas.
T h e relationship between physics (based on physical principles) and
mathematics (based on self-consistent structures) is now evident: To
solve a physical principle, physicists may require many self-consistent
structures. T h u s physics automatically unites many diverse branches of mathematics. Viewed in this light, we can u n d e r s t a n d how the great ideas in
theoretical physics evolved. For example, both mathematicians a n d physicists claim Isaac Newton as o n e of the giants of their respective professions. However, Newton did n o t begin the study of gravitation starting
with mathematics. By analyzing the motion of falling bodies, he was led
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to believe that t h e m o o n was continually falling toward the earth, b u t
never collided with it because the earth curved b e n e a t h it; the curvature
of t h e earth c o m p e n s a t e d for t h e falling of the m o o n . He was therefore
led to postulate a physical principle: the universal law of gravitation.
However, because he was at a loss to solve t h e equations for gravity,
Newton began a 30-year quest to construct from scratch a mathematics
powerful e n o u g h to calculate them. In the process, he discovered many
self-consistent structures, which are collectively called calculus. From this
viewpoint, the physical principle came first (law of gravitation), a n d t h e n
came the construction of diverse self-consistent structures necessary to
solve it (such as analytic geometry, differential equations, derivatives,
a n d integrals). In the process, the physical principle united these diverse
self-consistent structures into a c o h e r e n t body of mathematics (the calculus).
T h e same relationship applies to Einstein's theory of relativity. Einstein b e g a n with physical principles (such as t h e constancy of t h e speed
of light a n d t h e equivalence principle for gravitation) a n d t h e n , by
searching t h r o u g h the mathematical literature, found the self-consistent
structures (Lie groups, R i e m a n n ' s tensor calculus, differential g e o m e try) that allowed him to solve these principles. In t h e process, Einstein
discovered how to link these b r a n c h e s of mathematics into a c o h e r e n t
picture.
String theory also demonstrates this pattern, but in a startlingly different fashion. Because of its mathematical complexity, string theory has
linked vastly different b r a n c h e s of mathematics (such as Riemann surfaces, Kac-Moody algebras, super Lie algebras, finite groups, m o d u l a r
functions, a n d algebraic topology) in a way that has surprised the mathematicians. As with o t h e r physical theories, it automatically reveals the
relationship a m o n g many different self-consistent structures. However,
the underlying physical principle b e h i n d string theory is u n k n o w n . Physicists h o p e that o n c e this principle is revealed, new b r a n c h e s of mathematics will be discovered in the process. In o t h e r words, t h e reason why
the string theory c a n n o t be solved is that twenty-first-century mathematics has n o t yet b e e n discovered.
O n e c o n s e q u e n c e of this formulation is that a physical principle that
unites many smaller physical theories must automatically unite many
seemingly u n r e l a t e d b r a n c h e s of mathematics. This is precisely what
string theory accomplishes. In fact, of all physical theories, string theory
unites by far the largest n u m b e r of b r a n c h e s of mathematics into a single
c o h e r e n t picture. Perhaps o n e of the by-products of the physicists' quest
for unification will be the unification of mathematics as well.
Of course, t h e set of logically consistent mathematical structures is
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many times larger than the set of physical principles. Therefore, some
mathematical structures, such as n u m b e r theory (which some mathematicians claim to be the purest branch of mathematics), have never
b e e n incorporated into any physical theory. Some argue that this situation may always exist: Perhaps the h u m a n m i n d will always be able to
conceive of logically consistent structures that c a n n o t be expressed
t h r o u g h any physical principle. However, there are indications that
string theory may soon incorporate n u m b e r theory into its structure as
well.
Science and Religion
Because the hyperspace theory has o p e n e d up new, profound links
between physics and abstract mathematics, some p e o p l e have accused
scientists of creating a new theology based on mathematics; that is, we
have rejected the mythology of religion, only to embrace an even
stranger religion based on curved space-time, particle symmetries, and
cosmic expansions. While priests may chant incantations in Latin that
hardly anyone understands, physicists chant arcane superstring equations that even fewer understand. T h e "faith" in an all-powerful God is
now replaced by "faith" in q u a n t u m theory and general relativity. When
scientists protest that our mathematical incantations can be checked in
the laboratory, the response is that Creation c a n n o t be measured in the
laboratory, and h e n c e these abstract theories like the superstring can
never be tested.
This debate is n o t new. Historically, scientists have often b e e n asked
to debate the laws of n a t u r e with theologians. For example, the great
British biologist Thomas Huxley was the foremost defender of Darwin's
theory of natural selection against the c h u r c h ' s criticisms in the late
n i n e t e e n t h century. Similarly, q u a n t u m physicists have appeared on
radio debates with representatives of the Catholic C h u r c h concerning
whether the Heisenberg Uncertainty Principle negates free will, a question that may d e t e r m i n e whether o u r souls will enter heaven or hell.
But scientists usually are reluctant to engage in theological debates
about God and Creation. O n e problem, I have found, is that " G o d "
means many things to many people, and the use of loaded words full of
unspoken, h i d d e n symbolism only clouds the issue. To clarify this problem somewhat, I have found it useful to distinguish carefully between
two types of meanings for the word God. It is sometimes helpful to differentiate between the God of Miracles a n d the God of O r d e r .
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W h e n scientists use t h e word God, they usually m e a n the God of
O r d e r . For example, o n e of the most i m p o r t a n t revelations in Einstein's
early c h i l d h o o d took place when he read his first books on science. He
immediately realized that most of what he h a d b e e n t a u g h t a b o u t religion could n o t possibly be true. T h r o u g h o u t his career, however, he
clung to the belief that a mysterious, divine O r d e r existed in the universe. His life's calling, he would say, was to ferret o u t his thoughts, to
d e t e r m i n e whether he h a d any choice in creating the universe. Einstein
repeatedly referred to this God in his writings, fondly calling him " t h e
O l d M a n . " W h e n s t u m p e d with an intractable mathematical p r o b l e m ,
he would often say, " G o d is subtle, b u t not malicious." Most scientists,
it is safe to say, believe that there is s o m e form of cosmic O r d e r in the
universe. However, to the nonscientist, the word God almost universally
refers to the G o d of Miracles, a n d this is the source of miscommunication between scientists a n d nonscientists. T h e G o d of Miracles intervenes
in o u r affairs, performs miracles, destroys wicked cities, smites enemy
armies, drowns t h e P h a r a o h ' s troops, a n d avenges t h e p u r e a n d noble.
If scientists a n d nonscientists fail to c o m m u n i c a t e with each o t h e r
over religious questions, it is because they are talking past each other,
referring to entirely different Gods. This is because the foundation of
science is based on observing r e p r o d u c i b l e events, b u t miracles, by definition, are n o t reproducible. They h a p p e n only once in a lifetime, if at
all. Therefore, the God of Miracles is, in some sense, beyond what we
know as science. This is n o t to say that miracles c a n n o t h a p p e n , only
that they are outside what is commonly called science.
Biologist Edward O. Wilson of Harvard University has puzzled over
this question a n d asked whether t h e r e is any scientific reason why
h u m a n s cling so fiercely to their religion. Even trained scientists, he
found, who are usually perfectly rational a b o u t their scientific specialization, lapse into irrational a r g u m e n t s to defend their religion. Furt h e r m o r e , he observes, religion has b e e n used historically as a cover to
wage hideous wars a n d perform unspeakable atrocities against infidels
a n d h e a t h e n s . T h e sheer ferocity of religious or holy wars, in fact, rivals
the worst crime that any h u m a n has ever committed against any other.
Religion, notes Wilson, is universally found in every h u m a n culture
ever studied on earth. Anthropologists have found that all primitive
tribes have an " o r i g i n " myth that explains where they came from. Furt h e r m o r e , this mythology sharply separates " u s " from " t h e m , " provides
a cohesive (and often irrational) force that preserves the tribe, a n d suppresses divisive criticism of the leader.
This is not an aberration, b u t the n o r m of h u m a n society. Religion,
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Wilson theorizes, is so prevalent because it provided a definite evolutionary advantage for those early h u m a n s who a d o p t e d it. Wilson notes
that animals that h u n t in packs obey the leader because a pecking order
based on strength a n d d o m i n a n c e has b e e n established. But roughly 1
million years ago, when our apelike ancestors gradually became m o r e
intelligent, individuals could rationally begin to question the power of
their leader. Intelligence, by its very nature, questions authority by reason, and h e n c e could be a dangerous, dissipative force on the tribe.
Unless there was a force to counteract this spreading chaos, intelligent
individuals would leave the tribe, the tribe would fall apart, a n d all individuals would eventually die. Thus, according to Wilson, a selection pressure was placed on intelligent apes to suspend reason and blindly obey
the leader a n d his myths, since d o i n g otherwise would challenge the
tribe's cohesion. Survival favored the intelligent ape who could reason
rationally about tools and food gathering, b u t also favored the one who
could suspend that reason when it threatened the tribe's integrity. A
mythology was n e e d e d to define a n d preserve the tribe.
To Wilson, religion was a very powerful, life-preserving force for apes
gradually becoming m o r e intelligent, a n d formed a " g l u e " that held
t h e m together. If correct, this theory would explain why so many religions rely on "faith" over c o m m o n sense, a n d why the flock is asked to
suspend reason. It would also help to explain the i n h u m a n ferocity of
religious wars, and why the God of Miracles always seems to favor the
victor in a bloody war. T h e God of Miracles has one powerful advantage
over the God of O r d e r . T h e God of Miracles explains the mythology of
our purpose in the universe; on this question, the God of O r d e r is silent.
Our Role in Nature
Although the God of O r d e r c a n n o t give humanity a shared destiny or
purpose, what I find personally most astonishing a b o u t this discussion
is that we h u m a n s , who are just beginning o u r ascent up the technological scale, should be capable of making such audacious claims concerning the origin and fate of the universe.
Technologically, we are j u s t beginning to leave the earth's gravitational pull; we have only b e g u n to send crude probes to the outer planets. Yet imprisoned on o u r small planet, with only o u r minds a n d a few
instruments, we have b e e n able to decipher the laws that govern matter
billions of light-years away. With infinitesimally small resources, without
even leaving the solar system, we have b e e n able to d e t e r m i n e what
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h a p p e n s d e e p inside the nuclear furnaces of a star or inside the nucleus
itself.
According to evolution, we are intelligent apes who have only
recently left the trees, living on the third p l a n e t from a m i n o r star, in a
m i n o r spiral a r m of a m i n o r galaxy, in a m i n o r g r o u p of galaxies n e a r
the Virgo supercluster. If t h e inflation theory is correct, t h e n o u r entire
visible universe is b u t an infinitesimal bubble in a m u c h larger cosmos.
Even t h e n , given the almost insignificant role that we play in t h e larger
universe, it seems amazing that we should be capable of m a k i n g t h e
claim to have discovered the theory of everything.
Nobel laureate Isidor I. Rabi was once asked what event in his life
first set him on t h e long j o u r n e y to discover the secrets of n a t u r e . He
replied that it was when he checked o u t some books on t h e planets from
the library. What fascinated him was that the h u m a n m i n d is capable of
knowing such cosmic truths. T h e planets a n d t h e stars are so m u c h larger
t h a n t h e earth, so m u c h m o r e distant t h a n anything ever visited by
h u m a n s , yet the h u m a n m i n d is able to u n d e r s t a n d t h e m .
Physicist Heinz Pagels r e c o u n t e d his pivotal e x p e r i e n c e when, as a
child, he visited the Hayden Planetarium in New York. He recalled,
T h e drama a n d power of the dynamic universe o v e r w h e l m e d m e . I learned
t h a t s i n g l e g a l a x i e s c o n t a i n m o r e stars t h a n all t h e h u m a n b e i n g s w h o h a v e
e v e r l i v e d . . . . T h e reality o f t h e i m m e n s i t y a n d d u r a t i o n o f t h e u n i v e r s e
c a u s e d a kind of 'existential shock' that s h o o k t h e f o u n d a t i o n s of my b e i n g .
Everything that I h a d e x p e r i e n c e d or k n o w n s e e m e d insignificant p l a c e d
i n t h a t vast o c e a n o f e x i s t e n c e .
10
Instead of b e i n g overwhelmed by t h e universe, I t h i n k that p e r h a p s
o n e of the deepest experiences a scientist can have, almost a p p r o a c h i n g
a religious awakening, is to realize that we are children of the stars, a n d
that o u r m i n d s are capable of u n d e r s t a n d i n g t h e universal laws that they
obey. T h e atoms within o u r bodies were forged on the anvil of nucleosynthesis within an exploding star aeons before the birth of t h e solar
system. O u r atoms are older than t h e m o u n t a i n s . We are literally m a d e
of star dust. Now these atoms, in t u r n , have coalesced i n t o intelligent
beings capable of u n d e r s t a n d i n g t h e universal laws governing that
event.
W h a t I find fascinating is that the laws of physics that we have found
on o u r tiny, insignificant planet are the same as the laws found everywhere else in t h e universe, yet these laws were discovered without o u r
ever having left the earth. Without mighty starships or dimensional win-
334
MASTERS
OF
HYPERSPACE
dows, we have been able to d e t e r m i n e the chemical nature of the stars
and decode the nuclear processes that take place d e e p in their cores.
Finally, if ten-dimensional superstring theory is correct, t h e n a civilization thriving on the farthest star will discover precisely the same truth
about o u r universe. It, too, will wonder about the relation between marble and wood, and come to the conclusion that the traditional threedimensional world is " t o o small" to accommodate the known forces in
its world.
