Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimens ion

HYPERSPACE A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension Michio Kaku Illustrations by Robert O'Keefe

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a division of Bantam Doubleday Dell Publishing G r o u p , Inc. 1540 Broadway, New York, New York 10036 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 © 1960 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

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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 a b o u t the universe were correct, t h e n science would have solved the secrets of the universe thousands of years ago. T h e p u r p o s e of science is to peel back the layer of the appearance of objects to reveal their underlying n a t u r e . In fact, if a p p e a r a n c e a n d 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 a b o u t o u r world is that it is three dimensional. It goes without saying that length, width, a n d b r e a d t h suffice to describe all objects in o u r visible universe. Experiments with babies a n d animals have shown that we are b o r n with an innate sense that o u r world is t h r e e 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 p r o b e d , 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 o u r universe may coexist with others, is to invite certain scorn. Yet this deeply ingrained prejudice a b o u t o u r 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 a n d 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 p r o f o u n d conceptual a n d philosophical revolution in o u r 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 1

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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 t h r e e dimensions of space (length, width, a n d b r e a d t h ) 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 a n d 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 n u m ber 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 i m p o r t a n c e 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 c u r r e n t fascinating research on hyperspace. To explain why the hyperspace theory has g e n e r a t e d so m u c h excitem e n t within the world of theoretical physics, I have developed four fund a m e n t a l t h e m e s that r u n t h r o u g h this book like a thread. These four t h e m e s 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 n a t u r e 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 a d d i n g h i g h e r dimensions can simplify physical problems, consider the following example: To the a n c i e n t 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 w h o m 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 a n d 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 o u t e r space, it is clear that the earth's axis is tilted a b o u t 23 degrees from the vertical (the 'vertical" b e i n g the p e r p e n d i c u l a r to the plane of the earth's orbit a r o u n d the s u n ) . 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 d u r i n g o n e part of its orbit t h a n d u r i n g a n o t h e r part. H e n c e we have winter a n d s u m m e r . A n d since

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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

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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.

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In Part IV, we c o n c l u d e with a final, practical question: If the theory is proved correct, t h e n when will we be able to harness the power of the hyperspace theory? This is not j u s t an academic question, because in the past, the harnessing of just o n e of the four fundamental forces irrevocably c h a n g e d the course of h u m a n history, lifting us from the ignorance a n d 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 p r o f o u n d 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 E u r o p e . 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, h o u s e h o l d appliances, the t e l e p h o n e , microwaves, c o n s u m e r electronics, the electronic c o m p u t e r , lasers, a n d many o t h e r 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 a n d 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 a n d 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 m i g h t expect that any civilization that masters the hyperspace theory will b e c o m e 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 a n d time into a pretzel or to create wormholes linking distant parts of o u r universe. Unfortunately, the results are disappointing. T h e energy required far exceeds anything that o u r planet can muster. In fact, the energy is a quadrillion times larger than the energy of o u r largest a t o m smashers. We must wait centuries or even millennia until o u r civilization develops the technical capability of manipulating s p a c e - t i m e , or h o p e for contact with an advanced civilization that has already mastered hyperspace. T h e b o o k 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 n o r m a l , c o m m o n -

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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 w h o 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, a n d q u a n t u m field theory. This book, however, marks the first popular scie n c e 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 J e n n i f e r Trainer, who has b e e n my coaut h o r on two previous popular books. O n c e again, she has applied h e r considerable skills to make the presentation as s m o o t h 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 o t h e r individuals who have h e l p e d to strengthen and criticize earlier drafts of this book: Burt Solomon, Leslie Meredith, Eugene Mallove, a n d my agent, Stuart Krichevsky. Finally, I would like to t h a n k 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 a b o u t the revolutionary developments that have e x t e n d e d a n d embellished m u c h of his p i o n e e r i n g work.

Contents

Part I Entering the Fifth Dimension

1. Worlds Beyond Space a n d Time, 3 2. Mathematicians a n d Mystics, 30 3. T h e Man W h o " S a w " the Fourth Dimension, 55 T h e Secret of Light: Vibrations in the Fifth Dimension,

Part II Unification in Ten Dimensions

5. Q u a n t u m Heresy, 111 6. Einstein's Revenge, 136 7. Superstrings, 151 8. Signals from the T e n t h 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 t h a t 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

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WO incidents from my childhood greatly e n r i c h e d 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 G a r d e n in San Francisco. O n e of my happiest childhood m e m o r i e s is of c r o u c h i n g 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

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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 i m m e n s e chasm. T h e carp a n d I spent o u r lives in two distinct universes, never e n t e r i n g each o t h e r ' s world, yet were separated by only the thinnest barrier, the water's surface. I o n c e imagined that there may be carp "scientists" living a m o n g the fish. They would, I thought, scoff at any fish who p r o p o s e d 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 u n s e e n 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 p u s h i n g 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. U n a b l e to c o m p r e h 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, t h e r e 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 a p p e a r that a miracle h a d h a p pened. After collecting his wits, the "scientist" would tell a truly amazing story. " W i t h o u t w a r n i n g , " 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

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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 c a n n o t 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 h i g h e r dimensions because they c a n n o t 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 w o n d e r i n g 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 F o u n d a t i o n 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

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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 G a r d e n 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 c o o p e r wire on the high-school football field.

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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 h i g h e r mathematics of each of the forces of n a t u r e separately, in complete isolation from o n e a n o t h e r . I still r e m e m b 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 h i g h e r 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, discove r e d 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 " f o r c e s " to hide o u r ignorance of how objects can move each o t h e r without touching.

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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 t h e r e 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, e x p e r i m e n t s 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 s t u d e n t 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. T h e y 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 d o i n g , 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.

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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 t h o u s a n d 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 h i g h e r 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 ( a n d 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 t h r e e 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

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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 t h r e e 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 t h r e e 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 " f o r w a r d " 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.

Worlds Beyond Space and Time

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, c h a i r m a n 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 p r o g r a m s that allow us to m a n i p u l a t e higher-dimensional objects by projecting their shadows o n t o flat, twodimensional c o m p u t e r screens. Like the G r e e k 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 h a n d l e myriad emergencies in t h r e e 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 t h r e e dimensions h a d a distinct survival advantage over those who could not. Unfortunately, t h e r e 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.

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12

Traditionally, F r e u n d r e m i n d s us, scientists have viewed h i g h e r 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 d e c a d e 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 h i g h e r 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 h a s 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 c h e e t a h may be hyperspace.

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To illustrate how a d d i n g a h i g h e r 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. R o m e 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 h i g h e r 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 , p e r h a p s 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 a n o t h e r .

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 o n e , 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 o n e of the pivotal scientific discoveries in all human history.

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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 P e n t a g o n 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 d a r k e n , 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

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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 e l u d e d 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 o n c e reveal himself because of his e n o r m o u s s i z e . " 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 g o n e 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-

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o m i n g the n a t u r e of geometry. Now, scientists are realizing that the "useless" concepts of space a n d time may be the ultimate source of beauty a n d 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 h a d 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 a n d 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 a t o m smashers. Over the past 30 years, h u n d r e d s of subatomic particles have b e e n carefully cataloged a n d studied by physicists a m o n g the debris created by smashing together p r o t o n s 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 a n d 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 t h e s e 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 Congress i n O c t o b e r 1993) m i g h t have 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.

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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 s o m e t h i n g 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 u n s e e n 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 h i g h e r u n s e e n dimension, the third dimension. Although weary from his travels, Columbus discovers that t h e r e 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 d e m o n s t r a t e 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 t h r e e 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, p e r f o r m e d on the path of light beams, show that starlight is b e n t as it moves across the universe.

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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 c o n n e c t i n g 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 G e o r g 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 o n c e 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 m o d e l 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. W o r m h o l e s 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

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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 o p i n i o n 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-

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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 m a i n 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 p r e s e n t 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 " b a b y 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 c o n n e c t i n g 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-

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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 t h e r e 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 h i g h e r 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 n e a r 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 h i g h e r 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 t h r e e parallel planes: the earth, heaven, a n d hell. God himself, according to the theologian A r t h u r Willink, found his h o m e in a world far removed from these t h r e e planes; he lived in infinite-dimensional space. Interest in h i g h e r 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

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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 T h e o r y 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 G e o r g Bernh a r d R i e m a n n , who was the first to lay the mathematical foundation of geometries in higher-dimensional space. R i e m a n n c h a n g e d the course of mathematics for the next century by d e m o n s t r a t i n g 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 a n o t h e r . 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, t h e n , would n o t be alone, b u t would be o n e of

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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 a n d , 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 h i g h e r 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 h i g h e r 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

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Field Theory: The Language of Physics Fields were first i n t r o d u c e d 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 e x p e r i m e n t s 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 t h e r e 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 .

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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 a t o m smasher. Twisting s p a c e - t i m e 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 t e m p e r a t u r e s 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

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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 a t o m 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 h i g h e r 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 d e a t h of the universe will c o m e in the form of the Big Crunch. Gravitation will reverse the cosmic expan-

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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, t h e r e is o n e , a n d p e r h a p s 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 C r u n c h . 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 o n c e again, a n d interdimensional travel will b e c o m e possible. As all m a t t e r is crushed in the final m o m e n t s before doomsday, intelligent life forms may be able to t u n n e l into higher-dimensional space or an alternative universe, avoiding the seemingly inevitable d e a t h 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, t e m p e r a t u r e s 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 knowle d g e 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-

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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 p e r h a p s 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, R i e m a n n 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 h i g h e r dimensions was i n t r o d u c e d 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 F o u n d a t i o n 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 t h r e e dimensional, came tumbling down as a new R i e m a n n i a n 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 R i e m a n n i a n geometry to explain the creation of the universe a n d its evolution. And 130 years after his lecture, physicists 30

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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 p r o o f 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

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R i e m a n n a p o n d e r o u s b o o k to k e e p h i m occupied. T h e b o o k was Adrien-Marie L e g e n d r e ' 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 y o u n g 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 R i e m a n n answered perfectly. 2

Beset by the daily struggle to p u t food on the table, R i e m a n n ' s father m i g h t 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 r e n o w n e d 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 t h r e e 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 R i e m a n n , 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, R i e m a n n ' s studies were interrupted. He was inducted into the s t u d e n t 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 R i e m a n n ' 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-

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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, p e r h a p s 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 E u r o p e 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. M o u n t a i n 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, b r e a d t h , 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-

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oed by the p h i l o s o p h e r 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 t h r e e 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 t h r e e are all." F u r t h e r m o r e , in A . D . 150, the a s t r o n o m e r 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 t h r e e mutually perpendicular lines. For example, the c o r n e r of a cube consists of three mutually p e r p e n d i c u l a r lines. T h e n , he argued, try to draw a fourth line that is p e r p e n d i c u l a r to the o t h e r three lines. No matter how o n e tries, he reasoned, four mutually p e r p e n d i c u l a r 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 C h i m e r a or C e n t a u r e . . . . Length, Breadth, a n d Thickness, take up the whole of Space. N o r can Fansie imagine how t h e r e should be a Fourth Local Dimension beyond these T h r e e . " For several t h o u s a n d years, mathematicians would repeat this simple b u t fatal mistake, that the fourth dimension c a n n o t exist because we c a n n o t 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 R i e m a n n to p r e p a r e an oral presentation on the " f o u n d a t i o n of geometry." Gauss was keenly interested in seeing if his s t u d e n t 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-

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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 m a n , terrified of public speaking, was being asked by his m e n t o r to p r e p a r e 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 , R i e m a n n 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 R i e m a n n , 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 " f o u n d a t i o n 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

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R i e m a n n 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 c h a n g e the direction of a n o t h e r without even t o u c h i n g it. R i e m a n n developed a radically new physical picture. Like Gauss's " b o o k w o r m s , " R i e m a n n 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? R i e m a n n 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 a r g u e d 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 t h e m from moving in a straight line. T h e y 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 R i e m a n n , "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 s o m e t h i n g 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. R i e m a n n concluded that electricity, magnetism, a n d gravity are caused by the c r u m p l i n g 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

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By introducing the fourth spatial dimension, R i e m a n n 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 R i e m a n n 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 R i e m a n n ' 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 t h r e e 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 t h r e e 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

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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

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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 p a p e r , you can mathematically describe the properties of higher-dimensional objects that c a n n o t be visualized by o u r brains.) R i e m a n n 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 R i e m a n n 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-

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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 c r u m p l i n g of the sheet. No matter how c r u m p l e d the sheet of p a p e r , 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

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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, t h e r e 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.) R i e m a n n 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 b u g 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

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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 n o r m a l ; b u t when he tries to take a short-cut t h r o u g h the cut, he has a p r o b l e m . 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) c o n n e c t i n g two spaces. R i e m a n n ' 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 m a p p i n g . 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 c r u m p l e . 8

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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, R i e m a n n 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 n a t u r e ; that is, to him, electricity a n d magnetism as well as gravity were just effects caused by the c r u m p l i n g 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. R i e m a n n 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 . R i e m a n n , 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 y o u n g 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

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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, p e r h a p s the most famous G e r m a n physicist of his generation, was deeply affected by R i e m a n n ' 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-

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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 " s u c h 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 R i e m a n n ' 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). T h u s 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 u n a t t a c h e d 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 R i e m a n n 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

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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-

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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-

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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.

l a n d e r 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

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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 g o n e 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

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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 R i e m a n n (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 r o p e 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

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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 d e m o n s t r a t e 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 e x p e r i m e n t s 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

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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 p e o p l e 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 a s 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 i s 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 w o u l d 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,

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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 c o m m u n i t y 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 h i g h e r dimensions. As a c o n s e q u e n c e , the theory of h i g h e r 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 influe n c e of the fourth dimension would c o m e full circle a n d cross-pollinate the world of physics o n c e 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 d e c a d e of acrimonious debate, clergyman Edwin Abbot, h e a d m a s t e r 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 h i g h e r 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 d o d g e d 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

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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 h i g h e r 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 p e r m a n e n t l y t u r n e d upside down when he is visited by a mysterious Lord Sphere, a threedimensional sphere. Lord S p h e r e appears to Mr. Square as a circle that can magically c h a n g e 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 S p h e r e 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

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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 c h a n g e 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 h u r l e d 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 a p p e a r from n o w h e r e 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

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the differences in how these blobs change. Each person in h i g h e r 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 l a m p o o n s 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 a d o p t i n g 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."

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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. T h e y 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 p r o o f 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

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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 d i m e n s i o n s . " 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 e n a b l e an individual to

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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 m i r r o r 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 G o l d e n Years of the Fourth Dimension. It was a time d u r i n g which the ideas originated by Gauss a n d R i e m a n n 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 d e v e l o p m e n t because the rigorous results of R i e m a n n were now b e i n g 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. N o t 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

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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-

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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.) H a r o l d 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 b e g a n . 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

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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 t h o u g h 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 o n c e 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 t h o u g h 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 o n c e , 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 i n t r o d u c e d 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)

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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 n e x t 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 a r g u e d 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 t h e m t h r o u g h religion a n d spiritualism. To bolster their heretical views, the God-builders q u o t e d 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 e m b r a c e 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 . T h e y 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 t h r e e 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 p o p u larizing 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 H i n t o n , 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, b o r i n g 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 d e g r e e 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, r e p r i n t e d 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 M a u d e Weldon. T h e headmaster at U p p i n g h a m , noticing H i n t o n in the presence of his wife, Mary, a n d his mistress, Maude, h a d assumed that M a u d e 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 U n i t e d 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 m a c h i n e , 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 i m p o r t a n c e . 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 h o u r s contemplating a n d meditating on these cubes, until they attained the ability to mentally r e a r r a n g e 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 t h r e e dimensions, so that we have a series of six squares making a cross. Of course, a Flatlander c a n n o t reassemble the squares to m a k e a cube. In the second dimension, the j o i n t s 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 a r r a n g e d 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 c u b e 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 c u b e 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 s e c o n d 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. 71

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. 73

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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, c h a n g e 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 t h r e e m e t h o d s : by e x a m i n i n g 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 c o u n t e r p a r t s 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 m o t i o n 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 t h e r e 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 u n s e e n fourth dimension, which is essentially the viewpoint taken today by many theoretical physicists.) In the U n i t e d 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 D i m e n s i o n . " 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 r e a d e r 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 h i g h e r dimension? Perhaps the best way to explain the w o n d e r 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 h o o p l a 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 r e m a i n i n g 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 r e a d 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 a p p e a r 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 p o k e d 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 newspaperm a n , against the rules, takes flash pictures of the creature. T h e flash enrages the creature, which t h e n struggles so h a r d 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 d o o r s 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 c u b e 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 I f [relativity] s h o u l d p r o v e t o b e c o r r e c t , a s I e x p e c t i t will, h e 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 c o m p l a i n e d 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." 80

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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 l a n d e d 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 h i m . 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 a p p e a r s 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 o s c i l l a t o r y 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 -

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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 h a r d 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 a h e a d of us at the same velocity. In o t h e r words, a light b e a m is like the " g h o s t 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 b e e n 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 p r o f o u n d 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 h o u r , 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 a h e a d 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 h o u r . 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 h o u r . 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 a h e a d of us at 1 mile per h o u r , as expected, b u t speeding a h e a d 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 b e a m . 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 a p p r o a c h b u t never exceed 100 miles p e r h o u r . 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 w e r e n ' t moving at all. But this is absurd. How can b o t h p e o p l e 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 b e c o m e . 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 n e x t 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 c o m i n g 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 t h r e e 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. T h u s a d d i n g a h i g h e r 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 u n t h i n k a b l e . 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, t h e n space a n d time can rotate into each o t h e r a n d clocks can beat at different rates, d e p e n d ing on how fast they move. This classmate, for example, may have e n t e r e d 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 j u s t e n t e r e d the rocket, sped into o u t e r space for a few minutes, a n d t h e n l a n d e d back on earth in time for his twentieth high-school r e u n i o n after a short, pleasant j o u r n e y , 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 n a t u r e 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 o p a q u e . Maxwell's eight equations are clumsy a n d difficult to memorize because time a n d 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 a n d 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 h i g h e r symmetry; that is, space a n d 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, a n d the cornucopia of c o n s u m e r electronics that a p p e a r 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 n a t u r e .

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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 n e x t 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 o n c e 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 n o t h i n g . 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

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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 h i g h e r 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 h i g h e r dimensions have a purpose: to unify the principles of n a t u r e . By a d d i n g h i g h e r dimensions, he could unite physical concepts that, in a three-dimensional world, have no connection, such as m a t t e r 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 t h e r e was s o m e t h i n g missing. Einstein's key insight was to use the fourth dimension to unite the laws of n a t u r e by i n t r o d u c i n g 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 t h e r e 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 f o u n d e r of the q u a n t u m theory, advised the y o u n g 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, p l u n g e d 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 r o p e 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 5

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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 o n c e recalled, "I was sitting in a chair in the p a t e n t office at Bern when all of a s u d d e n a t h o u g h t occurred to m e : '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 t h e r e 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 o n c e 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 e x p e r i e n c e d 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

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time it takes for the light b e a m to move across the r o o m ) . Therefore, a r g u e d 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 c h a n g e 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 R i e m a n n , the b e n d i n g or warping of space causes the appearance of a force. T h u s 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 R i e m a n n ' s a p p r o a c h , however, was that he no idea specifically how gravity or electricity and magnetism caused warping of space. His a p p r o a c h 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 b e d s h e e t 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 t h e r e is an "instantaneous f o r c e " e m a n a t i n g 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 b e e n 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 c h a n g e 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 s u r r o u n d ing it. This is the essence of the physical principle t h a t R i e m a n n 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 m i n d . 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.

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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 R i e m a n n , 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 R i e m a n n 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. . . . R i e m a n n ' s achievement was the greatest o n e . " To his shock, Einstein found R i e m a n n ' s celebrated 1854 lecture to be the key to the p r o b l e m . He found that he could incorporate the entire body of R i e m a n n ' 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 R i e m a n n ' s famous 1854 lecture is now called general relativity, a n d Einstein's field equations rank a m o n g the most p r o f o u n d ideas in scientific history. 2

R i e m a n n ' s great contribution, we recall, was that he i n t r o d u c e d 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

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page of R i e m a n n ' 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 t h e r e 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 t h r o u g h the engine a n d so on. And then the peasant says: 'Yes, I understand all that, but where is the horse?' T h a t ' 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 s p a c e - t i m e in any dimension. However, he lacked the physical picture (that m a t t e r - e n e r g y d e t e r m i n e s 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 o n c e a t t e n d e d a hockey game in Boston. All the action, of course, was c o n c e n t r a t e d 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

\

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way or a n o t h e r . 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 m o t i o n of matter on its surface. A n o t h e r , 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 a h e a d , 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. T h u s 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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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 s o m e o n e ' s body. By t u r n i n g o u r heads the o t h e r way, we see s o m e o n e ' s left side. No matter how fast we t u r n o u r heads, the people a h e a d 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, t h e r e is a n o t h e r infinite sequence of people standing before you, a r r a n g e d in a straight line, all h o l d i n g 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, d o i n g exactly what we d o . 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 P l a n n e r 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 a h e a d of him. In fact, we would see an infinite chain of p e o p l e 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

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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 b e e n 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.

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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 R i e m a n n in several ways. Like R i e m a n n 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 i n d e e d 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 b e e n 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 " m a r b 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,

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a n d stars. But in the 1920s a n d 1930s, when Einstein was actively working on the unified field theory, the true n a t u r e of matter r e m a i n e d an unsolved mystery. Einstein's g r a n d strategy was to turn wood into m a r b l e — t h a t 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 u n d e r s t a n d i n g of the wood, this was impossible. By analogy, think of a magnificent, gnarled tree growing in the m i d d l e 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 a n d roots e m a n a t i n g from the tree. To paraphrase Mach's principle: T h e presence of the tree d e t e r m i n e s the pattern of the marble s u r r o u n d i n g it. But Einstein h a t e d this dichotomy between wood, which seemed to be ugly a n d complicated, a n d 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 n a t u r e simplify a n d unify in h i g h e r dimensions. Einstein correctly applied this principle twice, in special a n d general relativity. However, on his third try, he a b a n d o n e d this fundamental principle. Very little was known a b o u t 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 o t h e r words, everything we see a r o u n d us, from the trees a n d clouds to the stars in the heavens, was probably an illusion, some form of c r u m p l i n g 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 n e x t 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 u n k n o w n mathematician, T h e o d r Kaluza, at the University of Konigsberg in Germany, in what is Kaliningrad in the former Soviet U n i o n . 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 i n t r o d u c i n g 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 p r o p o s i n g 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 t h e m 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 u n k n o w n scientist was proposing to c o m b i n e , 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 n a t u r e ; 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 p e r m e a t e s the universe a n d binds the solar system. (Along with Weber a n d R i e m a n n , 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-

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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 u n c h a n g e able 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 a n d gravity are like oil a n d 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 a n d compelling that Einstein could n o t reject it. 1 1

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 a n d time from four to five. This was because, as we recall, t h e r e was no experimental evidence for the fourth spatial dimension. What astonished Einstein was that o n c e the fivedimensional field theory was b r o k e n down to a four-dimensional field theory, both Maxwell's a n d Einstein's equations r e m a i n e d . 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 b o t h of t h e m were part of a larger whole, a five-dimensional space. " L i g h t " was e m e r g i n g as the warping of the geometry of higherdimensional space. This was the theory that seemed to fulfill R i e m a n n ' 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 i m p o r t a n t theories up to that time, possessed "virtually unsurpassed formal unity." He furthermore insisted that the sheer simplicity a n d 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 a n d simplicity of the article. Like all great ideas, Kaluza's essential a r g u m e n t was elegant a n d compact. 12

T h e analogy with piecing together the parts of a jigsaw puzzle is a meaningful o n e . Recall that the basis of R i e m a n n ' 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 a r r a n g e d 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

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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.

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The Secret of Light: Vibrations in the Fifth Dimension 103 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 b o t h 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 b o t h 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 i m p o r t a n t 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 P r o b l e m of Physics." In the history of physics, no o n e h a d found any use for the fourth spatial dimension. Ever since R i e m a n n , 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 b o t h 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 p r e c e d e n t . 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 a n d 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 h i g h e r dimensions was p e r h a p s the single most p o p u l a r 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 R i e m a n n pollinated the world of arts a n d letters via H i n t o n a n d 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 b o t h Einstein's a n d Maxwell's theories. T h e fact that b o t h Kaluza a n d N o r d s t r o m i n d e p e n d e n t l y tried to exploit the fifth dimension indicates that the concepts widely circulating within p o p u l a r 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 a n d 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 h a d 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, a n d 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 h e a r i n g V r a n c e a n u discuss the i m p o r t a n t question: Why should light a n d 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 a n d 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

The Secret of Light: Vibrations in the Fifth Dimension 105 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, " W h a t a b o u t a d d i n g m o r e d i m e n s i o n s ? " " B u t how many m o r e d i m e n s i o n s ? " 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 t h r e e dimensions of space a n d o n e of time, the embarrassing question still r e m a i n e d . 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 h i g h e r 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 of s o m e imaginary p e o p l e living in L i n e l a n d , w h i c h consists of 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 p e o p l e could move off their universe a n d move p e r p e n d i c u l a r 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 a p p e a r i n g 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

The Secret of Light: Vibrations in the Fifth Dimension 107 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 d e a t h 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 R i e m a n n 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 111

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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 a t o m 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 m a c h i n e , they could predict all the chemical properties of matter found in the universe. To t h e m , chemistry would no longer be a fundamental science. From now on, it would be " a p p l i e d physics." N o t 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, y o u n g physicists began to whisper that Einstein was over the hill, that the q u a n t u m revolution h a d bypassed h i m completely. T h e younger generation rushed to read the latest papers written a b o u t q u a n t u m theory, n o t those a b o u t 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 " o l d relic." Einstein's d r e a m , we recall, was to create a universe m a d e of " m a r b 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 h o r r o r , 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 b l u n d e r , 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. T h e i r 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

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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 d a n c i n g 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 g e n e r a t e d 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 h i g h e r dimensions. However, with the a p p e a r a n c e of the q u a n t u m heresy after 1925, we see the first serious challenge to this t h e m e . 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 s t u n n i n g 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 c o m p o s e d 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 s o m e t h i n g 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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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 e x c h a n g e 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 a t o m 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 a t o m 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

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time is given by a probability wave. This wave, in turn, obeys a welldefined mathematical equation given by Schrodinger. What is so crazy a b o u t 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, h a d unsettling philosophical implications. T h e Newtonian vision held that the universe was a gigantic clock, wound at the b e g i n n i n g of time a n d ticking ever since because it obeyed Newton's t h r e e laws of motion; this picture of the universe was now replaced by uncertainty a n d 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 a n d 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 p h o t o g r a p h i c paper. According to nineteenth-century classical physics, t h e r e 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 p e r f o r m e d 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 c o m m o n l y associated with wavelike, n o t particlelike, behavior (Figure 5.1). ( T h e simplest way of creating an interference pattern is to take a quiet bath and t h e n 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 p h o tographic sheet corresponds to a wave that has p e n e t r a t e d b o t h 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 g o n e t h r o u g h b o t h 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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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 o n c e 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 o n c e 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 s t u n n i n g 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 p e n e t r a t e the walls of the box. If

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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, t h e r e 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 t u n n e l 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 t u n n e l i n g is the secret b e h i n d the t u n n e l 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 t u n n e l i n g t h r o u g h it. W h e n you listen to the beautiful s o u n d s 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 s p i n n i n g 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. T h u s the fact that we exist is living p r o o f 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 " i m p o s s i b l e " 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 a t o m 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 m o u n t a i n s 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 e x p e r i m e n t a l 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 n a m e s of all these particles, I would have b e c o m e a b o t a n i s t . " 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 a n d false starts in particle physics d u r i n g that period, o n e is r e m i n d e d of the story of the scientist a n d the flea. A scientist once trained a flea to j u m p whenever he rang a bell. Using a microscope, he t h e n 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 t h e n anesthetized a n o t h e r leg a n d t h e n r a n g the bell. T h e flea still j u m p e d . Eventually, the scientist anesthetized m o r e a n d 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 r a n g 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 e m e r g e d . In 1971, the key development that propelled a unified description of t h r e e of the q u a n t u m forces (excluding gravity) a n d c h a n g e d 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 p h o t o n s , the q u a n t a of light, physicists believed that the weak a n d strong forces were caused by the e x c h a n g e 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 i n t r o d u c e d 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 a n d

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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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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, p r o p o s e d 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 names have n o t h i n g to do with actual colors a n d flavors.) T h e r e are also the antimatter counterparts of the 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 turn, are held together by the exchange 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 that " g l u e s " the quarks p e r m a n e n t l y together. T h e gluon field is so powerful a n d binds the quarks so tightly together 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 example, the 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 three steel balls (quarks) h e l d together by a Y-shaped string (gluon) in the shape of a bola. O t h e r strongly interacting particles, such as the pi meson, can be c o m p a r e d to a q u a r k a n d an antiquark held together 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 Standard Model describing the strong force is called q u a n t u m chromodynamics ( Q C D ) — t h a t is, the q u a n t u m theory of the color force. The Weak Force In the Standard Model, the weak force governs the properties of "lept o n s , " such as the electron, the m u o n , a n d the tau meson, a n d their n e u t r i n o partners. Like the 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. These q u a n t a are also described mathematically by the 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 the leptons into a resonance, so we do n o t see an infinite n u m b e r of leptons 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 part of the Standard Model governing the interac-

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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 sentence, we have captured the essence of the 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 properties of matter. (Although it all seems so easy now, Nobel laureate Steven Weinberg, o n e of the creators of the Standard Model, o n c e reflected on how tortuous 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 me, that said the strong interactions [were] too complicated for the h u m a n mind." ) 4

Symmetry in Physics T h e details of the Standard Model are actually rather b o r i n g a n d unimportant. T h e most interesting feature of the Standard Model is that it is based on symmetry. W h a t has propelled 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 patterns in the Standard Model. Symmetry, of course, is n o t strictly the province of physicists. Artists, writers, poets, a n d mathematicians have long a d m i r e d the beauty that is to be found 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 fearful symmetry?"