O u r curiosity is part of the natural order. Perhaps we as h u m a n s want
to u n d e r s t a n d the universe in the same way that a bird wants to sing. As
the great seventeenth-century a s t r o n o m e r J o h a n n e s Kepler once said,
" W e do n o t ask for what useful purpose the birds do sing, for song is
their pleasure since they were created for singing. Similarly, we o u g h t
not to ask why the h u m a n m i n d troubles to fathom the secrets of the
heavens." Or, as the biologist T h o m a s H. Huxley said in 1863, " T h e
question of all questions for humanity, the p r o b l e m which lies b e h i n d
all others a n d is m o r e interesting than any of t h e m is that of the determination of m a n ' s place in Nature and his relation to the Cosmos."
Cosmologist Stephen Hawking, who has spoken of solving the problem of unification within this century, has written eloquently about the
n e e d to explain to the widest possible audience the essential physical
picture underlying physics:
[If] w e d o d i s c o v e r a c o m p l e t e t h e o r y , i t s h o u l d i n t i m e b e u n d e r s t a n d a b l e
i n b r o a d p r i n c i p l e b y e v e r y o n e , n o t j u s t a f e w scientists. T h e n w e shall all,
p h i l o s o p h e r s , scientists, a n d j u s t o r d i n a r y p e o p l e , b e a b l e t o take part i n
t h e d i s c u s s i o n o f t h e q u e s t i o n o f w h y i t i s t h a t w e a n d t h e u n i v e r s e exist.
I f w e f i n d t h e a n s w e r t o that, i t w o u l d b e t h e u l t i m a t e t r i u m p h o f h u m a n
reason—for then we would know the m i n d of G o d . "
On a cosmic scale, we are still awakening to the larger world a r o u n d
us. Yet the power of even our limited intellect is such that we can abstract
the deepest secrets of nature.
Does this give m e a n i n g or purpose to life?
Some people seek m e a n i n g in life t h r o u g h personal gain, through
personal relationships, or t h r o u g h personal experiences. However, it
seems to me that being blessed with the intellect to divine the ultimate
secrets of nature gives m e a n i n g e n o u g h to life.
Notes
Preface
1. T h e subject is so n e w that there is yet no universally a c c e p t e d term u s e d
by theoretical physicists w h e n referring to h i g h e r - d i m e n s i o n a l theories. T e c h nically speaking, w h e n physicists address t h e theory, they refer to a specific the-
hyperspace
hyper- is t h e
ory, s u c h as K a l u z a - K l e i n t h e o r y , s u p e r g r a v i t y , o r s u p e r s t r i n g , a l t h o u g h
is t h e t e r m p o p u l a r l y u s e d w h e n r e f e r r i n g t o h i g h e r d i m e n s i o n s , a n d
correct
scientific
prefix
for
higher-dimensional
a d h e r e d to popular c u s t o m a n d used the word
geometric
hyperspace
objects.
I
have
to refer to h i g h e r
dimensions.
Chapter I
1. H e i n z P a g e l s , Perfect Symmetry: The Search for the Beginning of Time ( N e w York:
Bantam, 1985), 324.
2. Peter Freund, interview with author, 1990.
3 . Q u o t e d in A b r a h a m Pais, Subtle Is the Lord: The Science and the Life of Albert
Einstein
(Oxford: O x f o r d University Press, 1 9 8 2 ) , 2 3 5 .
4 . T h i s i n c r e d i b l y s m a l l d i s t a n c e will c o n t i n u a l l y r e a p p e a r t h r o u g h o u t t h i s
b o o k . I t i s t h e f u n d a m e n t a l l e n g t h s c a l e t h a t typifies a n y q u a n t u m t h e o r y o f
gravity. T h e r e a s o n for t h i s i s q u i t e s i m p l e . I n a n y t h e o r y o f gravity, t h e s t r e n g t h
of the gravitational force is m e a s u r e d by N e w t o n ' s constant. H o w e v e r , physicists
u s e a simplified set of units w h e r e the s p e e d of light c is set e q u a l to o n e . T h i s
m e a n s that 1 s e c o n d is equivalent to 1 8 6 , 0 0 0 miles. Also, Planck's c o n s t a n t
divided by 2pi is also set equal to o n e , w h i c h sets a n u m e r i c a l relationship b e t w e e n
s e c o n d s a n d ergs of energy. In these strange but c o n v e n i e n t units, everything,
including Newton's constant, can be r e d u c e d to centimeters. W h e n we calculate
the length associated with Newton's constant, it is precisely the Planck length,
or 1 0
- 3 3
centimeter, or 1 0
1 9
b i l l i o n e l e c t r o n volts. T h u s all q u a n t u m g r a v i t a t i o n a l
335
Notes
336
effects are m e a s u r e d i n t e r m s o f this tiny d i s t a n c e . I n particular, t h e size o f t h e s e
u n s e e n higher dimensions is the Planck length.
5. L i n d a D a l r y m p l e H e n d e r s o n , The Fourth Dimension and Non-Euclidean Geom-
etry in Modern Art
( P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r s i t y P r e s s , 1 9 8 3 ) , x i x .
Chapter 2
1. E. T . B e l l , Men of Mathematics ( N e w York: S i m o n a n d S c h u s t e r , 1 9 3 7 ) , 4 8 4 .
2 . Ibid., 4 8 7 . T h i s i n c i d e n t m o s t likely s p a r k e d R i e m a n n ' s early i n t e r e s t i n
n u m b e r t h e o r y . Years later, h e w o u l d m a k e a f a m o u s s p e c u l a t i o n a b o u t a c e r t a i n
formula involving the zeta f u n c t i o n in n u m b e r theory. After 100 years of grapp l i n g with " R i e m a n n ' s h y p o t h e s i s , " t h e w o r l d ' s g r e a t e s t m a t h e m a t i c i a n s h a v e
f a i l e d t o o f f e r a n y proof. O u r m o s t a d v a n c e d c o m p u t e r s h a v e f a i l e d t o g i v e u s a
clue, a n d R i e m a n n ' s hypothesis has n o w g o n e d o w n i n history a s o n e o f the m o s t
f a m o u s u n p r o v e n t h e o r e m s i n n u m b e r t h e o r y , p e r h a p s i n all o f m a t h e m a t i c s .
B e l l n o t e s , " W h o e v e r p r o v e s o r d i s p r o v e s i t will c o v e r h i m s e l f w i t h g l o r y " ( i b i d . ,
488).
3 . J o h n Wallis,
Der Barycentrische Calcul ( L e i p z i g , 1 8 2 7 ) , 1 8 4 .
4. A l t h o u g h R i e m a n n is credited as having b e e n t h e driving creative force
w h o finally s h a t t e r e d t h e c o n f i n e s o f E u c l i d e a n g e o m e t r y , b y r i g h t s , t h e m a n w h o
s h o u l d h a v e d i s c o v e r e d t h e g e o m e t r y o f h i g h e r d i m e n s i o n s was R i e m a n n ' s a g i n g
m e n t o r , G a u s s himself.
I n 1 8 1 7 , a l m o s t a d e c a d e b e f o r e R i e m a n n ' s b i r t h , G a u s s privately e x p r e s s e d
his d e e p frustration with E u c l i d e a n g e o m e t r y . In a p r o p h e t i c letter to his friend
t h e a s t r o n o m e r H e i n r i c h O l b e r s , h e clearly s t a t e d that E u c l i d e a n g e o m e t r y i s
mathematically incomplete.
I n 1 8 6 9 , m a t h e m a t i c i a n J a m e s J . Sylvester r e c o r d e d t h a t G a u s s h a d s e r i o u s l y
c o n s i d e r e d t h e possibility o f h i g h e r - d i m e n s i o n a l s p a c e s . G a u s s i m a g i n e d t h e
p r o p e r t i e s o f b e i n g s , w h i c h h e c a l l e d " b o o k w o r m s , " that c o u l d live e n t i r e l y o n
two-dimensional sheets of paper. He t h e n generalized this c o n c e p t to i n c l u d e
"beings capable of realizing space of four or a greater n u m b e r of d i m e n s i o n s "
( q u o t e d in L i n d a D a l r y m p l e H e n d e r s o n , The Fourth Dimension and Non-Euclidean
Geometry in Modern Art
[ P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r s i t y Press, 1 9 8 3 ] , 1 9 ) .
B u t i f G a u s s was 4 0 years a h e a d o f a n y o n e e l s e i n f o r m u l a t i n g t h e t h e o r y o f
h i g h e r d i m e n s i o n s , t h e n w h y d i d h e m i s s this historic o p p o r t u n i t y t o s h a t t e r t h e
b o n d s of three-dimensional Euclidean geometry? Historians have n o t e d Gauss's
t e n d e n c y t o b e c o n s e r v a t i v e i n h i s work, his p o l i t i c s , a n d h i s p e r s o n a l life. I n fact,
h e n e v e r o n c e left G e r m a n y , a n d s p e n t a l m o s t h i s e n t i r e life i n o n e city. T h i s
a l s o a f f e c t e d h i s p r o f e s s i o n a l life.
In a revealing letter written in 1829, Gauss c o n f e s s e d to his friend Friedrich
B e s s e l that h e w o u l d n e v e r p u b l i s h h i s w o r k o n n o n - E u c l i d e a n g e o m e t r y for fear
o f t h e c o n t r o v e r s y i t w o u l d raise a m o n g t h e " B o e o t i a n s . " M a t h e m a t i c i a n Morris
Kline wrote, " [ G a u s s ] said in a letter to Bessel of January 27, 1829, that he
Notes
337
p r o b a b l y w o u l d n e v e r p u b l i s h h i s f i n d i n g s i n t h i s s u b j e c t b e c a u s e h e f e a r e d ridi c u l e , o r a s h e p u t it, h e f e a r e d t h e c l a m o r o f t h e B o e o t i a n s , a f i g u r a t i v e r e f e r e n c e
to a dull-witted Greek tribe"
(Mathematics and the Physical World [ N e w York: C r o w -
ell, 1 9 5 9 ] , 4 4 9 ) . Gauss was so i n t i m i d a t e d by the o l d guard, the n a r r o w - m i n d e d
" B o e o t i a n s " w h o b e l i e v e d i n t h e sacred nature o f t h r e e d i m e n s i o n s , that h e k e p t
secret s o m e of his finest work.
I n 1 8 6 9 , Sylvester, i n a n i n t e r v i e w w i t h G a u s s ' s b i o g r a p h e r S a r t o r i o u s v o n
W a l t e r s h a u s e n , w r o t e t h a t " t h i s g r e a t m a n u s e d t o say t h a t h e h a d l a i d a s i d e
s e v e r a l q u e s t i o n s w h i c h h e h a d t r e a t e d analytically, a n d h o p e d t o a p p l y t o t h e m
g e o m e t r i c a l m e t h o d s i n a f u t u r e state o f e x i s t e n c e , w h e n h i s c o n c e p t i o n s o f s p a c e
s h o u l d have b e c o m e a m p l i f i e d a n d e x t e n d e d ; for a s w e can c o n c e i v e b e i n g s (like
infinitely a t t e n u a t e d b o o k - w o r m s in an infinitely thin s h e e t of p a p e r ) w h i c h possess o n l y t h e n o t i o n o f space o f two d i m e n s i o n s , s o w e may i m a g i n e b e i n g s capable of realizing space of four or a greater n u m b e r of d i m e n s i o n s " ( q u o t e d in
H e n d e r s o n , Fourth Dimension and Non-Euclidean Geometry in Modern Art, 1 9 ) .
Gauss wrote to Olbers, "I am b e c o m i n g m o r e a n d m o r e c o n v i n c e d that the
(physical) necessity of o u r (Euclidean) g e o m e t r y c a n n o t be proved, at least n o t
b y h u m a n r e a s o n n o r f o r h u m a n r e a s o n . P e r h a p s i n a n o t h e r life w e will b e a b l e
to obtain insight into the nature of space, which is n o w unattainable. Until then,
w e m u s t p l a c e g e o m e t r y n o t i n t h e s a m e class w i t h a r i t h m e t i c , w h i c h i s p u r e l y a
priori, b u t with m e c h a n i c s " ( q u o t e d in Morris Kline,
Mathematical Thought from
Ancient to Modem Times [ N e w York: O x f o r d University Press, 1 9 7 2 ] , 8 7 2 ) .
I n fact, G a u s s w a s s o s u s p i c i o u s o f E u c l i d e a n g e o m e t r y t h a t h e e v e n c o n d u c t e d
a n i n g e n i o u s e x p e r i m e n t t o test it. H e a n d h i s assistants s c a l e d t h r e e m o u n t a i n
peaks: Rocken, H o h e h a g e n , a n d Inselsberg. From each m o u n t a i n peak, the
o t h e r t w o p e a k s w e r e clearly visible. B y d r a w i n g a t r i a n g l e b e t w e e n t h e t h r e e
p e a k s , G a u s s was a b l e t o e x p e r i m e n t a l l y m e a s u r e t h e i n t e r i o r a n g l e s . I f E u c l i d e a n
g e o m e t r y is correct, t h e n the a n g l e s h o u l d have s u m m e d to 180 d e g r e e s . To his
d i s a p p o i n t m e n t , h e f o u n d that the s u m was exactly 180 d e g r e e s (plus o r m i n u s
1 5 m i n u t e s ) . T h e c r u d e n e s s o f his m e a s u r i n g e q u i p m e n t d i d n o t a l l o w h i m t o
conclusively s h o w that Euclid was wrong. (Today, we realize that this e x p e r i m e n t
w o u l d h a v e t o b e p e r f o r m e d b e t w e e n t h r e e d i f f e r e n t star s y s t e m s t o d e t e c t a
sizable deviation f r o m Euclid's result.)
We s h o u l d also p o i n t o u t that the mathematicians Nikolaus I. Lobachevski
a n d J a n o s Bolyai i n d e p e n d e n t l y discovered the
non-Euclidean
mathematics
d e f i n e d o n c u r v e d s u r f a c e s . H o w e v e r , t h e i r c o n s t r u c t i o n was l i m i t e d t o t h e u s u a l
lower dimensions.
5. Q u o t e d i n B e l l ,
Men of Mathematics, 4 9 7 .
6 . T h e B r i t i s h m a t h e m a t i c i a n W i l l i a m Clifford, w h o t r a n s l a t e d R i e m a n n ' s
f a m o u s s p e e c h for
Nature
in 1 8 7 3 , a m p l i f i e d m a n y o f R i e m a n n ' s s e m i n a l i d e a s
a n d w a s p e r h a p s t h e f i r s t t o e x p a n d o n R i e m a n n ' s i d e a that t h e b e n d i n g o f s p a c e
is r e s p o n s i b l e for t h e force of electricity a n d m a g n e t i s m , thus crystallizing Riem a n n ' s work. Clifford s p e c u l a t e d that t h e two mysterious discoveries in m a t h e matics ( h i g h e r - d i m e n s i o n a l spaces) a n d physics (electricity a n d m a g n e t i s m ) are
Notes
338
really t h e s a m e t h i n g , that t h e f o r c e o f e l e c t r i c i t y a n d m a g n e t i s m i s c a u s e d b y
the b e n d i n g of higher-dimensional space.
T h i s i s t h e first t i m e that a n y o n e h a d s p e c u l a t e d that a " f o r c e " i s n o t h i n g b u t
t h e b e n d i n g o f s p a c e itself, p r e c e d i n g E i n s t e i n b y 5 0 years. Clifford's i d e a that
e l e c t r o m a g n e t i s m was c a u s e d b y v i b r a t i o n s i n t h e f o u r t h d i m e n s i o n a l s o p r e c e d e d t h e w o r k o f T h e o d r Kaluza, w h o w o u l d a l s o a t t e m p t t o e x p l a i n e l e c t r o m a g n e t i s m w i t h a h i g h e r d i m e n s i o n . Clifford a n d R i e m a n n t h u s a n t i c i p a t e d t h e
d i s c o v e r i e s o f t h e p i o n e e r s o f t h e t w e n t i e t h c e n t u r y , that t h e m e a n i n g o f h i g h e r d i m e n s i o n a l s p a c e i s i n its ability t o g i v e a s i m p l e a n d e l e g a n t d e s c r i p t i o n o f
forces. For t h e first time, s o m e o n e correctly isolated the true physical m e a n i n g
of h i g h e r d i m e n s i o n s , that a t h e o r y a b o u t space actually g i v e s us a u n i f y i n g p i c t u r e
of forces.
T h e s e p r o p h e t i c views w e r e r e c o r d e d b y m a t h e m a t i c i a n J a m e s Sylvester, w h o
wrote in 1869, "Mr. W. K Clifford has i n d u l g e d in s o m e remarkable s p e c u l a t i o n s
a s t o t h e possibility o f o u r b e i n g a b l e t o infer, f r o m c e r t a i n u n e x p l a i n e d p h e n o m e n a o f l i g h t a n d m a g n e t i s m , t h e fact o f o u r level s p a c e o f t h r e e d i m e n s i o n s
b e i n g in the act of u n d e r g o i n g in space of four d i m e n s i o n s . . . a distortion
a n a l o g o u s t o t h e r u m p l i n g o f a p a g e " ( q u o t e d i n H e n d e r s o n , Fourth Dimension
and Non-Euclidean Geometry in Modern Art,
19).
I n 1 8 7 0 , i n a p a p e r with t h e i n t r i g u i n g title " O n t h e S p a c e - T h e o r y o f M a t t e r , "
h e says e x p l i c i t l y that " t h i s variation o f t h e c u r v a t u r e o f s p a c e i s w h a t really
h a p p e n s i n that p h e n o m e n o n w h i c h w e call t h e motion o f matter, w h e t h e r p o n d e r a b l e o r e t h e r e a l " ( W i l l i a m Clifford, " O n t h e S p a c e - T h e o r y o f M a t t e r , " Proceedings of the Cambridge Philosophical Society 2
[1876]:
157-158).
7. M o r e precisely, in N d i m e n s i o n s the R i e m a n n metric tensor
is an N X
N m a t r i x , w h i c h d e t e r m i n e s t h e d i s t a n c e b e t w e e n t w o p o i n t s , s u c h that t h e infin2
11
i t e s i m a l d i s t a n c e b e t w e e n t w o p o i n t s is g i v e n by ds = S r f x f c , d£. In t h e l i m i t
o f flat s p a c e , t h e R i e m a n n m e t r i c t e n s o r b e c o m e s d i a g o n a l , t h a t is,
= 5 , and
|1V
h e n c e the formalism r e d u c e s back to the Pythagorean T h e o r e m in N - d i m e n s i o n s .
T h e d e v i a t i o n o f t h e m e t r i c t e n s o r f r o m 8^,,, r o u g h l y s p e a k i n g , m e a s u r e s t h e
d e v i a t i o n o f t h e s p a c e f r o m flat s p a c e . F r o m t h e m e t r i c t e n s o r , w e c a n c o n s t r u c t
t h e R i e m a n n c u r v a t u r e t e n s o r , r e p r e s e n t e d b y P?^.
T h e c u r v a t u r e o f s p a c e a t any g i v e n p o i n t c a n b e m e a s u r e d b y d r a w i n g a c i r c l e
a t t h a t p o i n t a n d m e a s u r i n g t h e a r e a i n s i d e that c i r c l e . I n flat t w o - d i m e n s i o n a l
2
space, the area inside the circle is pi r . H o w e v e r , if t h e curvature is positive, as in
2
a s p h e r e , t h e a r e a is less t h a n pi r If t h e c u r v a t u r e is n e g a t i v e , as in a s a d d l e or
2
t r u m p e t , t h e a r e a i s g r e a t e r t h a n pir .
Strictly s p e a k i n g , b y this c o n v e n t i o n , t h e c u r v a t u r e o f a c r u m p l e d s h e e t o f
p a p e r is zero. T h i s is b e c a u s e the areas of circles drawn on this c r u m p l e d s h e e t
2
o f p a p e r still e q u a l p i r . I n R i e m a n n ' s e x a m p l e o f f o r c e c r e a t e d b y t h e c r u m p l i n g
o f a s h e e t o f p a p e r , w e implicitly a s s u m e that t h e p a p e r i s d i s t o r t e d a n d s t r e t c h e d
a s well a s f o l d e d , s o that t h e c u r v a t u r e i s n o n z e r o .
8. Q u o t e d in Bell, Men of Mathematics, 5 0 1 .
9. Ibid., 14.
Notes
339
10. Ibid.
11. In 1917, physicist Paul Ehrenfest, a friend of Einstein, wrote a p a p e r entit l e d " I n W h a t W a y D o e s I t B e c o m e M a n i f e s t i n t h e F u n d a m e n t a l Laws o f P h y s i c s
t h a t S p a c e h a s T h r e e D i m e n s i o n s ? " E h r e n f e s t a s k e d h i m s e l f w h e t h e r t h e stars
a n d planets are possible in h i g h e r d i m e n s i o n s . For e x a m p l e , the light of a c a n d l e
g e t s d i m m e r a s w e m o v e f a r t h e r away f r o m it. Similarly, t h e g r a v i t a t i o n a l p u l l o f
a star g e t s w e a k e r a s w e g o f a r t h e r away. A c c o r d i n g t o N e w t o n , gravity g e t s w e a k e r
b y a n i n v e r s e s q u a r e law. I f w e d o u b l e t h e d i s t a n c e away f r o m a c a n d l e o r star,
t h e light or gravitational pull gets four times weaker. If we triple the d i s t a n c e , it
gets n i n e times weaker.
I f s p a c e w e r e f o u r d i m e n s i o n a l , t h e n c a n d l e l i g h t o r gravity w o u l d g e t w e a k e r
m u c h m o r e rapidly, a s t h e i n v e r s e c u b e . D o u b l i n g t h e d i s t a n c e f r o m a c a n d l e o r
star w o u l d w e a k e n t h e c a n d l e l i g h t o r gravity b y a f a c t o r o f e i g h t .
C a n solar systems exist in such a f o u r - d i m e n s i o n a l world? In principle, yes,
but t h e planets' orbits w o u l d n o t be stable. T h e slightest vibration w o u l d collapse
t h e o r b i t s o f t h e p l a n e t s . O v e r t i m e , all t h e p l a n e t s w o u l d w o b b l e away f r o m t h e i r
usual orbits a n d p l u n g e into t h e sun.
Similarly, t h e s u n w o u l d n o t b e a b l e t o e x i s t i n h i g h e r d i m e n s i o n s . T h e f o r c e
o f gravity t e n d s t o c r u s h t h e s u n . I t b a l a n c e s o u t t h e f o r c e o f f u s i o n , w h i c h t e n d s
to b l o w the s u n apart. T h u s t h e s u n is a delicate b a l a n c i n g act b e t w e e n n u c l e a r
forces that w o u l d cause it to e x p l o d e a n d gravitational forces that w o u l d c o n d e n s e i t d o w n t o a p o i n t . I n a h i g h e r - d i m e n s i o n a l u n i v e r s e , this d e l i c a t e b a l a n c e
w o u l d b e d i s r u p t e d , a n d stars m i g h t s p o n t a n e o u s l y c o l l a p s e .
1 2 . H e n d e r s o n , Fourth Dimension and
Non-Euclidean
Geometry in Modern Art, 2 2 .
13. Zollner h a d b e e n c o n v e r t e d to spiritualism in 1875 w h e n he visited the
laboratory of Crookes, the discoverer of the e l e m e n t thalium, inventor of the
c a t h o d e ray t u b e , a n d e d i t o r of t h e l e a r n e d Quarterly Journal of Science. C r o o k e s ' s
c a t h o d e ray t u b e r e v o l u t i o n i z e d s c i e n c e ; a n y o n e w h o w a t c h e s t e l e v i s i o n , u s e s
a c o m p u t e r m o n i t o r , plays a v i d e o g a m e , o r h a s b e e n x-rayed o w e s a d e b t t o
Crookes's famous invention.
C r o o k e s , i n t u r n , w a s n o c r a n k . I n fact, h e was a l i o n o f British s c i e n t i f i c
s o c i e t y , w i t h a wall full o f p r o f e s s i o n a l h o n o r s . H e w a s k n i g h t e d i n 1 8 9 7 a n d
r e c e i v e d t h e O r d e r o f M e r i t i n 1 9 1 0 . H i s d e e p i n t e r e s t i n s p i r i t u a l i s m was s p a r k e d
by t h e tragic d e a t h of his b r o t h e r Philip of yellow fever in 1867. He b e c a m e a
p r o m i n e n t m e m b e r ( a n d later p r e s i d e n t ) o f t h e S o c i e t y f o r P s y c h i c a l R e s e a r c h ,
w h i c h i n c l u d e d a n a s t o n i s h i n g n u m b e r o f i m p o r t a n t s c i e n t i s t s i n t h e late n i n e t e e n t h century.
14. Q u o t e d in R u d y R u c k e r , The Fourth Dimension ( B o s t o n : H o u g h t o n Mifflin,
1984), 54.
15. T o i m a g i n e h o w k n o t s c a n b e u n r a v e l e d i n d i m e n s i o n s b e y o n d t h r e e ,
i m a g i n e two rings that are intertwined. N o w take a two-dimensional cross s e c t i o n
o f this c o n f i g u r a t i o n , s u c h t h a t o n e r i n g lies o n this p l a n e w h i l e t h e o t h e r r i n g
b e c o m e s a p o i n t ( b e c a u s e i t lies p e r p e n d i c u l a r t o t h e p l a n e ) . W e n o w h a v e a
p o i n t inside a circle. In h i g h e r d i m e n s i o n s , we have t h e f r e e d o m of m o v i n g this
Notes
340
d o t c o m p l e t e l y o u t s i d e t h e c i r c l e w i t h o u t c u t t i n g a n y o f t h e rings. T h i s m e a n s
that t h e two rings have n o w c o m p l e t e l y separated, as desired. This m e a n s that
k n o t s i n d i m e n s i o n s h i g h e r t h a n t h r e e c a n always b e u n t i e d b e c a u s e t h e r e i s
" e n o u g h r o o m . " But also notice that we c a n n o t r e m o v e the d o t f r o m the ring
i f w e a r e i n t h r e e - d i m e n s i o n a l s p a c e , w h i c h i s t h e r e a s o n w h y k n o t s stay k n o t t e d
only in the third d i m e n s i o n .