5

To mathematician Lewis Carroll, symmetry r e p r e s e n t e d a familiar, almost playful concept. In the " T h e H u n t i n g of the Snark," he captured the essence of symmetry when 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 .

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In o t h e r words, symmetry is the preservation of the shape of an object even after we deform or rotate it. Several kinds of symmetries occur repeatedly in n a t u r e . T h e first is the symmetry of rotations a n d reflections. For example, a snowflake remains 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 are created by rotating the object t h r o u g h a dimension 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 game, where a huckster shuffles t h r e e shells with a pea h i d d e n b e n e a t h o n e of them. What makes the 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 are six different ways in which t h r e e shells can be shuffled. Since the pea is h i d d e n , these six configurations are identical to the 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 the shells with quarks, t h e n the equations of particle physics must remain the same if we shuffle the quarks a m o n g themselves. If we shuffle t h r e e colored quarks a n d the equations r e m a i n the 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 the fact that we have t h r e e types of colors, a n d the SU stands for a specific mathematical property of the symmetry.* We say that there 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 the properties of two particles, the electron a n d the n e u t r i n o . T h e symmetry that interchanges these particles, yet leaves the equation the same, is called SU(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 rotated into each other. Finally, the electromagnetic force has U ( l ) symmetry, which rotates the c o m p o n e n t s of the Maxwell field into itself. Each of these symmetries is simple a n d elegant. However, the most controversial aspect of the Standard Model is that it " u n i f i e s " the t h r e e fundamental forces by simply splicing all t h r e e theories into o n e large symmetry, SU(3) X SU(2) X U ( l ) , which is j u s t the p r o d u c t of the 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 together by h a n d . This is how the Standard Model is formed, by taping three distinct multiplets together. This may n o t be aesthetically pleasing, but at least the three jigsaw puzzles now h a n g together by tape.) Ideally, o n e m i g h t have expected that " t h e ultimate t h e o r y " would have all the particles inside j u s t a single multiplet. Unfortunately, the Standard 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 the Standard Model can say truthfully that it fits all known e x p e r i m e n t a l data. They can correctly p o i n t o u t that t h e r e are no experimental results that contradict the Standard Model. Nonetheless, nobody, 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 Standard Model does 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 Standard Model, the resulting theory gives nonsensical answers. W h e n we calculate, say, the probability of an electron being deflected by a gravitational field, the 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 together. Personally, I think that the Standard 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 elephant, a n d a whale. In fact, it is so ugly a n d contrived that even its creators are a bit embarrassed. They are the first to apologize for its shortcomings a n d admit that it c a n n o t be the final theory. This ugliness is obvious w h e n we write down the details of the quarks a n d leptons. To describe how ugly the theory is, let us list the various particles a n d forces within the Standard 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 antimatter c o u n t e r p a r t s to describe the strong interactions

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2. Eight Yang-Mills fields to describe the gluons, which b i n d the quarks 3. Four Yang-Mills fields to describe the weak a n d electromagnetic forces 4. Six types of leptons to describe the weak interactions (including the electron, m u o n , tau lepton, 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 the constants describing the particles 6. At least 19 arbitrary constants that describe the masses of the 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 the 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 than the 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 elementary these elementary particles really are.) T h e ugliness of the Standard Model can be contrasted to the 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 the aesthetic contrast between the Standard Model a n d Einstein's theory of general relativity, we must 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 experimental data with the most economical mathematical expressions T h e Standard 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 economical by any means. For example, Einstein's equations, written o u t in their entirety, are only a b o u t an inch long a n d wouldn't even fill up o n e line of this book. From this o n e line of equations, we can go beyond 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 quark

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, the 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 the Standard Model in its entirety would r e q u i r e two-thirds of this page a n d would look like a blizzard of complex 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, protein molecules, or black holes, it is sparing in its design. O r , as Nobel laureate C. N. Yang o n c e said, " N a t u r e seems to take advantage of the simple mathematical representations of the symmetry laws. W h e n o n e pauses to consider the elegance a n d the beautiful perfection of the mathematical reasoning involved a n d contrast it with the complex a n d far-reaching physical consequences, a d e e p sense of respect for the power of the symmetry laws never fails to develop." However, at the 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 assortment of particles, is o n e of the most disturbing features of the Standard Model, and raises a persistent p r o b l e m for physicists: Should the Standard 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 once a t t e n d e d a concert in Boston, where p e o p l e were visibly moved by the power a n d intensity of Beethoven's Ninth 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, where I noticed some p e o p l e staring in wonder at the sheet music left by the 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 the most moving musical piece must a p p e a r to be a raw mass of unintelligible squiggles, bearing m o r e resemblance to a chaotic j u m b l e of scratches than a beautiful work of art. However, to the ear of a trained musician, this mass of bars, clefs, keys, sharps, flats, a n d notes comes 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 sum 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 . " Not only is the definition sterile, b u t it is ultimately inaccurate because it fails to take into account the subtle interaction between the p o e m a n d the emotions that it evokes in the reader. Poems, because they crystallize a n d convey the essence of the feelings a n d images of the author, have a reality m u c h greater than the words p r i n t e d on a sheet of paper. A few short words of a haiku p o e m , for example, may transport the r e a d e r into a new realm of sensations a n d feelings.

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Like music or art, mathematical equations can have a natural progression a n d logic that can evoke rare passions in a scientist. Although the lay public considers mathematical equations to be rather o p a q u e , to a scientist an equation is very m u c h like a m o v e m e n t in a larger symphony. Simplicity. Elegance. These 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 the 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 , equations have a beauty a n d rhythm all their own. Physicist Richard Feynman expressed this when 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 right, it is o b v i o u s t h a t it is r i g h t — a t l e a s t if y o u h a v e a n y 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 c r a c k p o t s , a n d p e o p l e like that, m a k e g u e s s e s t h a t a r e 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 it sort of l o o k s as if it is all r i g h t , b u t I k n o w it is 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 mathematician H e n r i Poincare expressed it even m o r e frankly when he wrote, " T h e scientist does n o t study Nature 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 . They are short a n d are organized according 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 example, Maxwell's equations, we recall, originally consisted of eight equations. These equations are n o t "beautiful." They do 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 who has ever e a r n e d a living working with radar, radio, microwaves, lasers, or plasmas. These eight equations are what a tort is to a lawyer or a stethoscope is to a doctor. However, when rewritten using time as the fourth dimension, this r a t h e r awkward set of eight equations collapses into a single tensor equation. 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 the theory a n d can now explain vast a m o u n t s of experimental data with a single equation.

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As we have repeatedly seen, the addition of h i g h e r dimensions causes the laws of n a t u r e to simplify. O n e of the greatest mysteries confronting science today is the explanation of the origin of these symmetries, especially in the subatomic world. W h e n o u r powerful machines blow apart 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 according to these symmetries. Something rare a n d precious is unquestionably h a p p e n i n g when 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, but to explain t h e m . T h e fundamental 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 Standard Model fails. No matter how successful the theory is, physicists universally believe that it must be replaced by a higher 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 important, the Standard Model does n o t explain where these symmetries originally came from. They are j u s t spliced together 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, who discovered the nucleus of the atom, once said, "All science is either physics or stamp collecting." 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 collecting" or stamp 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 Standard Model is n o t real physics; it is m o r e like stamp collecting, arranging the subatomic particles according to some superficial symmetries, b u t without the vaguest hint of where the symmetries c o m e from. Similarly, when Charles Darwin n a m e d his b o o k On the Origin of Species, he was going far beyond taxonomy by giving the logical explanation for the diversity of animals in n a t u r e . What 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 found in n a t u r e . Because the Standard Model is so contrived, over the years attempts have b e e n m a d e to go beyond 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 Theory (GUT), popular in the late 1970s, which tried to unite the 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 example, S U ( 5 ) , O ( 1 0 ) , or E ( 6 ) ] . Instead of naively splicing the symmetry groups 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 the n u m b e r of particles beyond the Standard Model, b u t the advantage was that the 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, but 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 the GUTs was that they p u t the strongly interacting quarks a n d the weakly interacting leptons on the same footing. In S U ( 5 ) , for example, a multiplet of particles consisted of three colored 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 changing the physics. At first, GUTs were m e t with intense skepticism, because the energy at which the t h r e e fundamental forces were unified was a r o u n d 1 0 billion electron volts, j u s t a bit smaller than the Planck energy. This was far beyond 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 that in the Standard Model, a symmetry like SU(3) rotates three 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 turn into o n e of the o t h e r quarks u n d e r certain conditions (such as the exchange of a Y a n g Mills particle). However, quarks c a n n o t turn 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 the 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, turn a p r o t o n ( m a d e of quarks) into an electron or a n e u t r i n o . In o t h e r words, GUTs say that the p r o t o n , which was long held to be a stable particle with an infinite lifetime, is actually unstable. In principle, it also m e a n s that all atoms in the universe will eventually disintegrate into radiation.. If correct, it m e a n s that the chemical elements, which are taught in elementary 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

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31

p r o t o n to decay into leptons was calculated to be on the o r d e r of 1 0 years, far beyond the lifetime of the universe (15 to 20 billion years). Although this time scale was astronomically long, this d i d n ' t faze the experimentalists. Since an ordinary tank of water contains an astronomical 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 the test: Several expensive, multimillion-dollar experiments 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 detect p r o t o n decay involved highly expensive a n d sophisticated techniques. First, experimentalists n e e d e d to construct e n o r m o u s vats in which to detect 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 techniques in o r d e r to eliminate all impurities a n d contaminants. Most important, they t h e n h a d to bury these gigantic tanks d e e p in the earth 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. And finally, they h a d to construct thousands of highly sensitive detectors to record the 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 detector in J a p a n a n d the 1MB (Irvine, Michigan, Brookhaven) detector n e a r Cleveland, O h i o . They contained vast a m o u n t s of p u r e fluid (such as water) ranging in weight from 60 to 3,300 tons. ( T h e 1MB detector, for example, is the world's largest a n d is contained in a h u g e 20-meter cube hollowed out of a salt mine 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 turn would be picked up by some of the 2,048 photoelectric tubes.) To u n d e r s t a n d how these m o n s t r o u s detectors can measure the proton lifetime, by analogy think of the American p o p u l a t i o n . We know that the average American 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 minutes. Likewise, the simplest SU(5) G U T predicted that the half-life of the p r o t o n should be a b o u t 1 0 years; that is, after 1 0 years, half of the p r o t o n s in the universe will have 29

29

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decayed.* (By contrast, this is a b o u t 10 billion billion times longer than the life of the universe itself.) Although 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 protons. With that many protons, a handful of p r o t o n s were expected to decay every year. However, no m a t t e r how long the 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 must have a lifetime larger than 10 years, which rules out 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 excitement over the GUTs spilled over into the media. T h e quest for a unified theory of matter a n d the search for the decay of the p r o t o n c a u g h t the attention of science producers 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 the late 1980s. No matter how long physicists waited for the p r o t o n to decay, the 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 spent by various nations 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 theory," for several reasons. As with the Standard Model, GUTs make 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 make no sense. Like the Standard Model, GUTs are nonrenormalizable. Moreover, the theory is defined at t r e m e n d o u s energies, where we certainly expected gravitational effects to appear. T h u s the 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 plagued by the mysterious presence of three identical carbon 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 the q u a r k masses. GUTs lacked a larger physical principle that would fix the 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 stamp collecting. 29

32

T h e fundamental p r o b l e m was that the 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 .

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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 the world of m a r b l e . After half a century of dormancy, the time h a d c o m e for "Einstein's revenge."

6 Einstein's Revenge S u p e r s y m m e t r y i s t h e u l t i m a t e p r o p o s a l f o r 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 "Holy Grail" of physics, the quest to unite the q u a n t u m theory with gravity, thereby creating a Theory of Everything. This is the p r o b l e m that has frustrated the finest minds of the twentieth century. Without question, the person who solves this p r o b l e m will win the Nobel Prize. By the 1980s, physics was reaching an impasse. Gravity alone stubbornly stood apart a n d aloof from the o t h e r three forces. Ironically, a l t h o u g h the classical theory of gravity was the 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 problem, a n d all have failed. Einstein devoted the last 30 years of his life to his unified field theory. Even the great W e r n e r Heisenberg, o n e of the founders of q u a n t u m theory, s p e n t the last years of his life chasing after his version of a unified theory of fields, even publishing a b o o k on the subject. In 1958, H e i s e n b e r g even broadcast on radio that he a n d his colleague 136

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Wolfgang Pauli h a d finally succeeded in finding the unified field theory, a n d that only the technical details were missing. (When 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 send a letter to his collaborator, consisting of a blank sheet of p a p e r with the caption, " T h i s is to show the world that I can paint like Titian. Only technical details are missing." ) Later that year, when Wolfgang Pauli finally gave a lecture on the Heisenberg-Pauli unified field theory, many eager physicists were in the audience, 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 agreed 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 Julian Schwinger has said, "It's n o t h i n g m o r e than a n o t h e r symptom of the urge that afflicts every g e n e r a t i o n of physicist—the itch to have all the fundamental 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 the sense of frustration a m o n g j a d e d young physicists d u r i n g this period. Everyone sensed that the Standard Model was being killed by its own success. It was so successful that every international physics conference seemed like j u s t a n o t h e r r u b b e r stamp 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 experimental success for the Standard Model. At o n e physics conference, I glanced back at the a u d i e n c e a n d found 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 chart after c h a r t showing how the latest data could be fit according to the Standard 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 spent decades tediously filling up tables of figures for the spectral lines of various gases, or calculating the 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 imagined that physicists would spend decades finding the precise values of all 19 parameters. T h e time h a d c o m e for a revolution. What b e c k o n e d the n e x t generation of physicists was the world of marble. Of course, several p r o f o u n d p r o b l e m s stood in the 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 maddeningly weak. For example, it takes the entire 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 the gravitational pull of the entire planet. Similarly, if I were to try to construct an " a t o m " with electrons attracted to the 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 measure. 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 the 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 the Planck energy, or 10 billion electron volts, far beyond anything achievable on the planet earth in this century. This p e r p l e x i n g situation d e e p e n s when we try to construct a complete 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 the theory of gravity, you postulate that it functions by the e x c h a n g e of tiny packets of gravity, called gravitons. T h e rapid e x c h a n g e of gravitons between matter 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 perform simple calculations to calculate q u a n t u m corrections to Newton's a n d Einstein's laws of gravity, they found that the result is infinite, which is useless. 19

For example, 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 other. To calculate the Feynman diagrams for this theory, we have to make an approximation, 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 Riemann 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 , where 1 represents 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 equations in this way by breaking up the 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 performed, we arrive at a conventional-looking q u a n t u m theory. In Figure 6.1 (a), we see that the two neutral 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. 139

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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 the Yang-Mills field, we could use clever sleight-of-hand tricks to shuffle a r o u n d these infinite quantities until they either cancel or are absorbed into quantities that can'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, the 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 marble into pieces failed miserably. T h e n , in the early 1980s, a curious p h e n o m e n o n occurred. 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 began to overcome their prejudice about u n s e e n dimensions a n d hyperspace. They were ready for an alternative, a n d that was Kaluza-Klein theory. T h e late physicist Heinz Pagels summarized this excitement over the re-emergence 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 years i t lay d o r m a n t . B u t r e c e n t l y , a s physicists 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 , i n c o n t r a s t w i t h t h e 1 9 2 0 s , physicists a r e c h a l l e n g e d t o d o m o r e t h a n unify 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 the 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 measure experimentally these compactified dimensions that have curled u p , composed the following p o e m :

Steve W e i n b e r g , returning from Te 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 a r e r o l l e d u p i n a ball s o tiny i t n e v e r affects u s .

5

A l t h o u g h Kaluza-Klein theory was still nonrenormalizable, what sparked the intense interest in the theory was that it gave the h o p e of a

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theory m a d e of marble. T u r n i n g the ugly, confused j u m b l e of wood into the p u r e , elegant marble of geometry 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 the n a t u r e of wood. However, by the 1970s, the Standard Model h a d finally unlocked the secret of wood: that m a t t e r consists of quarks a n d leptons held together 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 marble. At first, that s e e m e d impossible. After all, these symmetries are the result of i n t e r c h a n g i n g point 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 the symmetry is SU(N). These symmetries s e e m e d to be exclusively the symmetries of wood, not 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 the 1960s, when physicists found, 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 the fifth dimension was curled u p , they saw that the Maxwell field p o p p e d o u t of R i e m a n n ' s metric. But when N dimensions were curled u p , physicists found the celebrated Yang-Mills field, the key to the Standard Model, p o p p i n g out 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 beach ball retains its shape. T h e symmetry of a beach ball, or a sphere, is called 0 ( 3 ) , or rotations in t h r e e dimension. Similarly, in higher dimensions, a h y p e r s p h e r e can also be rotated a r o u n d its center a n d maintain its shape. T h e h y p e r s p h e r e has a symmetry called O(N). Now consider vibrating the beach ball. Ripples form on the surface of the ball. If we carefully vibrate the beach ball in a certain way, we can induce regular vibrations on it that are called resonances. These resonances, unlike ordinary ripples, can vibrate at only certain frequencies. In fact, if we vibrate the beach ball fast e n o u g h , we can create musical tones of a definite frequency. These vibrations, in turn, can be cataloged by the symmetry 0 ( 3 ) . T h e fact that a m e m b r a n e , like a beach 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 chords in o u r throat,

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for example, 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 hearing. S o u n d waves of all types impinge 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 the 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 contained 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 resonances in the 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 the 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 hypersphere, the effect is the same. Like a m e m b r a n e , it can resonate at various frequencies, which in turn can be d e t e r m i n e d by its symmetry O(N). Alternatively, mathematicians have d r e a m e d up m o r e sophisticated surfaces in h i g h e r dimensions that are described by complex n u m b e r s . (Complex n u m b e r s use the square root of — 1 , '— 1.) T h e n it is straightforward to show that the symmetry corresponding 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 point is now this: If the wave function of a particle vibrates along this surface, it will inherit 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 explanation for the origin of the mysterious symmetries of wood: They are really the 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 dimensions, we will find that the equations split into two pieces. T h e first piece is Einstein's usual equations, which we retrieve as expected. But the second 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 the 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 marble. 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 the debris from a t o m smashers—emerge 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 help 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 geometric figures. However, clay may be pressed into 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

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the clay will also inherit the symmetry of the m o l d . Clay, like matter, inherits its symmetry because t h e mold, 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 that the strange symmetries we see a m o n g the 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 example, if the u n s e e n dimensions have the symmetry S U ( 5 ) , then 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 except 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 checkerboard. By separating o u t the fifth column a n d row of the checkerboard, we can split off Maxwell's field from Einstein's field. Now perform 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 columns a n d rows, t h e n you obtain a metric 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 perform this reduction was University of Texas physicist Bryce DeWitt, who has spent 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 the Yang-Mills field is straightforward. DeWitt felt that extracting the Yang-Mills field from Ndimensional 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 Freund that Oskar Klein h a d i n d e p e n d e n t l y discovered the Yang-Mills field in 1938, p r e c e d i n g the work of Yang, Mills, a n d others by several decades. In a conference held 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 unleashed by World War II a n d because Kaluza-Klein theory was buried by the excitement 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 forgotten. It is ironic that Kaluza-Klein theory was killed by the 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 excitement to develop q u a n t u m theory, physicists h a d ignored a central discovery c o m i n g from Kaluza-Klein theory.] Extracting the Yang-Mills field out of Kaluza-Klein theory was only the first step. A l t h o u g h the symmetries of wood could now be seen as arising from the h i d d e n symmetries of u n s e e n dimensions, the next step was to create wood itself ( m a d e of quarks a n d leptons) entirely out of marble. This n e x t step would be called supergravity.

Supergravity T u r n i n g wood into marble still faced formidable problems because, according to the Standard 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 turn, have 1/2 unit 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 Enrico Fermi, who first investigated their strange properties). However, forces are described by q u a n t a with integral spin. For example, the p h o t o n , the q u a n t u m of light, has o n e unit 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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Traditionally, q u a n t u m theory kept fermions a n d bosons strictly apart. I n d e e d , any a t t e m p t to turn wood into marble would inevitably come to grips with the fact that fermions a n d bosons are worlds apart in their properties. For example, 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 supposed to mix. It came as a shock, therefore, when a new symmetry, called supersymmetry, was discovered, that did exactly that. Equations that are supersymmetric allow the interchange of a fermion with a boson 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 the bosons a n d fermions within the same multiplet, the supersymmetric equations remain the same. This gives us the tantalizing possibility of putting all the particles in the universe into o n e multiplet! As Nobel laureate Abdus Salam has emphasized, "Supersymmetry 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 that would drive any schoolteacher insane. Most of the operations of multiplication a n d division that we take for granted 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 super 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 the 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 rather astonishing statement that a X a = 0 even when a =/ 0. Although these super 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 childhood, they can be shown to yield a self-consistent a n d highly nontrivial system. Remarkably, an entirely new system of super calculus can be based on t h e m . Soon, three physicists (Daniel F r e e d m a n , Sergio Ferrara, a n d Peter 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 attempt to construct a world m a d e entirely of marble. In a supersymmetric theory, all particles have super partners, 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 boson) a n d its spin-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 include the Standard Model, attempts were m a d e to couple the theory to m o r e complicated particles. T h e simplest way to include m a t t e r is to write down the supergravity theory in 11-dimensional space. In o r d e r to write down the super

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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 components within the R i e m a n n tensor vastly, which now becomes 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, the Yang-Mills field, a n d the matter fields into o n e supergravity field (Figure 6.3). T h e essential feature of this diagram is that matter, along with the Yang-Mills a n d Einstein equations, is now included 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 longer 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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Physicist Peter van Nieuwenhuizen, o n e of supergravity's creators, was deeply impressed by the 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 parameters. 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 Nature 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 important. 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 the c o n t i n u e d success of the supergravity theory. It seemed that we were finally on the verge of carrying out Einstein's d r e a m of a universe of marble 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 Stephen Hawking assumed the Lucasian Professorship (previously h e l d by s o m e of t h e immortals 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 Theoretical 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 him: " [ W ] e have m a d e a lot of progress in recent years a n d , as I shall describe, t h e r e are some 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, the c o n c e p t of "unification" is a central belief within the transcendental meditation m o v e m e n t . Its followers therefore published a large poster containing the c o m p l e t e equations describing 11-dimensional supergravity. Each term in the equation, they claimed, r e p r e s e n t e d something 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 on. (This poster hangs on the wall of the theoretical institute at Stony Brook. This is the first time that I am aware of that an abstract equation from theoretical physics has inspired a following a m o n g a religious group!)

Super Metric Tensors Peter van Nieuwenhuizen cuts a r a t h e r dashing 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 suntan lotion on television than o n e of the original creators of supergravity. He is a Dutch physicist who 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 the question of unification. He is o n e of the few physicists I have ever m e t with a truly inexhaustible capacity for mathematical p u n i s h m e n t . Working with supergravity requires an extraordinary a m o u n t of patience. 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 the 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 components. 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 compon e n t s to describe it, b u t this vastly increases the mathematical complexity of the 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 super metric many times larger t h a n anything 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 super metric tensors has m e a n t that the a m o u n t of mathematics 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 the past decade. 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 on

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it for the past ten years are enormously bright. Some of t h e m are brighter than anyone 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 equation can easily exceed a sheet of paper, he eventually started using large, oversize artist's sketch boards. I went to his house 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 pad, a n d start writing his equations 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 completed. 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 notepads on his shelf, as t h o u g h they were volumes of some scientific j o u r n a l . Peter's sketch pads gradually b e c a m e notorious a r o u n d campus. Soon, a fad started; all the g r a d u a t e students in physics began to buy these bulky artist's sketch pads a n d could be seen on c a m p u s hauling 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 Townsend (now at C a m b r i d g e University), a n d I were collaborating on an exceptionally difficult supergravity problem. 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 dining 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 sum up to exactly zero. (Usually, we theoretical physicists can "visualize" blocks of equations in o u r heads 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 sheer length a n d delicacy of this p r o b lem, we h a d to check every single minus sign in the calculation.) We t h e n divided the p r o b l e m into several large chunks. Sitting a r o u n d the dining-room table, each of us would busily calculate the same chunk. After an h o u r or so, we would t h e n cross-check o u r results. Usually two out of three would get it right, a n d the 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 the 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 into the 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 checked the last a n d final term. 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 exhausted even an indefatigable workhorse like Peter. After leaving my a p a r t m e n t , he promptly forgot where 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 doors of an a p a r t m e n t house, 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 began to see problems with supergravity. After an intensive search, sparticles were n o t seen in any experiment. For example, the spin-1/2 electron does n o t have any spin-0 partner. In fact, t h e r e is, at the present, n o t o n e shred of experimental 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 partners. 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 that 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 marble. 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 marble, 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, although it h a d fewer infinities t h a n the original Kaluza-Klein theory, was still nonrenormalizable. T h e r e were o t h e r problems. 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 the S t a n d a r d Model. Supergravity, it appeared, was just 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 came along that was p e r h a p s the strangest but most powerful physical theory ever proposed: the 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 the Institute for Advanced Study in Princeton, 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 the pack," the most brilliant high-energy physicist, who sets trends 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 Princeton, Samuel T r e i m a n , says, " H e ' s h e a d a n d shoulders above the rest. H e ' s started whole groups 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 then 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 when it comes to Witten . . . " '

Witten c o m e s f r o m a family of physicists. His f a t h e r is Louis Witten, professor of physics at t h e University of Cincinnati a n d a leading a u t h o r i t y on Einstein's t h e o r y of 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 greatest 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 151

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history major at Brandeis University with an intense interest in linguistics. After graduating in 1971, he worked on George McGovern's presidential campaign. McGovern even wrote h i m a letter of r e c o m m e n d a tion for graduate 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 person in the world is a liberal D e m o c r a t . " ) But o n c e Witten decided that physics was his chosen profession, he learned physics with a vengeance. He b e c a m e a graduate student at Princeton, taught at Harvard, a n d t h e n rocketed a full professorship at Princeton at the age of 28. He also received the 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 the 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 the time, however, Witten sits a n d stares out 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 head. His wife notes, " H e never does 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 minus sign, or a factor of t w o . " Witten says, "Most p e o p l e who haven'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 the 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 the concepts, the principles by which the world w o r k s . " Witten's next project is the most ambitious a n d daring of his career. A new theory called superstring theory has created a sensation in the world of physics, claiming to be the theory that can unite 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 uniqueness, is its unusual geometry: Strings can vibrate self-consistently only in ten a n d 26 dimensions. 2

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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 matter a n d s p a c e - t i m e — t h a t is, the n a t u r e of wood a n d marble. String

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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 nature. T h e d e e p e r we p r o b e into the n a t u r e of subatomic particles, the m o r e particles we find. T h e c u r r e n t " z o o " of 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 the string, about 100 billion billion times smaller than a p r o t o n , is vibrating; each m o d e of vibration represents a distinct resonance or particle. T h e string is so incredibly tiny that, from a distance, a resonance of a string a n d a particle are indistinguishable. Only when we somehow magnify the particle can we see that it is n o t a p o i n t at all, but a m o d e of a vibrating string. In this picture, each subatomic particle corresponds 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 example of singing in the 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 the shower walls are magnified many times, p r o d u c i n g that r e s o n a n t sound. 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 the 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, the string can vibrate at any of an infinite n u m b e r of different frequencies. We know that the notes themselves are n o t fundamental. T h e n o t e A is no m o r e fundamental than the n o t e B. However, what is fundamental 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 the properties of an infinite n u m b e r of musical notes. Likewise, the particles of the universe are not, by themselves, fundamental. An electron is no m o r e fundamental than a n e u t r i n o . They a p p e a r to be fundamental only because o u r microscopes are n o t powerful e n o u g h to reveal their structure. According to string theory, if we could somehow magnify a point particle, we would actually see a small vibrating string. In fact, according to this theory, m a t t e r is n o t h i n g b u t the harmonies 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 out of vibrating strings. This explains the richness of the particles in n a t u r e . Likewise, the 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 the 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, but that of s p a c e - t i m e as well. As a string moves in space-time, 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 point is that all these q u a n t u m corrections or loop 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. These 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 point particle. W h e n the constraints that the string places on space-time were first calculated, physicists were shocked to find Einstein's equations emerging from the string. This was remarkable; without assuming any of Einstein's equations, physicists found 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 fundamental; they could be derived from string theory. If correct, t h e n string theory solves the long-standing mystery about the n a t u r e of wood a n d marble. Einstein conjectured that marble 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 space-time, 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. These are two diametrically opposite points of view, a n d it was long t h o u g h t that a compromise between t h e m was impossible. T h e string, however, is precisely the "missing link" between wood a n d marble. String theory can derive the particles of m a t t e r as resonances vibrating on the string. And string theory can also derive Einstein's equations by d e m a n d i n g that the string move self-consistently in space-time. 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 example, they forbid the string to move in three or four dimensions. We will see that these self-consistency conditions force the string to move in a specific n u m b e r of dimensions. In fact, the only "magic 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 dimensions 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 geometric 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 included in string theory. T h e c u r r e n t interest in string theory stems from the work of J o h n Schwarz of the California Institute of Technology a n d his collaborator Michael Green 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 might possess defects that would prevent a fully self-consistent theory. T h e n in 1984, these two physicists proved that all self-consistency conditions on the string can be met. This, in turn, 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 the theory a n d win potential recognition. By the late 1980s, a veritable " g o l d r u s h " began a m o n g physicists. (The competition 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, who openly boasted that he was h o t on the 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 lunch with a Nobel Prize winner 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 came u p . W i t h o u t warning, he launched into a long personal discussion of why superstring theory was not the correct path 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, too