Chapter 3
1. A. T. Schofield wrote, "We c o n c l u d e , therefore, that a h i g h e r world than
o u r s i s n o t o n l y c o n c e i v a b l y p o s s i b l e , b u t p r o b a b l e ; s e c o n d l y that s u c h a w o r l d
m a y b e c o n s i d e r e d a s a w o r l d o f f o u r d i m e n s i o n s ; a n d thirdly, t h a t t h e spiritual
w o r l d a g r e e s largely i n its m y s t e r i o u s laws . . . w i t h w h a t b y a n a l o g y w o u l d b e t h e
laws, l a n g u a g e , a n d c l a i m s o f a f o u r t h d i m e n s i o n " ( q u o t e d i n R u d y R u c k e r , The
Fourth Dimension [ B o s t o n : H o u g h t o n Mifflin, 1 9 8 4 ] , 5 6 ) .
2. Arthur Willink wrote, " W h e n we have recognized the existence of Space
o f F o u r D i m e n s i o n s t h e r e i s n o g r e a t e r strain c a l l e d for i n t h e r e c o g n i t i o n o f t h e
e x i s t e n c e o f S p a c e o f Five D i m e n s i o n s , a n d s o o n u p t o S p a c e o f a n i n f i n i t e
n u m b e r o f D i m e n s i o n s " ( q u o t e d i n ibid., 2 0 0 ) .
3. H.
G.
Wells,
The
Time Machine:
An
Invention
(London:
Heinemann,
1895), 3.
4. L i n d a D a l r y m p l e H e n d e r s o n ,
The Fourth Dimension and Non-Euclidean Geom-
etry in Modern Art ( P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r s i t y P r e s s , 1 9 8 3 ) , x x i .
5 . Ibid. A c c o r d i n g t o H e n d e r s o n , " [ T ] h e f o u r t h d i m e n s i o n a t t r a c t e d t h e
n o t i c e o f s u c h literary f i g u r e s a s H . G . W e l l s , O s c a r W i l d e , J o s e p h C o n r a d , F o r d
M a d o x Ford, Marcel Proust, and Gertrude Stein. A m o n g musicians, A l e x a n d e r
S c r i a b i n , E d g a r V a r e s e , a n d G e o r g e A n t h e i l w e r e actively c o n c e r n e d with t h e
fourth dimension, and were encouraged to make bold innovations in the n a m e
o f a h i g h e r reality" ( i b i d . , x i x - x x ) .
6. L e n i n ' s Materialism and Empiro-Criticism is i m p o r t a n t t o d a y b e c a u s e it d e e p l y
affected m o d e r n Soviet a n d Eastern E u r o p e a n science. For e x a m p l e , Lenin's
celebrated phrase "the inexhaustibility of the e l e c t r o n " signified the dialectical
n o t i o n that w e f i n d n e w sublayers a n d c o n t r a d i c t i o n s w h e n e v e r w e p r o b e d e e p l y
i n t o t h e h e a r t o f m a t t e r . F o r e x a m p l e , g a l a x i e s are c o m p o s e d o f s m a l l e r star
systems, w h i c h i n turn c o n t a i n p l a n e t s , w h i c h are c o m p o s e d o f m o l e c u l e s , w h i c h
are m a d e o f a t o m s , w h i c h c o n t a i n e l e c t r o n s , w h i c h , i n t u r n , are " i n e x h a u s t i b l e . "
T h i s is a variation of the "worlds within w o r l d s " theory.
7. V l a d i m i r L e n i n ,
Materialism and Empiro-Criticism, in
Karl M a r x , F r i e d r i c h
E n g e l s , a n d V l a d i m i r L e n i n , On Dialectical Materialism ( M o s c o w : P r o g r e s s , 1 9 7 7 ) ,
305-306.
8. Ibid.
9. Q u o t e d in R u c k e r , Fourth Dimension, 6 4 .
1 0 . I m a g i n e a F l a t l a n d e r b u i l d i n g a s e q u e n c e o f six a d j a c e n t s q u a r e s , i n t h e
Notes
341
s h a p e of a cross. To a Flatlander, t h e squares are rigid. T h e y c a n n o t be twisted
or rotated along any of the sides c o n n e c t i n g the squares. N o w imagine, however,
that we grab the squares a n d d e c i d e to fold up the series of squares, f o r m i n g a
c u b e . T h e j o i n t s c o n n e c t i n g t h e squares, w h i c h w e r e rigid i n two d i m e n s i o n s ,
c a n b e e a s i l y f o l d e d i n t h r e e d i m e n s i o n s . I n fact, t h e f o l d i n g o p e r a t i o n c a n b e
p e r f o r m e d s m o o t h l y w i t h o u t a Flatlander e v e n n o t i c i n g that the f o l d i n g is taking
place.
N o w , if a Flatlander were inside the cube, he w o u l d notice a surprising thing.
Each square leads to a n o t h e r square. T h e r e is no " o u t s i d e " to the cube. Each
time a Flatlander m o v e s from o n e square to the next, he s m o o t h l y (without his
k n o w l e d g e ) b e n d s 9 0 d e g r e e s i n the third d i m e n s i o n a n d enters t h e n e x t square.
From the outside, the house is just an ordinary square. However, to s o m e o n e
e n t e r i n g the square, he w o u l d find a bizarre s e q u e n c e of squares, e a c h square
leading impossibly to the n e x t square. To h i m , it w o u l d s e e m impossible that the
interior of a single square c o u l d h o u s e a series of six squares.
Chapter 4
1. J a c o b B r o n o w s k i ,
The Ascent of Man
( B o s t o n : Little, B r o w n , 1 9 7 4 ) , 2 4 7
2 . Q u o t e d i n A b r a h a m Pais, Subtle Is the Lord: The Science and the Life of Albert
Einstein
( O x f o r d : O x f o r d University Press, 1 9 8 2 ) , 1 3 1 .
3 . N o r m a l l y , i t i s a b s u r d t o t h i n k t h a t t w o p e o p l e c a n e a c h b e taller t h a n t h e
other. However, in this situation we have two p e o p l e , e a c h correctly t h i n k i n g
that the o t h e r has b e e n c o m p r e s s e d . This is n o t a true contradiction b e c a u s e it
takes
time
i n w h i c h t o p e r f o r m a m e a s u r e m e n t , a n d t i m e as w e l l as s p a c e h a s
b e e n distorted. In particular, events that a p p e a r s i m u l t a n e o u s in o n e frame are
not simultaneous w h e n viewed in another frame.
F o r e x a m p l e , l e t ' s say that p e o p l e o n t h e p l a t f o r m t a k e o u t a r u l e r a n d , a s t h e
train p a s s e s by, d r o p t h e m e a s u r i n g stick o n t o t h e p l a t f o r m . A s t h e train g o e s by,
t h e y d r o p t h e t w o e n d s o f t h e stick s o t h a t t h e e n d s h i t t h e p l a t f o r m s i m u l t a n e o u s l y . I n t h i s way, t h e y c a n p r o v e t h a t t h e e n t i r e l e n g t h o f t h e c o m p r e s s e d
train, f r o m f r o n t t o b a c k , i s o n l y 1 f o o t l o n g .
N o w c o n s i d e r t h e s a m e m e a s u r i n g p r o c e s s f r o m t h e p o i n t o f v i e w o f t h e pass e n g e r s o n t h e train. T h e y t h i n k t h e y are a t rest a n d s e e t h e c o m p r e s s e d s u b w a y
station c o m i n g toward t h e m , with c o m p r e s s e d p e o p l e a b o u t to d r o p a c o m p r e s s e d ruler o n t o t h e platform. At first it s e e m s i m p o s s i b l e that s u c h a tiny ruler
w o u l d b e a b l e t o m e a s u r e t h e e n t i r e l e n g t h o f t h e train. H o w e v e r , w h e n t h e r u l e r
i s d r o p p e d , t h e e n d s o f t h e r u l e r d o not h i t t h e f l o o r s i m u l t a n e o u s l y . O n e e n d
o f t h e r u l e r h i t s t h e f l o o r j u s t a s t h e s t a t i o n g o e s b y t h e f r o n t e n d o f t h e train.
O n l y w h e n t h e s t a t i o n h a s m o v e d c o m p l e t e l y b y t h e l e n g t h o f t h e e n t i r e train
d o e s t h e s e c o n d e n d o f t h e r u l e r finally h i t t h e f l o o r . I n this way, t h e s a m e r u l e r
h a s m e a s u r e d t h e e n t i r e l e n g t h o f t h e train i n e i t h e r f r a m e .
T h e e s s e n c e o f this " p a r a d o x , " a n d m a n y o t h e r s that a p p e a r i n relativity
342
Note
theory, is that the measuring process takes time, and that both space and time
become distorted in different ways in different frames.
4. Maxwell's equations look like this (we set c = 1):
The second and last lines are actually vector equations representing three equations each. Therefore, there are eight equations in Maxwell's equations.
We can rewrite these equations relativistically. If we introduce the Maxwell
tensor F„ = d^i, - d A^, then these equations reduce to one equation:
v
which is the relativistic version of Maxwell's equations.
5. Quoted in Pais, Subtle Is the Lord, 239.
6. Ibid., 179.
7. Einstein's equations look like this:
where Tuv is the energy-momentum tensor that measures the matter-energy
content, while Ru is the contracted Riemann curvature tensor. This equation
says that the energy-momentum tensor determines the amount of curvature
present in hyperspace.
8. Quoted in Pais, Subtle Is the Lord, 212.
9. Quoted in K. C. Cole, Sympathetic Vibrations: Reflections on Physics as a Way
of Life (New York: Bantam, 1985), 29.
10. A hypersphere can be defined in much the same way as a circle or sphere.
A circle is defined as the set of points that satisfy the equation x + y2 = r in the
x-y plane. A sphere is defined as the set of points that satisfy x + y2 + z = r2 in
x-y-z space. A four-dimensional hypersphere is defined as the set of points that
satisfy x + y + z + u = r in x-y-z-u space. This procedure can easily be
extended to N-dimensional space.
11. Quoted in Abdus Salam, "Overview of Particle Physics," in The New Physics, ed. Paul Davies (Cambridge: Cambridge University Press, 1989), 487.
12. Theodr Kaluza, "Zum Unitatsproblem der Physik," Sitzungsberichte Preussische Akademie der Wissenschaften 96 (1921): 69.
13. In 1914, even before Einstein proposed his theory of general relativity,
v
2
2
2
2
2
2
2
2
2
Notes
343
p h y s i c i s t G u n n a r N o r d s t r o m t r i e d t o u n i f y e l e c t r o m a g n e t i s m w i t h gravity b y
i n t r o d u c i n g a f i v e - d i m e n s i o n a l M a x w e l l t h e o r y . I f o n e e x a m i n e s his t h e o r y , o n e
finds that it correctly c o n t a i n s Maxwell's theory of light in four d i m e n s i o n s , but
i t i s a s c a l a r t h e o r y o f gravity, w h i c h i s k n o w n t o b e i n c o r r e c t . A s a c o n s e q u e n c e ,
N o r d s t r o m ' s ideas w e r e largely f o r g o t t e n . I n s o m e s e n s e , h e p u b l i s h e d t o o s o o n .
H i s p a p e r w a s w r i t t e n 1 y e a r b e f o r e E i n s t e i n ' s t h e o r y o f gravity was p u b l i s h e d ,
a n d h e n c e i t was i m p o s s i b l e f o r h i m t o w r i t e d o w n a f i v e - d i m e n s i o n a l E i n s t e i n type t h e o r y o f gravity.
Kaluza's theory, in contrast to N o r d s t r o m ' s , b e g a n with a metric t e n s o r g ,
d e f i n e d in five-dimensional space. T h e n Kaluza identified
v
with the Maxwell
t e n s o r A^. T h e o l d f o u r - d i m e n s i o n a l E i n s t e i n m e t r i c w a s t h e n i d e n t i f i e d b y Kaluza's n e w m e t r i c o n l y i f p , a n d v d i d n o t e q u a l 5 . I n t h i s s i m p l e b u t e l e g a n t way,
b o t h t h e E i n s t e i n f i e l d a n d t h e M a x w e l l f i e l d w e r e p l a c e d i n s i d e Kaluza's f i v e dimensional metric tensor.
Also, a p p a r e n d y Heinrich M a n d e l a n d Gustav Mie p r o p o s e d five-dimensional
t h e o r i e s . T h u s t h e fact t h a t h i g h e r d i m e n s i o n s w e r e s u c h a d o m i n a n t a s p e c t o f
p o p u l a r culture probably h e l p e d to cross-pollinate t h e world of physics. In this
s e n s e , t h e w o r k o f R i e m a n n w a s c o m i n g full c i r c l e .
14. P e t e r F r e u n d , i n t e r v i e w w i t h a u t h o r , 1 9 9 0 .
15. I b i d .
Chapter 5
1. Q u o t e d in K. C. C o l e , Sympathetic Vibrations: Reflections on Physics as a Way
of Life ( N e w York: B a n t a m , 1 9 8 5 ) , 2 0 4 .
2. Q u o t e d i n N i g e l C a l d e r , The Key to the Universe ( N e w York: P e n g u i n , 1 9 7 7 ) ,
69.
3 . Q u o t e d i n R. P. C r e a s e a n d C. C. M a n n ,
The Second Creation
( N e w York:
Macmillan, 1986), 326.
4. Ibid., 2 9 3 .
5. William Blake, "Tyger! Tyger! b u r n i n g bright," from " S o n g s of Experie n c e , " in
The Poems of William Blake,
ed. W. B.Yeats (London: Routledge, 1905).