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o r t h o g o n a l to all the previous trends 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 are n o t fundamental concepts in n a t u r e . T h e c o n c e p t of orbits, for example, occurs repeatedly in n a t u r e in different variations; since the 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 the 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 themes. 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 found in the Standard 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 weapon in the arsenal of theoretical physics. All known forms of matter a n d energy have b e e n expressed in terms of field theory. Patterns, t h e n , like themes a n d variations in a symphony, are constantly r e p e a t e d . But strings? Strings do n o t seem to be a pattern favored by n a t u r e in designing the heavens. We do n o t see strings in outer space. In fact, my colleague explained to m e , we do n o t see strings anywhere. A m o m e n t ' s thought, however, will reveal that n a t u r e has reserved the string for a special role, as a basic building block for other forms. For example, the essential feature of life on earth is the stringlike DNA molecule, which contains the complex information a n d coding of life itself. W h e n building the stuff of life, as well as subatomic matter, strings seem to be the 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, reproducible structure. T h e distinguishing feature of a string is that it is o n e of the most compact ways of storing vast a m o u n t s of data in a way in which information can be replicated. For living things, n a t u r e uses the d o u b l e strands of the DNA molecule, which unwind a n d form duplicate copies of each other. Also, o u r bodies contain billions u p o n billions of protein 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 a r o u n d o u r bones.

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Currently, the most successful version of string theory is the o n e created by Princeton physicists David Gross, Emil Martinec, Jeffrey Harvey, a n d Ryan R o h m , who are sometimes called the Princeton string quartet. T h e most senior of t h e m is David Gross. At most seminars in Princeton, Witten may ask questions in his soft voice, b u t Gross's voice is unmistakable: loud, booming, a n d d e m a n d i n g . Anyone who gives a seminar at Princeton lives in fear of the sharp, rapid-fire questions that Gross will shoot at them. What is remarkable 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 the heterotic string. Today, it is precisely the 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 the 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 matter itself from geometry—that 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 matter 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 emerges from g e o m e t r y . " T h e most remarkable feature of string theory, as we have e m p h a sized, is that Einstein's theory of gravity is automatically contained in it. In fact, the graviton (the q u a n t u m of gravity) emerges as the 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 example, if we simply d r o p Einstein's theory of gravity as o n e vibration of the string, t h e n the theory b e c o m e s inconsistent a n d useless. This, in fact, is the reason why Witten was attracted to string theory in the 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 emerges from superstring theory from self-consistency r e q u i r e m e n t s alone. 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 include 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 . " 5

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Gross takes satisfaction in believing that Einstein, if he were alive, would love superstring theory. He would love the fact that the beauty a n d simplicity of superstring theory ultimately c o m e from a geometric 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 the real-

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ization.. . . He would have liked the fact that t h e r e is an underlying geometrical principle—which, unfortunately, we d o n ' t really understand." Witten even goes so far as to say that "all the really great ideas in physics" are "spinoffs" of superstring theory. By this, he means that all the great advances in theoretical physics are included 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 on p l a n e t E a r t h . " He claims that, somewhere in outer space, " o t h e r civilizations in the u n i v e r s e " might have discovered superstring theory first, a n d derived general relativity as a by-product. 7

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Compactification and Beauty String theory is such a promising candidate for physics because it gives a simple origin of the symmetries found in particle physics as well as general relativity. We saw in C h a p t e r 6 that supergravity was b o t h nonrenormalizable 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 both. As we shall soon see, it banishes the infinities found 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 alone would g u a r a n t e e that string theory should be taken as a serious candidate for a theory of the universe. However, t h e r e is an a d d e d bonus. W h e n we compactify some of the dimensions of the 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 the symmetries of the Standard 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, the fifth dimension was compactified by b e i n g wrapped up into a circle.) T h e heterotic string owes its n a m e to the fact that the clockwise a n d the counterclockwise vibrations live in two different dimensions but 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 the most interesting. In Kaluza-Klein theory, we recall that the compactified N-dimensional

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space can have a symmetry associated with it, m u c h like a beach ball. T h e n all the vibrations (or fields) defined on the N-dimensional space automatically inherit these symmetries. If the symmetry is SU(N), t h e n all the vibrations on the space must obey SU(N) symmetry (in the same way that clay inherits the symmetries of the m o l d ) . In this way, KaluzaKlein theory could a c c o m m o d a t e the symmetries of the Standard 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 the symmetries found in the Standard Model. This was sufficient to kill the supergravity theory as a realistic theory of matter a n d space-time. But when the Princeton string quartet analyzed the symmetries of the 16-dimensional space, they found that it is a monstrously large symmetry, called E(8) X E ( 8 ) , which is m u c h larger than 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 the symmetry of the 16-dimensional space, which was m o r e than e n o u g h to a c c o m m o d a t e the symmetry of the Standard Model. This, then, 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 dimensions. In this case, the 26-dimensional space of the counterclockwise vibrations of the heterotic string has r o o m e n o u g h to explain all the symmetries found in both Einstein's theory a n d q u a n t u m theory. So, for the first time,. p u r e geometry has given a simple explanation of why the 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 found in n a t u r e can ultimately be traced back to higher-dimensional space. For example, snowflakes create beautiful, hexagonal patterns, n o n e of which are precisely the same. These snowflakes a n d crystals, in turn, have inherited their structure from the way in which their molecules have b e e n geometrically arranged. 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 elements are d u e to the symmetries cataloged by the Standard Model, which in t u r n can be derived by compactifying the 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 geometric u n d e r s t a n d i n g of why forces 9

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can d e t e r m i n e the m o t i o n a n d the n a t u r e of matter. But they were missing a key ingredient in showing the relationship between wood a n d marble. This missing link is most likely superstring theory. With the tendimensional string theory, we see that the geometry of the string may ultimately be responsible for both 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 occurred, then the theory would n o t have b e e n discovered until the twenty-first century. This is because it is such a sharp d e p a r t u r e from all the ideas tried in this century. It is n o t a continuous 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 and 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 young theoretical physicists, Gabriel Veneziano 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 stumbled on the Euler beta function, a mathematical function written down in the nineteenth century by the mathematician L e o n h a r d Euler. They were aston-

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ished to find that the Euler beta function fit almost all the properties required to describe the strong interactions of elementary particles. Over lunch at the Lawrence Berkeley Laboratory in California, with a spectacular view of the sun blazing down over San Francisco h a r b o r , Suzuki once explained to me the thrill of discovering, quite by accident, a potentially i m p o r t a n t result. Physics was not supposed to h a p p e n that way. After finding the 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 young 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 Veneziano 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 published his result. T h e r e is a lesson to all this, I suspect: Never take too seriously the advice of your superiors.) In 1970, the mystery s u r r o u n d i n g the Veneziano-Suzuki 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 Nihon 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 not know the physical principle that underlies string theory. T h e last step in the evolution of the theory (and 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 it on p u r p o s e , it was i n v e n t e d in a l u c k y a c c i d e n t . By 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 h a v e h a d t h e p r i v i l e g e o f s t u d y i n g this t h e o r y . B y rights, s t r i n g t h e o r y s h o u l d n ' t h a v e b e e n i n v e n t e d u n t i l 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 .

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Loops T h e formula discovered by Veneziano a n d Suzuki, which they h o p e d would describe the properties of interacting subatomic particles, was still

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incomplete. It violated o n e of the properties of physics: unitarity, or the conservation of probability. By itself, the 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 interpretation 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 the 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. Although 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 example, when a b u m b l e b e e flies in space, its path can be described as a wiggly line. W h e n a piece of string drifting in the air moves in space, its path can be likened to an imaginary twodimensional sheet. W h e n a closed string floats in space, its path resembles a tube. Strings interact by breaking 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 out the configurations shown in Figure 7.1. Notice that two tubes come in from the left, with o n e tube fissioning in half, e x c h a n g e the middle tube, a n d t h e n veer off to the 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 mathematical expression. W h e n we calculate the numerical expression corres p o n d i n g to these diagrams, we get back the 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 break 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 diagram coming closer to the 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 mathematical expressions r e m a i n the same.) KSV claimed that the sum total of all these loop diagrams would yield the precise mathematical formula explaining how subatomic particles interact. However, the 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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Intrigued 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 machine-gun bullets at the time.

Boot Camp I r e m e m b e r clearly when the KSV p a p e r came out 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 than giving precise details. I decided 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, and the university campuses from Kent State to the University of Paris, were in a state of turmoil. I h a d graduated from Harvard the year before, when President Lyndon J o h n s o n revoked deferments for graduate students, s e n d i n g panic t h r o u g h o u t graduate schools in the country. Chaos gripped the campuses. Suddenly, my friends were d r o p p i n g out of college, teaching high school, packing their bags a n d h e a d i n g to Canada, or trying to ruin 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 good friends in physics from MIT vowed that he would go to jail rather than fight in Vietnam. He told us to send copies of the Physical Review to his jail cell so he could k e e p up with developments in the Veneziano model. O t h e r friends, who quit college to teach in high schools rather than fight in the war, t e r m i n a t e d promising scientific careers. (Many of t h e m still teach in these high schools.) T h r e e days after graduation, I left Cambridge a n d found myself in the U n i t e d States Army stationed at Fort Benning, Georgia (the largest infantry training center in the world), a n d later at Fort Lewis, Washington. T e n s of thousands 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 , replacing the 500 GIs who were dying every week. O n e day, while throwing live grenades u n d e r the grueling Georgia sun a n d seeing the deadly shrapnel scatter in all directions, my thoughts began to wander. How many scientists t h r o u g h o u t history h a d to face the p u n i s h i n g ravages of war? How many promising 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 the 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 found the basic solution to Einstein's equations used in every black hole calculation. ( T h e Schwarzschild radius of a black hole is n a m e d in his

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h o n o r . Einstein addressed the Prussian Academy in 1916 to c o m m e m orate Schwarzschild's work after his untimely d e a t h at the 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, the military does n o t necessarily want some wit who will question the sergeant's orders in the middle of a firefight. U n d e r s t a n d ing this, I decided to b r i n g along some physics papers. I n e e d e d something to keep my m i n d active while peeling potatoes in KP or firing machine guns, so I b r o u g h t along a copy of the KSV paper. During 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 machine-gun bullets, froglegging u n d e r barbed wire, a n d crawling t h r o u g h thick brown m u d . Because the automatic fire h a d tracers on t h e m , I could see the beautiful crimson streaks m a d e by thousands of machine-gun bullets sailing a few feet over my head. However, my thoughts kept 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, the language of topology. Never before in the history of physics h a d Mobius strips or Klein bottles b e e n used in a fundamental way. Because I rarely h a d any p a p e r or pencils while practicing with machine 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 the time I finished the 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 the University of California at Berkeley, where I furiously worked out the details that were racing in my head. 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 loop diagrams known at that time. However, I wasn't satisfied with this work. T h e KSV p r o g r a m consisted of a hodge-podge 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 next 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 the 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 than 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 Heisenberg, h a d labored for years to create a field theory that was n o t based on p o i n t particles. Elementary particles, they thought, m i g h t be pulsating blobs of matter, r a t h e r than points. However, no matter how h a r d they tried, field theories based on blobs always violated causality. If we were to shake the blob 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 "nonlocal 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 point particles could be consistent. Nonlocal field theories must violate relativity. T h e second 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 properties (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, Richard 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 problem. 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 the c o n c e p t of Faraday a n d postulate a field that 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 equations that the string obeyed. T h e field equation for a single string moving alone in s p a c e time was easy. As expected, o u r field equations r e p r o d u c e d an infinite series of string resonances, each c o r r e s p o n d i n g to a subatomic particle. Next, we found that the 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 the string, the vibrations traveled down the string at less t h a n the 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 the Veneziano a m p l i t u d e correctly. Duality a n d the c o u n t i n g of graphs given by Feynman for any field theory were in direct conflict. J u s t as the critics expected, the Feynm a n graphs were incorrect. This was disheartening. It a p p e a r e d that field theory, which h a d formed the foundation of physics for the past century, was fundamentally incompatible 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 the night. For hours, I began systematically to check all the possible alternatives to this p r o b l e m . But the conclusion that duality h a d to be b r o k e n seemed 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 Arthur 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 properties 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 the obvious fact that o n e can split the Veneziano-Suzuki formula into two pieces. Each p a r t t h e n corresponds to o n e of Feynman's diagrams, a n d each part violates duality, b u t the sum obeys all the correct properties 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 next 5 h o u r s checking a n d rechecking the calculation from all possible directions. T h e conclusion was inescapable: Field theory does violate duality, as everyone expected, b u t this is acceptable because the final sum reproduces the Veneziano-Suzuki formula. I had now solved most of the p r o b l e m . However, o n e m o r e Feynman diagram, representing the collision of four strings, was still lacking. T h a t year, I was teaching introductory electricity a n d magnetism 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, repeating the same steps p i o n e e r e d by Faraday in the n i n e t e e n t h century. Suddenly, it dawned on me that the squiggly lines that I was asking my students to draw h a d exactly the same topological structure as the 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 had found the correct configuration describing the collision of four strings. Was it that simple? I rushed h o m e to check my h u n c h , a n d I was right. By employing pictorial techniques 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 attempt to c o m b i n e string theory with the formalism of field theory. O u r field theory, although it correctly e m b o d i e d the entire information 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 example, the 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 the properties of o u r field theory, the model unexpectedly 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 dimensions, the theory completely loses all its beautiful mathematical properties. But no o n e believed that a theory defined in ten or 26 dimensions 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 hibernation. For 10 long years, the m o d e l was banished to obscurity. (Although most string physicists, myself included, a b a n d o n e d the 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 the m o d e l alive by steadily m a k i n g improvements. For example, string theory was originally t h o u g h t to be just 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 resonance of the q u a r k model. 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. From 1974 to 1984, when 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 the failures of these models also b e c a m e apparent. For example, 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 the growing list of models that were tried a n d t h e n discarded d u r i n g this period. 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 the 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 the supergravity theory was superstring theory. On the o t h e r h a n d , physicists slowly became accustomed to working in hyperspace. Because of the KaluzaKlein renaissance, the idea of hyperspace d i d n ' t seem that farfetched or forbidding anymore. Over time, even a theory defined in 26 dimensions d i d n ' t seem that outlandish. T h e original resistance to 26 dimensions 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 the field theory of strings, which many p e o p l e think is o n e of the most beautiful achievements of the theory. He showed that o u r old field theory could be derived using powerful mathematical a n d geometric 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, the true mathematical elegance of string field theory, which was concealed in o u r formalism, was revealed. Soon, almost a h u n d r e d scientific papers were written to explore the fascinating mathematical properties 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 the 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 rank a m o n g the most significant advances in physics in 2,000 years.