6. Q u o t e d in H e i n z P a g e l s , Perfect Symmetry: The Search for the Beginning of Time
( N e w York: B a n t a m , 1 9 8 5 ) , 1 7 7 .
7. Q u o t e d i n C o l e , Sympathetic Vibrations, 2 2 9 .
8. Q u o t e d in J o h n G r i b b e n ,
In Search of Schrodinger's Cat ( N e w York: B a n t a m ,
1 9 8 4 ) , 79.
Chapter 6
1. Q u o t e d i n R. P. C r e a s e a n d C. C. M a n n ,
The Second Creation
( N e w York:
Notes
344
2. Q u o t e d in Nigel Calder,
The Key to the Universe ( N e w York: P e n g u i n , 1 9 7 7 ) ,
15.
3. Q u o t e d in Crease a n d M a n n ,
Second Creation, 4 1 8 .
Bantam, 1985), 327.
5. Q u o t e d i n C r e a s e a n d M a n n ,
6. P e t e r v a n N i e u w e n h u i z e n , " S u p e r g r a v i t y , " i n Supersymmetry and Supergravity, e d . M . J a c o b ( A m s t e r d a m : N o r t h H o l l a n d , 1 9 8 6 ) , 7 9 4 .
7. Q u o t e d i n C r e a s e a n d M a n n ,
Chapter 7
1. Q u o t e d i n K. C. C o l e , " A T h e o r y o f E v e r y t h i n g , "
New York Times Magazine,
18 O c t o b e r 1987, 20.
2 . J o h n H o r g a n , " T h e P i e d P i p e r o f S u p e r s t r i n g s , " Scientific American, N o v e m ber 1991, 42, 44.
3. Q u o t e d in Cole, "Theory of Everything," 25.
4. Edward Witten, Interview, in
Superstrings: A Theory of Everything? e d . P a u l
D a v i e s a n d J . B r o w n ( C a m b r i d g e : C a m b r i d g e U n i v e r s i t y Press, 1 9 8 8 ) , 9 0 - 9 1 .
Superstrings, e d . D a v i e s a n d B r o w n , 1 5 0 .
6. W i t t e n , Interview, i n Superstrings, e d . D a v i e s a n d B r o w n , 9 5 .
5. D a v i d G r o s s , Interview, i n
W i t t e n stresses that E i n s t e i n was l e d t o p o s t u l a t e t h e g e n e r a l t h e o r y o f relativity starting f r o m a physical p r i n c i p l e , t h e e q u i v a l e n c e p r i n c i p l e ( t h a t t h e gravi t a t i o n a l m a s s a n d inertial m a s s o f a n o b j e c t are t h e s a m e , s o that all b o d i e s , n o
m a t t e r h o w l a r g e , fall a t t h e s a m e rate o n t h e e a r t h ) . H o w e v e r , t h e c o u n t e r p a r t
o f t h e e q u i v a l e n c e p r i n c i p l e for s t r i n g t h e o r y h a s n o t yet b e e n f o u n d .
A s W i t t e n p o i n t s o u t , "It's b e e n c l e a r that s t r i n g t h e o r y d o e s , i n fact, g i v e a
logically
consistent
framework,
encompassing
both
gravity
and
quantum
m e c h a n i c s . A t t h e s a m e t i m e , t h e c o n c e p t u a l f r a m e w o r k i n w h i c h this s h o u l d b e
properly u n d e r s t o o d , a n a l o g o u s to the principle of equivalence that Einstein
f o u n d i n h i s t h e o r y o f gravity, h a s n ' t y e t e m e r g e d " (ibid., 9 7 ) .
T h i s is why, a t p r e s e n t , W i t t e n is f o r m u l a t i n g w h a t a r e c a l l e d topological field
theories—that is, t h e o r i e s that are totally i n d e p e n d e n t o f t h e way w e m e a s u r e
d i s t a n c e s . T h e h o p e i s that t h e s e t o p o l o g i c a l field t h e o r i e s m a y c o r r e s p o n d t o
s o m e " u n b r o k e n p h a s e o f s t r i n g t h e o r y " — t h a t is, s t r i n g t h e o r y b e y o n d t h e
Planck length.
7. Gross, Interview, i n
Superstrings, e d . D a v i e s a n d B r o w n , 1 5 0 .
8. Horgan, "Pied Piper of Superstrings," 42.
9 . L e t u s e x a m i n e c o m p a c t i f i c a t i o n i n t e r m s o f t h e full h e t e r o t i c s t r i n g , w h i c h
h a s t w o k i n d s o f vibrations: o n e v i b r a t i n g i n t h e full 2 6 - d i m e n s i o n a l s p a c e - t i m e ,
a n d the o t h e r in the usual ten-dimensional s p a c e time. S i n c e 26 — 10 — 16, we
n o w a s s u m e t h a t 1 6 o f t h e 2 6 d i m e n s i o n s h a v e c u r l e d u p — t h a t is, " c o m -
Notes
345
p a c t i f i e d " i n t o s o m e m a n i f o l d — l e a v i n g us with a t e n - d i m e n s i o n a l theory. Anyo n e w a l k i n g a l o n g a n y o f t h e s e 1 6 d i r e c t i o n s will w i n d u p p r e c i s e l y a t t h e s a m e
spot.
I t w a s P e t e r F r e u n d w h o s u g g e s t e d t h a t t h e s y m m e t r y g r o u p o f this 1 6 - d i m e n s i o n a l c o m p a c t i f i e d s p a c e was t h e g r o u p E ( 8 ) X E ( 8 ) . A q u i c k c h e c k s h o w s t h a t
this s y m m e t r y i s vastly l a r g e r a n d i n c l u d e s t h e s y m m e t r y g r o u p o f t h e S t a n d a r d
Model, given by S U ( 3 ) X S U ( 2 ) X U ( l ) .
In s u m m a r y , t h e key relation is 26 — 10 = 16, w h i c h m e a n s that if we c o m pactify 1 6 o f t h e o r i g i n a l 2 6 d i m e n s i o n s o f t h e h e t e r o t i c s t r i n g , w e are left w i t h
a 16-dimensional c o m p a c t space with a leftover symmetry called E ( 8 )
X E(8).
H o w e v e r , i n K a l u z a - K l e i n t h e o r y , w h e n a p a r t i c l e i s f o r c e d t o live o n a c o m p a c tified s p a c e , i t m u s t n e c e s s a r i l y i n h e r i t t h e s y m m e t r y o f t h a t s p a c e . T h i s m e a n s
that the vibrations of the string must rearrange themselves a c c o r d i n g to the
symmetry group E(8) X E(8).
A s a result, w e c a n c o n c l u d e t h a t g r o u p t h e o r y r e v e a l s t o u s t h a t this g r o u p i s
m u c h larger than the symmetry g r o u p a p p e a r i n g in the Standard Model, a n d
can thus i n c l u d e the Standard M o d e l as a small subset of the ten-dimensional
theory.
10. A l t h o u g h t h e s u p e r g r a v i t y t h e o r y i s d e f i n e d i n 1 1 d i m e n s i o n s , t h e t h e o r y
i s still t o o s m a l l t o a c c o m m o d a t e all p a r t i c l e i n t e r a c t i o n s . T h e l a r g e s t s y m m e t r y
g r o u p for supergravity is 0 ( 8 ) , w h i c h is t o o small to a c c o m m o d a t e the Standard
Model's symmetries.
At first, it appears that the 11-dimensional supergravity has m o r e d i m e n s i o n s ,
a n d h e n c e m o r e s y m m e t r y , t h a n t h e t e n - d i m e n s i o n a l s u p e r s t r i n g . T h i s i s a n illusion because the heterotic string begins by compactifying 26-dimensional space
d o w n to t e n - d i m e n s i o n a l space, leaving us with 16 c o m p a c t i f i e d d i m e n s i o n s ,
which yields the g r o u p E ( 8 ) X E ( 8 ) . This is m o r e than e n o u g h to a c c o m m o d a t e
the Standard Model.
11. Witten, Interview, in
Superstrings,
e d . Davies a n d Brown, 102.
12. N o t e that o t h e r alternative nonperturbative a p p r o a c h e s to string theory
have b e e n p r o p o s e d , b u t they are n o t as a d v a n c e d as string field theory. T h e
m o s t a m b i t i o u s i s " u n i v e r s a l m o d u l i s p a c e , " w h i c h tries t o a n a l y z e t h e p r o p e r t i e s
of string surfaces with an infinite n u m b e r of h o l e s in t h e m . (Unfortunately, no
o n e k n o w s h o w t o calculate with this k i n d o f surface.) A n o t h e r i s the r e n o r m a l i z a t i o n g r o u p m e t h o d , w h i c h c a n s o far r e p r o d u c e o n l y s u r f a c e s w i t h o u t a n y
h o l e s ( t r e e - t y p e d i a g r a m s ) . T h e r e i s a l s o t h e m a t r i x m o d e l s , w h i c h s o far c a n b e
d e f i n e d o n l y i n t w o d i m e n s i o n s o r less.
13. To u n d e r s t a n d this mysterious factor of two, c o n s i d e r a light b e a m that
h a s t w o p h y s i c a l m o d e s o f v i b r a t i o n . P o l a r i z e d l i g h t c a n v i b r a t e , say, e i t h e r h o r i z o n t a l l y o r vertically. H o w e v e r , a relativistic M a x w e l l
field
has four c o m p o -
n e n t s , w h e r e u. = 1,2,3,4. We are a l l o w e d to subtract two of t h e s e four c o m p o nents using the g a u g e symmetry o f Maxwell's equations. Since 4 - 2 = 2 , the
o r i g i n a l f o u r M a x w e l l f i e l d s h a v e b e e n r e d u c e d b y t w o . Similarly, a relativistic
s t r i n g v i b r a t e s i n 2 6 d i m e n s i o n s . H o w e v e r , t w o o f t h e s e vibratory m o d e s c a n b e
Notes
346
r e m o v e d w h e n w e b r e a k t h e s y m m e t r y o f t h e string, l e a v i n g u s w i t h 2 4 vibratory
m o d e s , w h i c h are t h e o n e s t h a t a p p e a r i n t h e R a m a n u j a n f u n c t i o n .
1 4 . Q u o t e d i n G o d f r e y H . H a r d y , Ramanujan ( C a m b r i d g e : C a m b r i d g e U n i versity Press, 1 9 4 0 ) , 3 .
1 5 . Q u o t e d in J a m e s N e w m a n ,
The World of Mathematics ( R e d m o n d , W a s h . :
T e m p u s Books, 1988), 1: 363.
16. H a r d y ,
Ramanujan,
9.
17. I b i d . , 10.
18. Ibid., 11.
1 9 . Ibid., 1 2 .
2 0 . J o n a t h a n B o r w e i n a n d P e t e r B o r w e i n , " R a m a n u j a n a n d P i , " Scientific
American, F e b r u a r y 1 9 8 8 , 1 1 2 .
Chapter 8
1. D a v i d G r o s s , Interview, in Superstrings: A Theory of Everything? ed. P a u l D a v i e s
a n d J . B r o w n ( C a m b r i d g e : C a m b r i d g e U n i v e r s i t y Press, 1 9 8 8 ) , 1 4 7 .
2 . S h e l d o n G l a s h o w , Interactions ( N e w York: W a r n e r , 1 9 8 8 ) , 3 3 5 .
3 . Ibid., 3 3 3 .
4. Ibid., 330.
5. S t e v e n W e i n b e r g , Dreams of a Final Theory ( N e w York: P a n t h e o n , 1 9 9 2 ) ,
218-219.
6. Q u o t e d in J o h n D. B a r r o w a n d Frank J. T i p l e r , The Anthropic Cosmological
Principle ( O x f o r d : O x f o r d U n i v e r s i t y Press, 1 9 8 6 ) , 3 2 7 .
7. Q u o t e d in F. W i l c z e k a n d B. D e v i n e , Longing for the Harmonies ( N e w York:
Norton, 1988), 65.
8. J o h n U p d i k e , " C o s m i c G a l l , " in Telephone Poles and Other Poems ( N e w York:
Knopf, 1960).
9. Q u o t e d in K. C. C o l e , "A T h e o r y of E v e r y t h i n g , " New York Times Magazine,
18 O c t o b e r 1987, 28.
1 0 . Q u o t e d in H e i n z P a g e l s , Perfect Symmetry:
The Search for the Beginning of
Time ( N e w York: B a n t a m , 1 9 8 5 ) , 1 1 .
1 1 . Q u o t e d in K. C. C o l e , Sympathetic Vibrations: Reflections on Physics as a Way
of Life ( N e w York: B a n t a m , 1 9 8 5 ) , 2 2 5 .
Chapter 9
1. Q u o t e d in E. H a r r i s o n , Masks of the Universe ( N e w York: M a c m i l l a n , 1 9 8 5 ) ,
211.
2 . Q u o t e d i n C o r e y S . P o w e l l , " T h e G o l d e n A g e o f C o s m o l o g y , " Scientific
American, J u l y 1 9 9 2 , 17.