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After the 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 the theory would be completely solved within a few years a n d the Nobel Prizes h a n d e d o u t by the d o z e n ) , a certain degree of cold realism has set in. Although the theory is well defined mathematically, no o n e has b e e n able to solve the 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 nonperturbative 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 techniques that are currently beyond 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 the 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 remark 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, the fault is n o t in the theory, but in o u r primitive mathematics. T h e reason for this pessimism is that o u r main calculational tool, p e r t u r b a t i o n theory, fails. Perturbation 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 shape 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 dimensions that would uniquely describe the known spectrum 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 drowning in these perturbative solutions. T h e fundamental p r o b l e m that has stalled progress in superstring theory in the past few years is that no o n e knows how to select the correct solution o u t of the 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 modest assumptions, it is easy to extract the Standard Model as o n e vibration of the string. Several groups 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 our universe. In some of these solutions, the universe has no quarks or too many quarks. In most of them, life as we know it c a n n o t exist. O u r universe may be lost somewhere a m o n g the millions of possible universes that have b e e n found in string theory. To find the 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 the o n e true solution to the theory. T h e r e is some r o o m for optimism, however. Nonperturbative solutions that have b e e n found for m u c h simpler theories show that many 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 the 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 found 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 nineteenth-century physicists might react if a portable c o m p u t e r were given to t h e m . They could easily learn to t u r n the dials a n d press the buttons. They could learn to master video games or watch educational p r o g r a m s on the monitor. Being a century b e h i n d in technology, they would marvel at the fantastic calculational ability of the computer. Within its m e m o r y could easily be stored all known scientific knowledge of that century. In a short period of time, they could learn to perform mathematical feats that would amaze any of their colleagues. However, once they decide 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 experience to c o m p a r e with the electronic c o m p u t e r . It would be beyond their ken. They 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 reference 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 turn a few dials a n d press a few buttons with the theory, a n d out p o p s the supergravity theory, Kaluza-Klein theory, a n d the Standard 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 mathematics 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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mathematics 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 deepest secrets of string theory, which is still n o t well understood, 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 the geometry of higher 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 properties of the theory. Fortunately, these unwanted terms a p p e a r multiplied by (N — 10). Therefore, to make these anomalies vanish, we have no choice b u t to fix N to be ten. String theory, in fact, is the only known q u a n t u m theory that specifically d e m a n d s that the dimension 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 dimensions are singled out. T h e answer lies d e e p within mathematics, in an area called modular functions. Whenever we manipulate the KSV loop 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, where the n u m b e r ten appears in the strangest places. These m o d u l a r functions are as mysterious as the m a n who investigated t h e m , the mystic from the East. Perhaps if we better u n d e r s t o o d the work of this Indian 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 Ramanujan was the strangest m a n in all of mathematics, probably in the entire 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 him. Working in total isolation from the main currents of his field, he was able to rederive 100 years' worth of Western mathematics 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 found in mathematics. They r e a p p e a r in the most distant a n d u n r e l a t e d branches 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 term raised to the twenty-fourth power. In the work of Ramanujan, the n u m b e r 24 appears repeatedly. This is an example of what mathematicians call magic n u m b e r s , which continually appear, where we least expect t h e m , for reasons that no o n e understands. Miraculously, Ramanujan's function also appears in string theory. T h e n u m b e r 24 a p p e a r i n g in Ramanujan's function is also the origin of the miraculous cancellations occurring 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. Whenever the string executes its complex motions in space-time 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 mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan. (Since physicists add two m o r e dimensions 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 space-time must have 24 + 2 = 26 s p a c e - t i m e dimensions. ) 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 dimension. T h e string vibrates in ten dimensions because it requires these generalized Ramanujan functions in o r d e r to remain 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 numerology being 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 dimension of s p a c e - t i m e to be ten. In the final analysis, the origin of the ten-dimensional theory is as mysterious as Ramanujan himself. W h e n asked by audiences 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 dimension of s p a c e - t i m e must be selected (or else the 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. Perhaps the answer lies waiting to be discovered in Ramanuj a n ' s lost notebooks.

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Reinventing 100 Years of Mathematics Ramanujan 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 , the highest of the H i n d u castes, they were destitute, living off the m e a g e r wages of Ramanujan'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 him, 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 trigonometric 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 the fascination of observing a compass needle. For Riemann, it was r e a d i n g L e g e n d r e ' s book on n u m b e r theory. For Ramanujan, it was when he stumbled on an obscure, forgotten book on mathematics by George Carr. This b o o k has since b e e n immortalized by the fact that it m a r k e d Ramanujan's only known exposure to m o d e r n Western mathematics. According to his sister, " I t was this book which awakened his genius. He set himself to establish the formulae given therein. 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 . . . . Ramanujan used to say that the goddess of Namakkal inspired him 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 and intensely p r e o c c u p i e d with the equations that were constantly dancing 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 return, but only to fall ill a n d fail his examinations again. With the help of friends, Ramanujan m a n a g e d to b e c o m e a low-level clerk in the Port Trust of Madras. It was a menial j o b , paying a paltry £20 a year, b u t it freed Ramanujan, like Einstein before him at the Swiss p a t e n t office, to follow his d r e a m s in his spare time. Ramanujan then mailed some of the results of his " d r e a m s " to t h r e e well-known British mathematicians, h o p i n g for contact with o t h e r mathematical minds. Two of the mathematicians, receiving this letter written by an u n k n o w n Indian clerk with no formal education, promptly threw it away. T h e third o n e was the brilliant Cambridge mathematician Godfrey H. Hardy. Because of his stature in England, Hardy was accustomed to receiving crank mail a n d t h o u g h t dimly of the letter. Amid the dense scribbling he noticed many t h e o r e m s of mathematics that were already well known. 14

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T h i n k i n g it the obvious work of a plagiarist, he also threw it away. But something wasn't quite right. S o m e t h i n g nagged at Hardy; he c o u l d n ' t help w o n d e r i n g a b o u t this strange letter. At d i n n e r that night, J a n u a r y 16,1913, Hardy a n d his colleague J o h n Littlewood discussed this o d d letter a n d decided to take a second look at its contents. It began, 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 the Accounts D e p a r t m e n t of the Port Trust 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 contained t h e o r e m s that were totally u n k n o w n to Western mathematicians. In all, it contained 120 t h e o r e m s . Hardy was s t u n n e d . He recalled that proving some 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 that they could only be written down by a mathematician of the highest class." 1 5

16

Littlewood a n d Hardy r e a c h e d the 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 of E u r o p e a n mathematics. " H e h a d b e e n carrying an impossible h a n d icap, a p o o r a n d solitary H i n d u pitting his brains against the accumulated 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 Cambridge in 1914. For the first time, Ramanujan could communicate 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 began a burst of activity: 3 short, intense years of collaboration with Hardy at Trinity College in Cambridge. Hardy later tried to estimate the mathematical skill that Ramanujan possessed. He rated David Hilbert, universally recognized as o n e of the greatest Western mathematicians of the n i n e t e e n t h century, an 80. To Ramanujan, he assigned a 100. (Hardy rated himself a 25.) Unfortunately, n e i t h e r Hardy n o r Ramanujan s e e m e d interested in the psychology or thinking process by which Ramanujan discovered these incredible theorems, especially when this flood of material came p o u r i n g out of his " d r e a m s " with such frequency. Hardy n o t e d , " I t seemed ridiculous to worry h i m a b o u t how he h a d found this or that known t h e o r e m , when he was showing me half a dozen new ones 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

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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 , " h e r e p l i e d , "it i s a very i n t e r e s t i n g n u m b e r ; i t i s t h e s m a l l e s t 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 the sum 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 the spot, he could recite complex theorems in arithmetic that would require 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 the war-torn British economy prevented Ramanujan from maintaining his strict vegetarian diet, and he was constantly in a n d o u t of sanitariums. After collaborating with Hardy for 3 years, Ramanujan fell ill a n d never recovered. World War I i n t e r r u p t e d travel between England 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 died a year later.

Modular Functions Ramanujan'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 theorems 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 paper, containing the o u t p u t of the last year of his life, was discovered by accident in a box at Trinity College. This is now called Ramanujan's " L o s t N o t e b o o k . " C o m m e n t i n g on the Lost Notebook, mathematician 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 mathematician. What 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 the " 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 knowledge no mathematical redaction of this scope or difficulty has ever been attempted." 20

Looking at the progression of Ramanujan's equations, it's as t h o u g h we have b e e n trained for years to listen to the Western music of Beethoven, a n d t h e n suddenly we are exposed 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. Perhaps 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 haven't b e e n invited t o . "

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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 performing a long a n d difficult calculation, a n d t h e n suddenly having thousands 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 , underlying 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 deforming the string's world sheet. This is precisely where Ramanujan's work comes in. In o r d e r to protect 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 Ramanujan's modular function. In summary, we have said that o u r fundamental premise is that the laws of n a t u r e simplify when expressed in h i g h e r dimensions. 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 dimensions. T h e addition of the word self-consistently is crucial. This constraint forces us to use Ramanujan's m o d u l a r functions, which fixes the dimension of s p a c e - t i m e to be ten. This, in turn, may give us the 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, God h a d no choice. Self-consistency alone, they claim, must have forced God to create the universe as he did. Although the mathematical sophistication i n t r o d u c e d by superstring theory has reached dizzying heights a n d has startled the mathematicians, 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 not really a theory at all! T h e main problem, as we have p o i n t e d out, is theoretical r a t h e r than experimental. If we were smart e n o u g h , we could solve the theory exactly a n d find the true nonperturbative solution of the theory. However, this does not excuse us from finding some m e a n s by which to verify the theory experimentally. To test the theory, we must wait for signals from the tenth dimension. 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 discove 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 fundamental p r o b l e m is that an e x p e r i m e n t a l test of the theory seems beyond o u r present-day technology. In fact, the theory predicts that the unification of all forces occurs at the Planck energy, or 1 0 billion electron 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 generating this fantastic energy, says, " T h e r e is not e n o u g h m o n e y in the treasuries of all the countries in the world p u t together. It's truly astronomical."' This is disappointing, because it m e a n s that experimental verification, t h e e n g i n e that drives progress in physics, is no longer possible with o u r c u r r e n t g e n e r a t i o n of machines or with any generation of machines 19

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in the conceivable future. This, in turn, m e a n s that the ten-dimensional theory is n o t a theory in the usual sense, because it is untestable given the present 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, the answer is a r e s o u n d i n g n o . They derisively call these theories "theatrical physics" or "recreational m a t h e m a t i c s . " T h e most caustic of the critics is Nobel Prize winner Sheldon Glashow of Harvard University. He has assumed the role of gadfly in this debate, leading the charge against the claims of o t h e r physicists that h i g h e r dimensions may exist. Glashow rails against these physicists, c o m p a r i n g the 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 the c u r r e n t bandwagon effect with former President Reagan's Star Wars program:

H e r e ' s a r i d d l e : N a m e two g r a n d d e s i g n s t h a t a r e i n c r e d i b l y 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 be 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 , i n b o t h c a s e s , t h e R u s s i a n s a r e trying d e s p e r a t e l y t o c a t c h 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 admit that he is often o u t n u m b e r e d on this question. He regrets, "I find myself a dinosaur in a world of upstart m a m m a l s . " (Glashow's views are certainly n o t shared by o t h e r Nobel laureates, such as Murray Gell-Mann a n d Steven Weinberg. Physicist Weinberg, in fact, says, "String theory provides o u r only p r e s e n t source of candidates for a final theory—how could anyone expect that many of the brightest y o u n g theorists would not work on it?" ) To u n d e r s t a n d the implications of this debate c o n c e r n i n g the uni4

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fication of all forces, a n d also the problems with its experimental verification, it is instructive to consider the following analogy, the " p a r a b l e of the g e m s t o n e . " In the beginning, 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 dimensions. However, this gemstone was unstable. O n e day, it burst apart a n d sent fragments in all directions; they eventually rained down on the 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 the original explosion the Big Bang, but 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 on o n e side, a n d Flatlanders c o m p a r e d t h e m to " 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 the years, the Flatlanders divided into two camps. T h e first c a m p began to piece together the polished fragments. Slowly, some of the 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 operating. These Flatlanders called their partially assembled piece "relativity." T h e second g r o u p devoted their efforts to assembling the jagged, irregular fragments. They, too, h a d limited success in finding patterns a m o n g these fragments. However, the 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 the 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 there 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 declared that the 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 dimension. Most Flatlanders were bewildered by this new a p p r o a c h , because no o n e could u n d e r s t a n d what " u p " meant. However, he 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 some 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 both sets of fragm e n t s were assembled in the third dimension, the Flatlanders gasped at what was revealed in the computer: a dazzling g e m s t o n e with perfect three-dimensional symmetry. In o n e stroke, the artificial distinction between the two sets of fragments was resolved by p u r e geometry. This solution, however, left several questions unanswered. Some Flat-

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landers still wanted 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 concrete n u m b e r for the energy it would take to build powerful machines that could hoist these fragments " u p " off Flatland a n d assemble the pieces in three-dimensional space. But the energy r e q u i r e d was about a quadrillion times the largest energy source available to the Flatlanders. For some, the theoretical calculation was sufficient. Even lacking experimental 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 pointed out, that the solution to the 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 the best minds 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 good thing. As the 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 better to d e b a t e a question without settling it than 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 must 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 duplicated over a n d over, in different locations a n d at different times with the same results, the theory is unreliable. But how can o n e perform 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 concerning 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 the p r o b l e m identified by H u m e in the e i g h t e e n t h century. T h e p r o b l e m remains the same: T h e energy necessary to re-create Creation exceeds anything available on the planet earth. However, although direct experimental

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verification of the 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 the 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 the superstring, such as supersymmetry. Although the SSC could n o t have p r o b e d the Planck energy, it might have given us strong, indirect evidence of the 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 , the last of its type. W h e n completed outside Dallas, Texas, a r o u n d the 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 m a c h i n e . T h e p u r p o s e of the SSC was to whip two beams of p r o t o n s a r o u n d inside this tube until they r e a c h e d a velocity very close to the speed of light. Because these beams 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 the tube when 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 smashed 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 occurred since the Big Bang itself ( h e n c e the n i c k n a m e for the SSC: "window on creat i o n " ) . A m o n g the debris, physicists h o p e d to find exotic subatomic particles that would have shed 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 the limits of known technology. Because the magnetic fields necessary to b e n d the p r o t o n s a n d antiprotons within the tube are so exceptionally large (on the o r d e r of 100,000 times the earth's magnetic 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 maintain t h e m . For example, to r e d u c e the heating a n d electrical resistance within the wires, the magnets 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 the 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 the past, the sites for atom smashers were decided by u n a b a s h e d political horse trading. For example, the state of Illinois was able to land the Fermilab accelerator in Batavia, j u s t outside Chicago, because (according 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 Vietnam War. T h e SSC was probably no different. Although many states vigorously c o m p e t e d for the project, it probably came as no surprise that in 1988 the great state of Texas l a n d e d the SSC, especially when b o t h the president-elect of the United States a n d the Democratic vice-presidential candidate came from Texas. Although billions of dollars have b e e n spent on the SSC, it will never be completed. To the h o r r o r of the physics community, the H o u s e of Representatives voted in 1993 to cancel the project completely. Intense 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 plum, generating thousands of j o b s a n d billions of dollars in federal subsidies for the 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 generates no direct c o n s u m e r benefits. In lean times, they argue, 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 , funding for the SSC project must be p u t into p r o p e r perspective. Star Wars funding 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. And a single B-2 stealth b o m b e r costs almost $1 billion.) Although the SSC is dead, what might we have discovered with it? At the very least, scientists h o p e d to find exotic particles, such as the mysterious Higgs particle predicted by the Standard Model. It is the Higgs particle that generates symmetry breaking a n d is therefore the origin of the mass of the quarks. T h u s we h o p e d that the SSC would have found 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 chance that the SSC would find exotic particles beyond the Standard Model. (Possibilities included " 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 matter problem.) But p e r h a p s the most exciting possibility was the sparticles, which are the supersymmetric p a r t n e r s of ordinary particles. T h e gravitino, for example, is the supersymmetric p a r t n e r of the graviton. T h e supersymmetric p a r t n e r s of the q u a r k a n d lepton, respectively, are the squark a n d the 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 the 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 the discovery of supergravity. In fact, the superstring is probably the 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 the potential discovery of sparticles will n o t prove the correctness of superstring theory, it will h e l p to quiet the skeptics who 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 the SSC will never be built, a n d h e n c e will never detect particles that are low-energy resonances of the superstring, then a n o t h e r possibility is to measure the 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 beyond o u r galaxy. For example, although no o n e knows where they c o m e from, cosmic rays have energies m u c h larger than anything found 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 on 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 hose water or waiting for a rainstorm. T h e hose water is m u c h m o r e convenient: We can turn it on any time we please, we can adjust the intensity of the water at will, a n d all the water travels at the same uniform velocity. Water from a fire hydrant therefore corresponds to p r o d u c i n g controlled beams 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 than 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 experiments performed by the Jesuit priest T h e o d o r Wulf atop the Eiffel Tower in Paris. F r o m the 1900s to the 1930s, courageous 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 began to fade d u r i n g the 1930s, when Ernest Lawr e n c e invented the cyclotron a n d p r o d u c e d controlled beams in the laboratory m o r e energetic t h a n most cosmic rays. For example, cosmic rays, which are as energetic as 100 million electron volts, are as c o m m o n as rain drops; 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 square inch p e r second. However, Lawrence's invention spawned giant m a c h i n e s that could exceed that energy by a factor of 10 to 100. Cosmic-ray experiments, fortunately, have c h a n g e d dramatically since Father Wulf first placed electrified j a r s on the Eiffel Tower. Rockets

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a n d even satellites can now send radiation c o u n t e r s high above the earth's surface, so that atmospheric effects are minimized. W h e n a highly energetic cosmic ray strikes the a t m o s p h e r e , it shatters the atoms in its wake. These fragments, in turn, create a shower of b r o k e n atoms, or ions, which can t h e n be detected 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 the most ambitious cosmic-ray project yet, a vast array of 1,089 detectors scattered over a b o u t a square mile of desert, waiting for the 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 Grounds, 80 miles southwest of Salt Lake City, Utah. T h e Utah detector is sensitive e n o u g h to identify the 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. They are probably large, s p i n n i n g n e u t r o n stars, or even black holes, that are 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 example, protons) into outer space. So far, the most energetic cosmic ray ever detected h a d an energy of 1 0 electron volts. This figure is an incredible 10 million times 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 generate 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. Although this fantastic energy is still 100 million times smaller than the energy necessary to p r o b e the t e n t h dimension, 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 the structure of these energy sources a n d detect energies even larger t h a n this. According to o n e favored theory, the largest energy source within o u r Milky Way galaxy—far beyond anything 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 that the ultimate p r o b e for exploring the t e n t h dimension 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 many times when physicists have solemnly declared certain p h e n o m e n a to be " u n t e s t a b l e " or " 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 that scientists can take c o n c e r n i n g

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the inaccessibility of the Planck energy—unforeseen 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 the n i n e t e e n t h century, some scientists declared that the composition of the stars would forever be beyond the reach of experiment. In 1825, the F r e n c h p h i l o s o p h e r a n d social critic Auguste Comte, writing in Cours de philosophie, declared that we would never know the stars other 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 machines of the n i n e t e e n t h century, or any century, he argued, were n o t powerful e n o u g h to escape from the earth a n d reach the stars. A l t h o u g h d e t e r m i n i n g what the stars were m a d e of seemed beyond the capabilities of any science, ironically at almost the same time, the G e r m a n physicist J o s e p h von Fraunhofer 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 the chemical composition of those stars. Since each chemical within the stars emits a characteristic "fingerprint," or s p e c t r u m of light, it was easy for Fraunhofer to perform the "impossible" a n d to d e t e r m i n e that hydrogen is the most a b u n d a n t e l e m e n t in the stars. This, in turn, 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 the energy necessary to reach the stars via rockets was far beyond anything available to C o m t e (or, for that matter, anything available to m o d e r n science), the 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 than 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 (perhaps from cosmic rays or perhaps 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 tenth 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 the existence of atoms. In the 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 thermodynamics. However, many physicists refused to believe that atoms actually exist. P e r h a p s they were j u s t a mathematical device that, by accident, gave the correct description of the world. For e x a m p l e , the 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 unable to take direct pictures of the atom because of the 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, although indirect, evidence of the existence of atoms when 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 suspended in a liquid) can be explained as r a n d o m collisions between the particles a n d atoms in the liquid. By analogy, we m i g h t h o p e for experimental confirmation of the physics of the tenth 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 some way. T h e third " u n t e s t a b l e " idea in physics was the existence of the 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 the 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 matter 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 Centauri a n d placed it in the path of a b e a m of neutrinos, some would still c o m e out the o t h e r e n d . They can p e n e t r a t e the earth as t h o u g h it d o e s n ' t even exist, a n d , in fact, trillions of neutrinos emitted from the sun are always p e n e t r a t i n g your body, even at night. Pauli admitted, "I have c o m m i t t e d the ultimate sin, I have predicted the 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 the n e u t r i n o that it even inspired a p o e m by J o h n Updike, called "Cosmic Gall": N e u t r i n o s , t h e y a r e very s m a l l . 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 ball 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 hall O r p h o t o n s t h o u g h a s h e e t o f 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 in h i s stall, A n d s c o r n i n g b a r r i e r s o f class, Infiltrate 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

Although 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 considered the ultimate " u n t e s t a b l e " idea, today we regularly p r o d u c e beams of n e u t r i n o s in atom smashers, perform experiments with the n e u t r i n o s emitted from a nuclear reactor, a n d detect their presence within mines far below the earth's surface. (In fact, when a spectacular supernova lit up the sky in the southern h e m i s p h e r e in 1987, physicists noticed a burst of neutrinos streaming t h r o u g h their detectors 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 make crucial astronomical measurements.) Neutrinos, in 3 short 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 Taking the long view on the history of science, p e r h a p s there 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 w a s h o p e l e s s l y i n a c c e s s i b l e . I f y o u t o l d a 1 9 t h - c e n t u r y physicist t h a t b y t h e 2 0 t h c e n t u r y y o u w o u l d b e a b l e t o c a l c u l a t e this, i t w o u l d h a v e 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 difficult 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 years.

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 whether scientists were n o t overstating the case when they insisted that everything should 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." Nobel 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 equations than to have t h e m fit e x p e r i m e n t . " " Or, in the words of CERN physicist J o h n Ellis, " i n the words of a candy wrapper 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 experimental situation looks bleak. I share, along with the skeptics, the idea that the best we can h o p e for is indirect tests of ten-dimensional theory into the twenty-first century. This is because, in the 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 our laboratories. 10

Personally, I d o n ' t think that we have to wait a century until o u r accelerators, space probes, a n d cosmic-ray counters will be powerful e n o u g h to p r o b e the tenth dimension 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 either verify or disprove the 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 bright physicist solves the field theory of strings a n d derives the known properties of o u r universe, t h e r e is still the practical p r o b l e m of when we might be able to harness the power of the hyperspace theory. T h e r e are two possibilities: 1. Wait until o u r civilization attains the ability to master energies trillions of times larger t h a n anything we can p r o d u c e today 2. E n c o u n t e r extraterrestrial civilizations that have mastered the art of manipulating hyperspace We recall that it took a b o u t 70 years, between the work of Faraday a n d Maxwell to the work of Edison a n d his co-workers, to exploit the electromagnetic force for practical purposes. Yet m o d e r n civilization d e p e n d s crucially on the harnessing of this force. T h e nuclear force was discovered n e a r the turn of the century, a n d 80 years later we still do not have the m e a n s to harness it successfully with fusion reactors. T h e next leap, to harness the power of the 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 fundamental 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, when its " n a t u r a l h o m e " lies at the 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 cheetah, 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 the c h e e t a h is the vast plains of Africa. T h e real " h o m e " of superstring theory is the 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 "laboratory" 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 ( I n d i a ,

ninth

century)

ID God have a m o t h e r ? " Children, when told that God m a d e the heavens a n d the earth, innocently ask w h e t h e r God 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 embarrassed the finest theologians, precipitating some of the thorniest theological debates over the 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 the logical paradoxes i n h e r e n t in the questions that even children ask. God may have created the heavens a n d the earth in 7 days, b u t what h a p p e n e d before the first day? If o n e concedes that God h a d a m o t h e r , then 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 on, forever. However, if God 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 God c o m e from? Was God always in existence since eternity, or is God beyond time itself?

"D

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Over the centuries, even great painters commissioned by the c h u r c h grappled with these ticklish theological debates in their works of art: W h e n depicting God or A d a m a n d Eve, do you give t h e m belly buttons? Since the navel marks the 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 God n o r A d a m a n d Eve could be painted with belly buttons. For example, Michelangelo faced this d i l e m m a in his celebrated depiction of Creation a n d the expulsion of Adam a n d Eve from the Garden of E d e n when he painted the ceiling of the Sistine Chapel. T h e answer to this theological question is to be found h a n g i n g in any large museum: God a n d A d a m a n d Eve simply have no belly buttons, 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, decided to raise the level of theological d e b a t e from the vagueness of mythology to the rigor of logic. He proposed to solve these ancient questions in his celebrated "proofs of the existence of G o d . " Aquinas summarized 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

(The 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 the fifth is called the teleological proof. T h e moral p r o o f is by far the weakest, because morality can be viewed in terms of evolving social customs.) Aquinas's "cosmological" a n d "teleological" proofs of the existence of God 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. Although 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 ingenious for their time a n d show the influence of the Greeks, who were the 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 began the cosmological p r o o f by postulating that God was the First Mover a n d First Maker. He artfully d o d g e d the 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 had 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 turn must have h a d s o m e t h i n g push it, a n d so on. But what started the first push? Imagine, for the m o m e n t , idly sitting in the 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 pushing the wagon. You wait a m o m e n t , only to find a n o t h e r wagon pushing the first wagon. Curious, you wait a bit longer for the child, b u t there 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 sequence of wagons stretching into the horizon, each wagon pushing the others, with no child at all. If it takes a child to push a wagon, then 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 sequence of wagons push itself? No. Therefore, God must exist. T h e teleological proof is even m o r e persuasive. It states that t h e r e has to be a First Designer. For example, imagine walking on the sands of Mars, where the winds a n d dust storms have worn even the m o u n t a i n s and giant craters. Over tens of millions of years, n o t h i n g has escaped the corrosive, grinding effect of the sand storms. T h e n , to your surprise, you find a beautiful c a m e r a lying in the 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 conclude 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 the surface of Mars some m o r e , you c o m e across a rabbit. Obviously, the eye of the rabbit is infinitely m o r e intricate than 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 camera. 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 the camera. This m a k e r must therefore be God. Now imagine the machines on the earth. T h e r e is no question that these machines 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 machine. Therefore, the person who created us must be infinitely m o r e complicated t h a n we are. So therefore God must exist. In 1078, St. Anselm, the archbishop of Canterbury, cooked 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 alone. He defined God as the most perfect, most powerful being 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 perform miracles, such as parting the rivers a n d raising the dead. Obviously, the second God (who exists) is m o r e powerful a n d m o r e perfect than the first God (who does n o t exist). However, we defined God to be the most perfect a n d powerful being imaginable. By the definition of God, the second God (who exists) is the m o r e powerful a n d m o r e perfect o n e . Therefore, 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 than the second God, a n d therefore does n o t fit the definition of God. H e n c e God must exist. In o t h e r words, if we define God as " t h a t b e i n g n o t h i n g greater t h a n which can be conceived," then God must exist because if he d i d n ' t , it's possible to conceive of a m u c h greater God who 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 the existence of God lasted for over 700 years, defying the 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 the potential flaws in these proofs. T h e flaw in the cosmological proof, for example, 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 example, gas molecules may b o u n c e against the walls of a c o n t a i n e r without r e q u i r i n g anyone or anything to get t h e m moving. In principle, these molecules can move forever, requiring 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 long as mass a n d energy are conserved. For the 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 ones t h r o u g h natural selection a n d chance. Ultimately, we can trace the origin of life itself back to the s p o n t a n e o u s formation of protein molecules in the early earth's oceans without appealing to a higher 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 containing m e t h a n e , a m m o n i a , a n d o t h e r gases found in the early earth'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 protein 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 out 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 the 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 example, this proof can be used to prove the existence of the unicorn. If we define the unicorn to be the most perfect horse imaginable, a n d if unicorns d o n ' t exist, t h e n it's possible to imagine a u n i c o r n that does exist. But saying that it exists does 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. Therefore, unicorns do n o t necessarily have to exist. And neither does 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 the 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 supplanted 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 the p r o b l e m of what came before God by defining him as the First Mover. Today, we are still struggling with the question of what h a p p e n e d before the Big Bang. Unfortunately, Einstein's equations break down at the enormously small distances a n d large energies found at the origin of the universe. At distances on the o r d e r of 1 0 centimeter, 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 must 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 add 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 breakdown of the 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 turn, is the t h e m e of this book in reverse. It is no wonder, therefore, that piecing together the dynamics of the Big Bang has b e e n so difficult. In effect, by going backward in time, we are reassembling the pieces of the ten-dimensional universe. - 3 3