Notes
347
3. T h e orbifold theory is actually t h e c r e a t i o n of several individuals, i n c l u d i n g
L. D i x o n , J. Harvey, a n d Edward Witten of P r i n c e t o n .
4. Years a g o , m a t h e m a t i c i a n s asked themselves a simple question: Given a
curved surface in N - d i m e n s i o n a l space, h o w m a n y kinds of vibrations can exist
o n it? F o r e x a m p l e , t h i n k o f p o u r i n g s a n d o n a d r u m . W h e n t h e d r u m i s v i b r a t e d
at a certain frequency, the particles of sands d a n c e on t h e d r u m surface a n d
form beautiful symmetrical patterns. Different patterns of sand particles corres p o n d t o d i f f e r e n t f r e q u e n c i e s a l l o w e d o n t h e d r u m s u r f a c e . Similarly, m a t h e maticians have calculated the n u m b e r a n d kind of resonating vibrations allowed
on the surface of a curved
N-dimensional
surface. T h e y e v e n calculated the
n u m b e r a n d kind of vibrations that an electron c o u l d have on s u c h a hypothetical
surface. To the m a t h e m a t i c i a n s , this was a c u t e intellectual e x e r c i s e . No o n e
t h o u g h t i t c o u l d p o s s i b l y h a v e a n y p h y s i c a l c o n s e q u e n c e . A f t e r all, e l e c t r o n s , t h e y
t h o u g h t , d o n ' t vibrate o n N - d i m e n s i o n a l surfaces.
This large body of mathematical t h e o r e m s can n o w be b r o u g h t to bear on
t h e p r o b l e m o f G U T f a m i l i e s . E a c h G U T family, i f s t r i n g t h e o r y i s c o r r e c t , m u s t
b e a r e f l e c t i o n o f s o m e v i b r a t i o n o n a n o r b i f o l d . S i n c e t h e v a r i o u s k i n d s o f vibrat i o n s h a v e b e e n c a t a l o g e d b y m a t h e m a t i c i a n s , all p h y s i c i s t s h a v e t o d o i s l o o k i n
a m a t h b o o k t o tell t h e m h o w m a n y i d e n t i c a l f a m i l i e s t h e r e a r e ! T h u s t h e o r i g i n
o f t h e f a m i l y p r o b l e m i s topology. I f s t r i n g t h e o r y i s c o r r e c t , t h e o r i g i n o f t h e s e
three duplicate families of G U T particles c a n n o t be u n d e r s t o o d unless we
e x p a n d our consciousness to ten dimensions.
O n c e w e h a v e c u r l e d u p t h e u n w a n t e d d i m e n s i o n s i n t o a tiny ball, w e c a n
t h e n c o m p a r e t h e theory with e x p e r i m e n t a l data. For e x a m p l e , the lowest excit a t i o n o f t h e s t r i n g c o r r e s p o n d s t o a c l o s e d s t r i n g w i t h a very s m a l l r a d i u s . T h e
particles that o c c u r in the vibration of a small c l o s e d string are precisely those
f o u n d i n s u p e r g r a v i t y . T h u s w e retrieve all t h e g o o d r e s u l t s o f s u p e r g r a v i t y , w i t h o u t t h e b a d results. T h e s y m m e t r y g r o u p o f this n e w supergravity i s E ( 8 ) X E ( 8 ) ,
which is m u c h larger than the symmetry of the Standard M o d e l or even the G U T
theory. T h e r e f o r e , the superstring c o n t a i n s b o t h t h e G U T a n d t h e supergravity
theory (without many of the bad features of either theory). Instead of wiping
o u t its rivals, t h e s u p e r s t r i n g s i m p l y e a t s t h e m u p .
T h e p r o b l e m with these orbifolds, however, is that we can construct h u n d r e d s
of thousands of t h e m . We have an embarrassment of riches! Each o n e of t h e m ,
i n p r i n c i p l e , d e s c r i b e s a c o n s i s t e n t u n i v e r s e . H o w d o w e tell w h i c h u n i v e r s e i s
the correct o n e ? A m o n g these t h o u s a n d s of solutions, we find m a n y that predict
exactly three g e n e r a t i o n s or families of quarks a n d leptons. We can also predict
t h o u s a n d s o f solutions w h e r e there are m a n y m o r e than three g e n e r a t i o n s . T h u s
while G U T s c o n s i d e r three g e n e r a t i o n s to be t o o many, m a n y solutions of string
t h e o r y c o n s i d e r t h r e e g e n e r a t i o n s t o b e t o o few!
5. D a v i d G r o s s , I n t e r v i e w , in Superstrings: A Theory of Everything? e d . P a u l D a v i e s
a n d j . Brown (Cambridge: C a m b r i d g e University Press, 1 9 8 8 ) , 1 4 2 - 1 4 3 .
6. I b i d .
Notes
348
Chapter 10
1 . M o r e p r e c i s e l y , t h e Pauli e x c l u s i o n p r i n c i p l e states t h a t n o two e l e c t r o n s
c a n o c c u p y t h e s a m e q u a n t u m state w i t h t h e s a m e q u a n t u m n u m b e r s . T h i s
m e a n s that a w h i t e dwarf c a n be a p p r o x i m a t e d as a F e r m i s e a , or a g a s of e l e c t r o n s o b e y i n g t h e Pauli p r i n c i p l e .
S i n c e e l e c t r o n s c a n n o t b e i n t h e s a m e q u a n t u m state, a n e t r e p u l s i v e f o r c e
p r e v e n t s t h e m f r o m b e i n g c o m p r e s s e d d o w n t o a p o i n t . I n a w h i t e d w a r f star, i t
i s this r e p u l s i v e f o r c e t h a t u l t i m a t e l y c o u n t e r a c t s t h e gravitational f o r c e .
T h e s a m e l o g i c a p p l i e s t o n e u t r o n s i n a n e u t r o n star, s i n c e n e u t r o n s also o b e y
t h e Pauli e x c l u s i o n p r i n c i p l e , a l t h o u g h t h e c a l c u l a t i o n i s m o r e c o m p l i c a t e d
b e c a u s e o f o t h e r n u c l e a r a n d g e n e r a l relativistic effects.
2. J o h n M i c h e l l , in Philosophical Transactions of the Royal Society 74 ( 1 7 8 4 ) : 3 5 .
3. Q u o t e d in H e i n z P a g e l s , Perfect Symmetry: The Search for the beginning of Time
( N e w York: B a n t a m , 1 9 8 5 ) , 5 7 .
Chapter II
1 . Q u o t e d i n A n t h o n y Z e e , Fearful Symmetry ( N e w York: M a c m i l l a n , 1 9 8 6 ) , 6 8 .
2. K. Godel, "An E x a m p l e of a N e w Type of Cosmological Solution of Eins t e i n ' s F i e l d E q u a t i o n s of G r a v i t a t i o n , " Reviews of Modern Physics 21
(1949): 447.
3. F. T i p l e r , "Causality V i o l a t i o n in A s y m p t o t i c a l l y Flat S p a c e - T i m e s , " Physical
Review Utters 37
(1976): 979.
4 . M . S . M o r r i s , K . S . T h o r n e , a n d U . Yurtsever, " W o r m h o l e s , T i m e M a c h i n e s ,
a n d t h e W e a k E n e r g y C o n d i t i o n , " Physical Review Utters 61
(1988): 1446.
5. M. S. Morris and K. S. T h o r n e , " W o r m h o l e s in Spacetime a n d T h e i r U s e
for I n t e r s t e l l a r Travel: A T o o l for T e a c h i n g G e n e r a l Relativity," American Journal
of Physics 56 ( 1 9 8 8 ) : 4 1 1 .
6 . F e r n a n d o E c h e v e r r i a , G u n n a r K l i n k h a m m e r , a n d K i p S . T h o r n e , "Billiard
Balls i n W o r m h o l e S p a c e t i m e s w i t h C l o s e d T i m e l i k e Curves: Classical T h e o r y , "
Physical Review D 44 ( 1 9 9 1 ) :
1079.
7 . Morris, T h o r n e , a n d Yurtsever, " W o r m h o l e s , " 1 4 4 7 .
Chapter 12
1. S t e v e n W e i n b e r g , " T h e C o s m o l o g i c a l C o n s t a n t P r o b l e m , " Reviews of Modern Physics 61 ( 1 9 8 9 ) : 6.
Bantam, 1985), 377.
3 . Ibid., 3 7 8 .
4. Q u o t e d in A l a n L i g h t m a n a n d R o b e r t a Brawer,
Origins: The Lives and
Notes
349
Worlds of Modern Cosmologists (Cambridge, Mass.: Harvard University Press, 1990),
479.
5. Richard Feynman, Interview, in Superstrings: A Theory of Everything? ed. Paul
Davies and J. Brown (Cambridge: Cambridge University Press, 1988), 196.
6. Weinberg, "Cosmological Constant Problem," 7.
7. Quoted in K. C. Cole, Sympathetic Vibrations: Reflections on Physics as a Way
of Life (New York: Bantam, 1985), 204.
8. Quoted in John Gribben, In Search of Schrodinger's Cat (New York: Bantam,
1984), vi.
9. Quoted in Heinz Pagels, The Cosmic Code (New York: Bantam, 1982), 113.
10. Quoted in E. Harrison, Masks of the Universe (New York: Macmillan, 1985),
246.
11. F. Wilczek and B. Devine, Longing for the Harmonies (New York: Norton,
1988), 129.
12. Pagels, Cosmic Code, 155.
13. Quoted in David Freedman, "Parallel Universes: The New Reality—From
Harvard's Wildest Physicist," Discover Magazine, July 1990, 52.
14. Ibid., 48.
15. Ibid., 49.
16. Ibid., 51.
17. Ibid., 48.
Chapter 13
1. Paul Davies, Superforce: The Search for a Grand Unified Theory of Nature (New
York: Simon and Schuster, 1984), 168.
2. Freeman Dyson, Disturbing the Universe (New York: Harper & Row, 1979),
76.
3. Freeman Dyson, Infinite in All Directions (New York: Harper & Row, 1988),
196-197.
4. Dyson, Disturbing the Universe, 212.
5. Carl Sagan, Cosmos (New York: Random House, 1980), 306-307.
6. In fact, aeons ago it was even easier to self-destruct. In order to make an
atomic bomb, the fundamental problem facing any species is to separate uranium-235 from its more abundant twin, uranium-238, which cannot sustain a
chain reaction. Only the uranium-235 will sustain a chain reaction. But uranium235 is only 0.3% of naturally occurring uranium. To sustain a runaway chain
reaction, you need an enrichment level of at least 20%. In fact, weapons-grade
uranium has a 90% or more enrichment rate. (This is the reason why uranium
mines do not suffer from spontaneous nuclear detonations. Because naturally
occurring uranium in a uranium mine is only 0.3% enriched, it contains far too
low a concentration of U-235 to sustain a runaway nuclear chain reaction.)
350
Notes
B e c a u s e u r a n i u m - 2 3 5 i s relatively s h o r t - l i v e d c o m p a r e d w i t h its m o r e a b u n d a n t twin, u r a n i u m - 2 3 8 , a e o n s a g o , t h e naturally o c c u r r i n g e n r i c h m e n t rate i n
o u r u n i v e r s e was m u c h l a r g e r t h a n 0 . 3 % .
I n o t h e r w o r d s , i t was far e a s i e r t h e n for a n y civilization t o f a b r i c a t e a n a t o m i c
b o m b b e c a u s e t h e n a t u r a l l y o c c u r r i n g e n r i c h m e n t rate was m u c h l a r g e r t h a n i t
is t o d a y .
7. H e i n z P a g e l s ,
The Cosmic Code
( N e w York: B a n t a m , 1 9 8 2 ) , 3 0 9 .
8 . S a g a n , Cosmos, 2 3 1 .
9 . Q u o t e d i n M e l i n d a B e c k a n d D a n i e l Glick, " A n d I f t h e C o m e t M i s s e s , "
Newsweek, 2 3 N o v e m b e r 1 9 9 2 , 6 1 .
Chapter 14
1. Q u o t e d i n J o h n D . B a r r o w a n d Frank J. T i p l e r ,
The Anthropic Cosmological
2 . Q u o t e d in H e i n z P a g e l s , Perfect Symmetry: The Search for the Beginning of Time
( N e w York: B a n t a m , 1 9 8 5 ) , 3 8 2 .
3. Ibid., 2 3 4 .
4. Astronomers J o h n D. Barrow of the University of Sussex in E n g l a n d a n d
J o s e p h Silk o f t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y s e e s o m e h o p e i n this
d i s m a l s c e n a r i o . T h e y w r i t e , "If life, i n a n y s h a p e o r f o r m , i s t o survive this
u l t i m a t e e n v i r o n m e n t a l crisis, t h e n t h e u n i v e r s e m u s t satisfy c e r t a i n basic r e q u i r e m e n t s . T h e basic p r e r e q u i s i t e f o r i n t e l l i g e n c e t o survive i s a s o u r c e o f e n e r g y .
" T h e anisotropics in the cosmic e x p a n s i o n , the evaporating black holes, the
r e m n a n t n a k e d s i n g u l a r i t i e s are all life p r e s e r v e r s o f a sort. . . . A n i n f i n i t e
a m o u n t o f i n f o r m a t i o n i s p o t e n t i a l l y available i n a n o p e n u n i v e r s e , a n d its assimi l a t i o n w o u l d b e t h e p r i n c i p a l g o a l o f a n y surviving n o n c o r p o r e a l i n t e l l i g e n c e "
(The Left, Hand of Creation [ N e w York: Basic B o o k s , 1 9 8 3 ] , 2 2 6 ) .
5 . Ibid.
6. G e r a l d F e i n b e r g , Solid Clues ( N e w York: S i m o n a n d S c h u s t e r , 1 9 8 5 ) , 9 5 .
Chapter 15
1. Q u o t e d i n H e i n z P a g e l s ,
The Cosmic Code ( N e w York: B a n t a m B o o k s , 1 9 8 2 ) ,
173-174.
2 . E d w a r d W i t t e n , Interview, i n
1
Superstrings: A Theory of Everything ? e d . P a u l
D a v i e s a n d J . B r o w n ( C a m b r i d g e : C a m b r i d g e U n i v e r s i t y Press, 1 9 8 8 ) , 1 0 2 .
3 . Q u o t e d i n J o h n D . B a r r o w a n d F r a n k J. T i p l e r ,
The Anthropic Cosmological
4 . P a g e l s , Cosmic Code, 3 8 2 .
5. J a m e s Trefil,
The Moment of Creation ( N e w York: M a c m i l l a n , 1 9 8 3 ) , 2 2 0 .
Notes
351
6. John Ellis, Interview, in Superstrings, ed. Davies and Brown, 161.
7. Quoted in R. P. Crease and C. C. Mann, The Second Creation (New York:
8. Quoted in Anthony Zee, Fearful Symmetry (New York: Macmillan, 1986),
122.
9. Ibid., 274.
10. Heinz Pagels, Perfect Symmetry: The Search for the Beginning of Time (New
York: Bantam, 1985), xiii.
11. Stephen Hawking, A Brief History of Time (New York: Bantam, 1988), 175.
References and Suggested Reading
Abbot, E. A. Flatland: A Romance of Many Dimensions. New York: New American
Library, 1984.
Barrow.J. D., and F.J. Tipler. The Anthropic Cosmological Principle. Oxford: Oxford
University Press, 1986.
Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1937.
Calder, N. The Key to the Universe. New York: Penguin, 1977.
Chester, M. Particles. New York: Macmillan, 1978.
Crease, R., and C. Mann. The Second Creation. New York: Macmillan, 1986.
Davies, P. The Forces of Nature. Cambridge: Cambridge University Press, 1979.
Davies, P. Superforce: The Search for a Grand Unified Theory of Nature. New York:
Simon and Schuster, 1984.
Davies, P., and J. Brown, eds. Superstrings: A Theory of Everything? Cambridge:
Cambridge University Press, 1988.
Dyson, F. Disturbing the Universe. New York: Harper & Row, 1979.
Dyson F. Infinite in All Directions. New York: Harper & Row, 1988.
Feinberg, G. Solid Clues. New York: Simon and Schuster, 1985.
Feinberg, G. What Is the World Made Of? New York: Doubleday, 1977.
French, A. P. Einstein: A Centenary Volume. Cambridge, Mass.: Harvard University
Press, 1979.
Gamow, G. The Birth and Death of Our Sun. New York: Viking, 1952.
Glashow, S. L. Interactions. New York: Warner, 1988.
Gribben.J. In Search of Schrodinger's Cat. New York: Bantam, 1984.
Hawking, S. W. A Brief History of Time. New York: Bantam, 1988.
Heisenberg, W. Physics and Beyond. New York: Harper Torchbooks, 1971.
Henderson, L. D. The Fourth Dimension and Non-Euclidean Geometry in Modern Art.
Princeton, N.J.: Princeton University Press, 1983.
Kaku, M. Introduction to Superstrings. New York: Springer-Verlag, 1988.
Kaku, M., and J. Trainer. Beyond Einstein: The Cosmic Quest for the Theory of the
Universe. New York: Bantam, 1987.
Kaufmann, W. J. Black Holes and Warped Space-Time. San Francisco: Freeman,
1979.
353
References and Suggested Reading
354
L e n i n , V. Materialism and Empiro-Criticism. In K. M a r x , F. E n g e l s , a n d V. L e n i n ,
On Dialectical Materialism. M o s c o w : P r o g r e s s ,
1977.
P a g e l s , H. The Cosmic Code. N e w York: B a n t a m , 1 9 8 2 .
P a g e l s , H. Perfect Symmetry: The Search for the Beginning of Time. N e w York: B a n t a m ,
1985.
Pais, A. Subtle Is the Lord: The Science and the Life of Albert Einstein. O x f o r d : O x f o r d
U n i v e r s i t y Press, 1 9 8 2 .
P e n r o s e , R. The Emperor's New Mind. O x f o r d : O x f o r d U n i v e r s i t y Press, 1 9 8 9 .
P o l k i n g h o r n e , J. C.
The Quantum World. P r i n c e t o n , N.J.: P r i n c e t o n U n i v e r s i t y
Press, 1 9 8 4 .
R u c k e r , R.
Geometry, Relativity, and the Fourth Dimension. N e w York: D o v e r , 1 9 7 7 .
R u c k e r , R. The Fourth Dimension. B o s t o n : H o u g h t o n Mifflin, 1 9 8 4 .
S a g a n , C . Cosmos. N e w York: R a n d o m H o u s e , 1 9 8 0 .
Silk, J.
The Big Bang: The Creation and Evolution of the Universe. 2 n d e d . S a n Francisco: F r e e m a n , 1988.
Trefil, J. S. From Atoms to Quarks. N e w York: S c r i b n e r , 1 9 8 0 .
Trefil, J. S. The Moment of Creation. N e w York: M a c m i l l a n , 1 9 8 3 .
W e i n b e r g , S.
The First Three Minutes: A Modern View of the Origin of the Universe.
N e w York: Basic B o o k s , 1 9 8 8 .
W i l c z e k , F., a n d B. D e v i n e . Longing for the Harmonies. N e w York: N o r t o n , 1 9 8 8 .
Z e e , A. Fearful Symmetry. N e w York: M a c m i l l a n , 1 9 8 6 .
Index
Abbot, Edwin, 55-58
Bronowski, Jacob, 81
Alvarez, Luis, 296
B u l l e r , A . H . R., 2 3 3
Alvarez, Walter, 296
B u s h , I a n D . , 186
A n t h e i l , George, 22
Anthropic principle, 257-259
C a p r a , F r i t j h o f , 319
A n t i m a t t e r , 1 2 2 - 1 2 3 , 126
C a r r o l l , Lewis (Charles D o d g s o n ) , 22, 42,
Aristotle, 34
6 2 , 124
A s i m o v , Isaac, 5 , 2 7 9 , 3 1 0
Casimir, H e n r i k , 250
A s k e y , R i c h a r d , 176
Casimir effect, 250
Astrochicken, 2 8 0 - 2 8 1 , 309
Causality, 2 3 4 - 2 3 5
Averaged weak energy c o n d i t i o n ( A W E C ) ,
250-251
Chandrasekhar, Subrahmanyan, 94, 226
Chew, Geoffrey, 324
Aztecs, 2 8 5 - 2 8 6 , 299, 305
Clifford, William, 337n.6
Closed time-like curve ( C T C ) , 240, 248
Banchoff, Thomas, 11
C o l e m a n , Sidney, 266-268
B a r r e t t , S i r W . F., 5 3
C o m p a c t i f i e d d i m e n s i o n , 105, 1 5 8 - 1 5 9
B a r r o w , J o h n D., 306, 3 0 8 - 3 1 0 , 350n.4
C o m p t e , A u g u s t e , 186
Bayeux Tapestry, 6 3 - 6 4
Conrad, Joseph, 22
Bell, E. T, 31
Cosmic Background Explorer (COBE), 1 9 9 -
B i g B a n g t h e o r y , x , 27, 180, 1 9 5 - 1 9 7 , 213,
202
C o s m i c rays, 1 8 4 - 1 8 5
218, 303, 310
B i g C r u n c h , 28, 303, 307
Cosmological constant, 267-268
B i n d i n g energy curve, 218-219
Cosmological p r o o f of G o d , 192-194
B l a c k b o d y r a d i a t i o n , 197
Crookes, W i l l i a m , 50, 3 3 9 n . l 3
Black holes, 22, 2 1 7 - 2 1 8 , 2 2 2 - 2 2 7 , 245,
Curvature, 40
253, 306
B l a k e , W i l l i a m , 124
D a l i , S a l v a d o r , 70
B o h r , Niels, 137, 260
D a r k matter, 304
Bolsheviks, 65, 6 7 - 6 8
D a r w i n , Charles, 28, 1 3 1 , 302
Bolyai.Janos, 377n.4
Davies, Paul, 273
B o n d , Nelson, 75
D e W i t t , B r y c e , 144, 2 6 2
Borges, Jorge Luis, 262
D i r a c , P . A . M . , 112, 1 4 7 , 1 8 9 , 3 2 7
B o r w e i n , J o n a t h a n , 176
D i r k s o n , E v e r e t t , 182
B o r w e i n , P e t e r , 176
D i x o n , L., 3 4 7 n . 3
B o s e , S a t y e n d r a , 144
Dostoyevsky, F y o d o r , 22, 6 5 - 6 7
B o s o n , 144
D o y l e , S i r A r t h u r C o n a n , 167
355
Index
356
Drake, Frank, 283-284
G o d e l , K u r t , 240, 242-243
Duchamp, Marcel, 22
G o l d s m i t h , D o n a l d , 283
Dyson, Freeman, 258, 2 8 0 - 2 8 1 , 285
Grand Unified Theories (GUTs), 131-134,
1 4 3 , 157, 159, 2 0 6 , 2 1 3 , 2 6 7 , 3 0 5 , 3 1 9 ,
Ehrenfest, Paul, 3 3 9 n . l l
E i n s t e i n , A l b e r t , 6 , 1 0 , 13, 1 5 , 7 9 , 8 0 - 1 0 7 ,
325, 347n.4
G r a v i t i n o , 145, 183
1 1 2 - 1 1 3 , 1 3 3 , 138, 142, 154, 1 5 7 , 1 7 7 ,
G r a v i t o n , 1 3 8 - 1 3 9 , 1 5 4 , 183
2 0 1 , 233, 2 4 3 - 2 4 6 , 266, 303, 314,
G r a v i t y , 1 4 - 1 5 , 9 0 - 9 3 , 9 5 , 1 0 0 - 1 0 1 , 126,
327-328, 342nn.7, 13
Einstein-Rosen bridge, 224-226
E l e c t r o m a g n e t i c i n t e r a c t i o n s , 13, 1 0 1 , 122,
125, 3 3 8 n . 6
1 3 8 - 1 3 9 , 1 4 6 - 1 4 8 , 154, 183, 2 5 3 ,
335n.4
G r e e n , M i c h a e l , 16, 1 5 5 , 169
G r o s s , D a v i d , 157, 1 7 8 , 2 0 6 , 3 1 5 - 3 1 6
E l l i s , J o h n , 189, 3 2 6
Grossman, Marcel, 93
Entropy death, 304-305
G u t h , A l a n , 20, 26, 2 0 1 , 259
Equivalence principle, 89
E r i k s o n , Erik, 209
H a l f - l i f e , 134
Escape v e l o c i t y , 2 2 3
Hardy, Godfrey, 174-175
Euclidean geometry, 33, 38
H a r t l e , James, 253
Everett, H u g h , 262
H a r v e y , J e f f r e y , 157, 3 4 7 n . 3
H a w k i n g , S t e p h e n , 147, 235, 2 5 2 - 2 5 4 ,
F a m i l y p r o b l e m , 127, 2 0 6
F a r a d a y , M i c h a e l , 2 5 , 7 9 , 1 0 0 - 1 0 1 , 168,
189
Faraday's Law, 35
F e i n b e r g , G e r a l d , 28, 307-308
F e r m i , E n r i c o , 1 1 8 , 144
267, 334
H e i n l e i n , Robert, 77, 236-237
H e i s e n b e r g , W e r n e r , 1 1 1 , 136, 1 6 6 , 2 6 0 ,
324
H e i s e n b e r g U n c e r t a i n t y P r i n c i p l e , 114,
187
F e r m i o n s , 144
H e n d e r s o n , L i n d a D a l r y m p l e , 22, 62
F e r r a r a , S e r g i o , 145
H e r n q u i s t , L a r s , 299
F e y n m a n , R i c h a r d , 130, 2 5 9
Heterotic string, 158-159, 345n.l0
Feynman diagrams, 119-120, 138-139,
H i g g s b o s o n , 1 2 7 , 183
1 6 6 , 325
F i e l d t h e o r y , 2 3 , 2 5 , 3 9 , 7 9 , 9 3 - 9 4 , 156,
166-168
F l a t l a n d , 4 6 - 4 8 , 7 0 - 7 4 , 106, 1 8 0 - 1 8 1 ,
340n.l0
H i n t o n , Charles, 54, 6 8 - 7 9 , 84, 88
H i n t o n ' s cubes, 6 9 - 7 0
Holism, 318-321
H o r o w i t z , Paul, 282
H u b b l e , E d w i n , 196
F r e e d m a n , D a n i e l , 145
H u b b l e ' s L a w , 196
F r e u n d , Peter, 1 1 - 1 2 , 1 0 4 - 1 0 5 , 345n.9
H u m e , D a v i d , 181
Huxley, Thomas H., 330
Gamow, George, 197-198, 238
H y p e r c u b e , 70, 7 7 - 7 8
Gauss, C a r l F r i e d r i c h , 3 2 , 6 2 , 3 3 6 n . 4
Hyperdoughnut, 96-97
Geller, U r i , 53
Hypersphere, 95, 3 4 2 n . l 0
G e l l - M a n n , M u r r a y , 179
G e n e r a l relativity, 9 1 - 9 5 , 1 0 0 - 1 0 1 , 1 3 8 -
I n f l a t i o n , 201
150, 251
Generation p r o b l e m , 127-128, 206
James, W i l l i a m , 22
G e o r g i , H o w a r d , 140
Jeans, Sir James, 304
Gladsone, William, 25
J o h n s o n , L y n d o n , 1 6 4 , 182
G l a s h o w , S h e l d o n , 1 2 1 , 179
G l u o n s , 15, 1 2 2 - 1 2 3
G o d , 191-193, 330-332
cosmological p r o o f of, 1 9 2 - 1 9 4
Kaluza, T h e o d r , 9 9 - 1 0 0 , 105-107, 338n.6,
343n.l3
K a l u z a - K l e i n t h e o r y , v i i , 8 , 16, 9 9 - 1 0 3 ,
ontological p r o o f of, 193-194
1 4 0 - 1 4 4 , 146, 1 5 4 - 1 5 5 , 169, 2 0 7 , 3 1 3 ,
teleological p r o o f of, 192-194
322, 3 3 5 n . l , 3 4 5 n . 9
Index
Kardashev, N i k o l a i , 277
K e p l e r , J o h a n n e s , 334
357
N e w t o n , Isaac, x i , 8 5 , 115, 1 4 7 , 2 4 2 , 3 2 9 ,
339n.ll
K e r r , Roy, 226
Newton's constant, 335n.4
Kikkawa, Keiji, 162, 166, 207
Non-Euclidean geometry, 3 4 - 3 6
K l e i n , O s k a r , 1 0 6 - 1 0 7 , 144, 207
N o n r e n o r m a l i z a b l e t h e o r y , 126, 150
N o r s t r o m , G u n n a r , 104, 3 4 3 n . l 3
L a w r e n c e , E r n e s t , 184
N U T solution, 244
L e n a r d , P h i l i p , 314
L e n i n , Vladimir, 22, 67-68, 87, 340n.6
Leonardo da Vinci, 64
L e p t o n s , 123, 1 2 7 , 1 4 2 , 1 4 3 , 146, 183
L i t t l e w o o d , J o h n , 175
L o b a c h e v s k i , N i k o l a u s I., 3 3 7 n . 4
L o d g e , Sir O l i v e r , 53
L o v e l a c e , C l a u d e , 168
Ontological p r o o f of God, 193-194
O o r t c l o u d , 297
O p p e n h e i m e r , J . R o b e r t , 112
Orbifolds, 202-204, 206, 2 1 1 , 347nn.3, 4
O s t r i k e r , J e r e m i a h P., 199
O u s p e n s k y , P. D . , 65
O w e n , Tobius, 283
Pagels, H e i n z , 9 , 140, 2 5 9 , 2 8 9 , 333
M a c h , Ernst, 67
M a c h ' s p r i n c i p l e , 9 1 , 242
Mandel, Heinrich, 343n.l3
M a n d e l s t a m , S t a n l e y , 165
Many-worlds theory, 262
M a r s d e n , B r i a n , 294
M a r t i n e c , E m i l , 157
Marx, Karl, 32
M a x w e l l , J a m e s C l e r k , x i , 7 , 8 6 , 1 0 1 , 189,
314
M a x w e l l ' s e q u a t i o n s , 1 0 1 - 1 0 3 , 123, 130,
137, 1 4 2 , 143, 2 7 6 , 3 4 2 n . 4 , 3 4 5 n . l 3
M c D o n a l d , George, 62
M c G o v e r n , G e o r g e , 152
M i c h e l , H e l e n , 296
M i c h e l l . J o h n , 223
Microwave background, 197-200
P a u l i , W o l f g a n g , 1 0 6 - 1 0 7 , 1 3 7 , 187
Pauli exclusion principle, 348n.l
P e n z i a s , A r n o , 197
P e r t u r b a t i o n t h e o r y , 119
Phase t r a n s i t i o n , 2 1 0 - 2 1 4
P h o t o n , 113
Piaget,Jean, 210
Picasso, P a b l o , 6 5
Planck, Max, 88
P l a n c k e n e r g y , 107, 138, 177, 185, 2 6 9
P l a n c k l e n g t h , 16, 269, 3 3 5 n . l
P l a n c k ' s c o n s t a n t , 113, 3 3 5 n . l
P o i n c a r e , H e n r i , 130, 3 2 7
P r o t o n decay, 1 3 3 - 1 3 4
Proust, Marcel, 22
Ptolemy, 34
Pulsar, 220
M i e , Gustav, 3 4 3 n . l 3
M i l l s , R. L., 2 6 , 118
P y t h a g o r e a n T h e o r e m , 37, 338n.7
M i s s i n g mass, 3 0 4
Mobius, August, 51
Q u a n t a , 113
Mobius strip, 6 0 - 6 1 , 96
Q u a n t u m c h r o m o d y n a m i c s ( Q C D ) , 122
Modular functions, 172-173, 176-177
Q u a n t u m e l e c t r o d y n a m i c s ( Q E D ) , 123
More, Henry, 21
Q u a n t u m theory, 112-115
M o r r i s , M i c h a e l , 245
Quarks, 15,122-123,125,142,143,183,213
M u l l e r , R i c h a r d , 297
M u l t i p l y c o n n e c t e d spaces, 1 8
M u o n , 128
b o t t o m q u a r k , 128
c h a r m e d q u a r k , 128
c o l o r e d q u a r k s , 122, 128
f l a v o r e d q u a r k s , 1 2 2 , 128
N a m b u , Y o i c h i r o , 161
s t r a n g e q u a r k , 128
N a n o p o u l o u s , D . V . , 155
s u p e r q u a r k s , 183
N a p p i , C h i a r a , 151
t o p q u a r k , 128
Nemesis theory, 296
R a b i , I s i d o r I., x i i , 333
N e u t r i n o , 125, 128, 1 8 7 - 1 8 8
R a m a n u j a n , Srinivasa, 1 7 2 - 1 7 7
N e u t r o n star, 2 2 0 - 2 2 1 , 3 4 8 n . l
Ramanujan function, 346n.l3
N e w m a n , Ezra, 243
R a u p , D a v i d , 297
Index
358
Red giant, 218
Supernova, 220, 295
R e d s h i f t , 196
Superstrings, viii, 16, 1 5 2 - 1 8 3 , 3 3 5 n . l ,
Reductionism, 318-321
R e i s s n e r - N o r d s t r o m s o l u t i o n , 225
345n.l0
S u p e r s y m m e t r y , 1 4 5 , 183
R e s o n a n c e , 1 4 1 , 153
Susskind, L e o n a r d , 268
Riemann, Georg Bernhard, 22-23, 30-45,
S u z u k i , M a h i k o , 1 6 0 - 1 6 1 , 167, 3 2 5
6 2 , 79, 9 0 - 9 1 , 107, 2 4 3 , 3 2 6 , 329,
336nn.2, 4, 337n.6, 3 4 3 n . l 3
R i e m a n n ' s m e t r i c tensor, 3 9 - 4 1 , 79, 9 3 ,
Swift-Tuttle comet, 294
Symmetry, 86, 1 2 4 - 1 3 0 , 2 0 9 - 2 1 3
Symmetry breaking, 209-213
1 0 1 , 1 4 3 - 1 4 4 , 146, 1 4 7 , 1 4 8 , 3 3 8 n . 7
R o h m , R y a n , 157
T a m b u r i n o , Louis, 243
Russell, B e r t r a n d , 28, 302
Tau lepton, 127-128
R u t h e r f o r d , E r n e s t , 131
Teleological p r o o f of G o d , 192-194
Sagan, Carl, 246, 295, 298
T h e r m o d y n a m i c s , s e c o n d law o f , 3 0 4
S a k i t a , B u n j i , 162
T h o m a s A q u i n a s , 192
S a l a m , A b d u s , 145, 211
T h o m p s o n , J. J., 50
Tesseract, 7 0 - 7 1 , 7 7 - 7 8
Schapiro, Meyer, 65
' t H o o f t , G e r a r d , 1 1 8 - 1 1 9 , 1 2 1 , 148, 3 2 5
Schell, J o n a t h a n , 287
T h o r n e , K i p , 20, 2 4 5 - 2 4 9
S c h e r k , J o e l , 168
T i m e travel, 18-20, 232-251
Schofield, A. T., 55, 3 4 0 n . l
Tipler, Frank, 244, 3 0 8 - 3 1 0
S c h r o d i n g e r , E r w i n , 111
T o w n s e n d , Paul, 149
S c h r o d i n g e r ' s cat, 2 6 0 - 2 6 1
T r a i n e r , J e n n i f e r , x i , x i i , 322
S c h w a r z , J o h n , 16, 155, 157, 1 6 8 - 1 6 9
T r e f i l , J a m e s S., 3 1 9
S c h w a r z s c h i l d , K a r l , 164
T r e i m a n , S a m u e l , 151
S c h w i n g e r , J u l i a n , 137
T u n n e l i n g , 116, 2 0 8
Scriabin, Alexander, 22
T y p e I , I I , I I I civilizations, 277-279, 2 9 0 -
Search f o r extraterrestrial intelligence
292, 301-303
( S E T I ) , 283
Sepkoski.John, 297
U n i f i e d f i e l d t h e o r y , 6 , 9 8 , 112
Sheehy, Gail, 209
U n t i , T h e o d o r e , 243
Silk, J o s e p h , 3 0 6 , 3 5 0 n . 4
U p d i k e , J o h n , 187
Singer, Isadore A., 327
Slade, H e n r y , 49, 52
V a c u u m , false, 2 0 9 , 2 1 1
S l e p t o n , 183
Vafa, C u m r u m , 202
S-matrix t h e o r y , 3 2 4 - 3 2 6
v a n N i e u w e n h u i z e n , P e t e r , 145, 1 4 7 - 1 5 0
Smoot, George, 199-200
van S t o c k u m , W. J . , 244
S n o w , C. P., 3 0 4
V e l t m a n , M a r t i n u s , 119, 148
Space w a r p , 9 0 - 9 2
V e n e z i a n o , G a b r i e l , 1 6 0 - 1 6 1 , 1 6 7 , 170,
Sparnaay, M . J . , 250
325
Special relativity, 8 2 - 8 5
V i r a s o r o , M i g u e l , 162
Spielberg, Steven, 18
v o n F r a u n h o f e r , J o s e p h , 186
S p i n , 144, 150
v o n H e l m h o l t z , H e r m a n n , 10, 4 4 - 4 5 , 3 1 4
Standard M o d e l , 121-127, 131-134, 137,
v o n N e u m a n n , J o h n , 309
150, 1 5 3 , 155, 1 7 0 - 1 7 1 , 2 1 1 , 2 6 7 , 3 1 3 ,
Vranceanu, George, 104-105
319, 345n.9, 347n.4
S t e f a n - B o l t z m a n n law, 197
Stein, Gertrude, 22
Wave f u n c t i o n o f t h e universe, 2 5 4 - 2 5 5 ,
264-265
S t r o n g i n t e r a c t i o n s , 14, 1 1 4 , 1 2 1 , 2 1 3
W b o s o n s , 114, 122
S u p e r c o n d u c t i n g s u p e r c o l l i d e r (SSC), 16,
W e a k i n t e r a c t i o n s , 1 4 , 1 1 4 , 1 2 2 , 196, 2 1 3
1 8 2 - 1 8 5 , 187, 274, 316
S u p e r g r a v i t y , v i i , 16, 1 4 4 - 1 4 8 , 1 5 0 , 1 8 3 ,
3 3 5 n . I , 3 4 5 n . l 0 , 347n.4
Weber, W i l h e l m , 35, 50
W e i n b e r g , S t e v e n , 9 , 1 2 1 , 124, 140, 1 4 8 ,
179, 2 5 9
Index
Weisskopf, Victor, 9 4 , 315
Welles, H. G., 20, 22, 5 9 - 6 1 , 84, 96, 232,
249
Wetherill, George W., 283
White dwarf, 220
Whitehead, Alfred North, 327
Wigner, Eugene, 328
Wilczek, Frank, 2 6 2 - 2 6 3
Wilde, Oscar, 22, 59
Willink, Arthur, 21, 55, 340n.2
Wilson, Edward O., 331
Wilson, Robert, 197
Witten, Edward, 1 5 1 - 1 5 2 , 161, 179, 188,
207, 316, 344n.6, 347n.3
Witten, Louis, 151
Vtorld line, 2 3 7 - 2 3 9
Wormholes, x, xi, 17, 24, 213, 2 2 4 - 2 2 6 ,
2 2 8 - 2 3 1 , 2 4 6 - 2 4 7 , 256, 2 6 6 - 2 6 8
Wulf, T h e o d o r , 184
Wyndham, J o h n , 265
X e n o p h a n e s , 257
Yang, C. N., 26, 118, 129
Yang-Mills field, 26, 118, 121-123, 132,
134, 140, 142, 143, 3 2 5
Yu, Loh-ping, 165
Yukawa, Hideki, 166
Yurtsever, Ulvi, 245
7, boson, 122
Zollner, J o h a n n , 4 9 - 5 3 , 84, 3 3 9 n . l 3
359
ABOUT THE AUTHOR
Michio Kaku is professor of theoretical physics at the City College of
the City University of New York. He graduated from Harvard and
received his Ph.D. from the University of California, Berkeley. He is
author of Beyond Einstein (with Jennifer Trainer), Quantum Field Theory:
A Modern Introduction, and Introduction to Superstrings. He has also
hosted a weekly hour-long science program on radio for the past ten
years.

Michio Kaku`s Hyperspace

Transcription

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