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Experimental Evidence for the Big Bang Every year, we find m o r e experimental 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 experimental results. First, the 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 measuring the distortion of their starlight (called the red shift). ( T h e starlight of a receding star is shifted to longer wavelengths—that is, toward the red e n d of the spectrum—in the same way that the whistle of a r e c e d i n g train sounds h i g h e r than normal when a p p r o a c h i n g a n d lower when receding. This is called the Doppler effect. Also, H u b b l e ' s Law states that the farther from us the star or galaxy, the faster it is receding 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 the 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 the distribution of the 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 the Big Bang a n d in the stars. In the original Big Bang, because of the e n o r m o u s heat, elemental hydrogen nuclei banged 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 element: helium. T h e Big Bang theory predicts that the ratio of helium to hydrogen in the universe should be approximately 2 5 % helium to 7 5 % hydrogen. This agrees with the observational result for the abund a n c e of h e l i u m in the universe. Third, the earliest objects in the universe date back 10 to 15 billion years, in a g r e e m e n t with the r o u g h estimate for the Big Bang. We do n o t see any evidence for objects older t h a n the Big Bang. Since radioactive materials decay (for example, via the weak interactions) at a precisely known rate, it is possible to tell the age of an object by calculating the 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 carbon. O t h e r radioactive elements (like uranium-238, with a halflife of over 4 billion years) allow us to d e t e r m i n e the age of m o o n rocks (from the Apollo mission). T h e oldest rocks a n d m e t e o r s found on earth date to a b o u t 4 to 5 billion years, which is the 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 that the oldest stars in o u r galaxy date back about 10 billion years.

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Fourth, a n d most important, 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 should be measurable by o u r instruments. In fact, A r n o Penzias a n d R o b e r t Wilson of the Bell T e l e p h o n e Laboratories won the Nobel Prize in 1978 for detecting this e c h o of the Big Bang, a microwave radiation that p e r m e a t e s the known universe. T h e fact that the e c h o of the Big Bang should be circulating a r o u n d the universe billions of years after the event was first predicted by George Gamow a n d his students Ralph Alpher a n d Robert H e r m a n , b u t no o n e took t h e m seriously. T h e very idea of m e a s u r i n g the e c h o of Creation s e e m e d outlandish when they first p r o p o s e d this idea soon after World War II. Their logic, however, was very compelling. Any object, when heated, 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 the iron, the h i g h e r the frequency of radiation it emits. A precise mathematical 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 the 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 emitted radiation also decreases, until the iron no longer emits in the visible range. T h e iron returns to its n o r m a l color, b u t it continues to emit invisible infrared radiation. This is how the army's night 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 engines may be concealed in the darkness, b u t they c o n t i n u e to emit invisible blackbody radiation in the 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 the s u m m e r . Sunlight penetrates the glass of your car a n d heats the interior. As it gets hot, it begins to emit 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 the a t m o s p h e r e , caused by the b u r n i n g of fossil fuels, can trap the infrared blackbody radiation of the earth a n d thereby gradually h e a t the planet.) Gamow reasoned that the Big Bang was initially quite hot, a n d h e n c e would be an ideal blackbody emitter of radiation. A l t h o u g h the 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 predict that o n e day o u r instruments would be sensitive e n o u g h to detect 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 the p o i n t where atoms 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 atoms that would no longer be broken up by the intense radiation p e r m e a t i n g the universe. Before this time, the universe was so h o t that atoms were continually r i p p e d apart by radiation as soon as they were formed. This m e a n t that the 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, the radiation was no longer sufficiently strong to break up the atoms, a n d h e n c e light could travel long distances without being scattered. In o t h e r words, the 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 the early universe was n o t transp a r e n t at all, b u t filled with turbulent, o p a q u e radiation.) After 300,000 years, electromagnetic radiation no longer interacted so strongly with matter, a n d h e n c e b e c a m e blackbody radiation. Gradually, as the universe cooled, the frequency of this radiation decreased. Gamow a n d his students calculated that the radiation would be far below the infrared range, into the microwave region. Gamow reasoned that by s c a n n i n g the heavens for a uniform, isotropic source of microwave radiation, o n e should be able to detect this microwave radiation a n d discover the e c h o of the Big Bang. Gamow's prediction 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 found a mysterious b a c k g r o u n d radiation permeating all space when 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 unwanted radiation was d u e to electrical static caused by contaminants, such as bird d r o p p i n g s on their a n t e n n a . But when they disassembled a n d cleaned large portions of the a n t e n n a , they found that the "static" persisted. At the same time, physicists Robert Dicke a n d J a m e s Peebles at Princeton 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 informed of the Princeton 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 might 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 the universe!" They 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 predicted years earlier by George Gamow a n d his collaborators if the Big Bang had left a residual blanket of radiation that h a d cooled down to 3°K.

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COBE and the Big Bang Perhaps the most spectacular scientific confirmation of the Big Bang theory came in 1992 with t h e results of the COBE (Cosmic Background Explorer) satellite. On April 23, newspaper headlines across the country h e r a l d e d the findings of a team of scientists at the University of California at Berkeley, led by George Smoot, w h o a n n o u n c e d the 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 the "face of G o d " in their dispatches. T h e COBE satellite was able to improve vastly the earlier work of Penzias, Wilson, Peebles, a n d Dicke by many orders of m a g n i t u d e , sufficient to rule o u t all d o u b t that the fossil radiation emitted by the Big Bang h a d b e e n conclusively found. Princeton cosmologist J e r e m i a h P. Ostriker declared, " W h e n fossils were found in the rocks, it m a d e the origin of species absolutely clear-cut. Well, COBE found 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 the structure of the microwave backg r o u n d radiation first postulated by George 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 found that it was exceptionally smooth, with no a p p a r e n t ripples, kinks, or blotches. In fact, it was too smooth. 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 the universe, so uniform that scientists h a d difficulty reconciling it with known astronomical data. In the 1970s, astronomers 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 portions of the sky. To their surprise, they found 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 , containing billions of galaxies at a time, and the voids stretched across millions of light-years. But h e r e lay a cosmic mystery: If the Big Bang was exceptionally smooth 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 clumpiness that we see a m o n g the galactic clusters. T h e gross mismatch between the original s m o o t h Big Bang a n d the lumpiness of the universe 1 billion years later was a nagging 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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doubt; what was in trouble was o u r u n d e r s t a n d i n g of the post-Big Bang evolution 1 billion years after Creation. But without sensitive satellites that could measure the cosmic b a c k g r o u n d radiation, the p r o b l e m festered 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 began to write sensational articles saying erroneously that scientists h a d found a fatal flaw in the Big Bang theory itself. Many journalists 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 the press. Even the New York Times published a major article saying that 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 the a n n o u n c e m e n t of the COBE data all the m o r e interesting. With unprece d e n t e d accuracy, capable of detecting variations as small as o n e part in 100,000, the COBE satellite was able to scan the heavens a n d radio back the most accurate m a p of the cosmic b a c k g r o u n d radiation ever constructed. T h e COBE results reconfirmed 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 problems. For example, they had to subtract carefully the effect of the earth's m o t i o n in the background radiation. T h e solar system drifts at a velocity of 370 kilometers p e r second relative to the b a c k g r o u n d radiation. T h e r e is also the relative motion of the solar system with respect to the galaxy, a n d the galaxy's complex motions 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 came out 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 experimental n u m b e r s ) to within 0 . 1 % (Figure 9.1). T h e solid line represents the prediction; the x's mark t h e data points m e a s u r e d by the COBE satellite. W h e n this graph was flashed on the screen for the first time to a m e e t i n g of about a thousand astronomers, everyone in the r o o m e r u p t e d in a standing ovation. This was p e r h a p s the first time in the history of science that a simple graph received such a t h u n d e r o u s applause from so many distinguished scientists. Second, Smoot'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 and voids found 1 billion years after the Big Bang. (If these blotches had not b e e n found by COBE, t h e n a major revision in the post-Big Bang analysis would have h a d to be made.) Third, the results were consistent with, b u t did n o t prove, the so-

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Intensity

Frequency of cosmic 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 there was a m u c h m o r e explosive expansion of the universe at the initial instant of Creation t h a n the usual Big Bang scenario; it holds 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 occurred. General relativity, if taken to its limits, ultimately yields nonsensical answers. Einstein, realizing that general relatively simply breaks down at those enormously small dis-

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tances, tried to extend general relativity into a m o r e comprehensive theory that could explain these p h e n o m e n a . At the 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 Bang is the ten-dimensional superstring theory. Scientists are just now conjecturing how the ten-dimensional universe split into a four- and a six-dimensional universe. W h a t does o u r twin universe look like? O n e physicist who is struggling with these cosmic questions is Cumr u m Vafa, a Harvard professor who has spent 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 Cambridge, 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 the 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 troubled region of the world, all the way to the far reaches of six-dimensional space, long before the tumult 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 the 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 the left side of the screen, at exactly the 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 top of the screen. T h u s , Vafa explains, t h e r e is an entirely self-contained universe in that 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 shaped like. Vafa points out, surprisingly e n o u g h , that the topology of the video screen is that of an i n n e r tube! T h i n k of the video screen as a sheet of paper. 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 together 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 the right side of the tube. O n e way to glue these two e n d s is to b e n d the tube carefully into a circle, a n d seal the two o p e n e n d s together with glue (Figure 9.2). W h a t we have d o n e is to turn 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 reappears on the o t h e r side of the screen, this corresponds to the rocket ship moving across the glued j o i n t of the inner 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 the 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 the topology of an orbifold seems to fit the observed data rather well. To visualize an orbifold, think of moving 360 degrees in a circle. Everyone knows that we c o m e back to the same point. In o t h e r words, if I d a n c e 360 degrees a r o u n d a May pole, I know that I will come back to the same spot. In an orbifold, however, if we move less than 360 degrees a r o u n d the May pole, we will still c o m e back to the same point. Although this may s o u n d preposterous, it is easy to construct orbifolds. T h i n k of Flatlanders living on a c o n e . If they move less than 360 degrees a r o u n d the apex of the c o n e , they arrive at the 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 some Flatlanders live on what is called a Z-orbifold, which is equivalent to the surface of a square bean bag (like those found 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 returns 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 degrees (not 360 degrees), they return to the same place from which they started. T h e remarkable thing a b o u t Vafa's orbifolds is that, with just 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 wrong with this construction. If n o n e of the solutions of string theory contains the Standard Model, t h e n we must throw away superstring theory as a n o t h e r promising but ultimately incorrect theory. However, physicists are excited by the fact that it is possible to obtain solutions that are tantalizingly close to the Standard 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 out the properties of these weird surfaces in h i g h e r dimensions, ever since the French mathematician H e n r i Poincare p i o n e e r e d the subject of topology in the early twentieth century. T h u s the ten-dimensional theory is able to incorporate a large body of m o d e r n mathematics that previously seemed quite useless. Why Are There Three Generations? In particular, the rich storehouse of mathematical t h e o r e m s compiled by mathematicians over the past century are now being 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 the G U T s . Vafa a n d his co-workers have discovered many promising solutions to the string equations 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 the 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 fundamental 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 a r e m i l l i o n s a n d m i l l i o n s o f s o l u t i o n s t h a t h a v e 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 of 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 it 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 like 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 w a s a s o u r c e of e x c i t e m e n t two years a g o .

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Gross cautions that although some of these solutions are very close to the Standard Model, o t h e r solutions p r o d u c e undesirable physical properties: " I t is, however, slightly embarrassing that we have so many solutions b u t no good way of choosing 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 properties, a few potentially disastrous p r o p e r t i e s . " A layperson, hear6

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ing this for the 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 the answer. T h e p r o b l e m is that the p e r t u r b a t i o n theory, o n e of the m a i n tools in physics, is of no use. Perturbation theory (which adds up increasingly small q u a n t u m corrections) fails to break the ten-dimensional theory down to four a n d six dimensions. 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 the 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 improved 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 he warned me a b o u t the c o m p u t e r revolution, which eventually might 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 they?" To him, it was only a matter of time before mathematicians p u t all physics questions in the c o m p u t e r a n d physicists got on the u n e m p l o y m e n t line. I was taken aback by the 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, the 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 the 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. Although 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 decide which of the millions of possibilities was the 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 tunneling, 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 the 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, the answer to this question has to do with q u a n t u m theory. Perhaps the most startling (and complex) 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 example, I am now sitting in a chair. T h e t h o u g h t of my body suddenly zapping t h r o u g h the molecules of the wall next to me and reassembling, uninvited, in s o m e o n e else's living r o o m is an unpleasant o n e . Also an unlikely o n e . However, q u a n t u m mechanics postulates that t h e r e is a finite (although small) probability that even the most unlikely, bizarre events—such as waking up o n e m o r n i n g a n d finding o u r b e d in the m i d d l e of the Amazon jungle—will actually h a p p e n . All events, no matter 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 sounds 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 the laboratory and, in fact, solves the riddle of radioactive decay. Normally, the 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 the nuclear force. However, t h e r e is a small probability that the nucleus m i g h t fall apart, that the p r o t o n s a n d n e u t r o n s might escape by t u n n e l i n g past the large energy barrier, the nuclear force, that binds the nucleus 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 when they s h o u l d n ' t ; in fact, the conservation of energy law is briefly violated as the n e u t r o n s in the nucleus 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 tunneling t h r o u g h a wall within the lifetime of the 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 sixdimensional universe. To u n d e r s t a n d this form of tunneling, think of an imaginary Charlie Chaplin film, in which Chaplin is trying to stretch a bed sheet a r o u n d an oversize bed. T h e sheet is the 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 the mattress, o n e at a time. He grins with satisfaction o n c e he has stretched the b e d sheet smoothly a r o u n d all four corners of the bed. But the strain is too great; o n e elastic b a n d pops off o n e corner, a n d the b e d sheet curls u p . Frustrated, he pulls this elastic a r o u n d the corner, 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 corner, a n o t h e r elastic p o p s off a n o t h e r corner. This process is called symmetry breaking. T h e smoothly stretched bed sheet possesses a high d e g r e e of symmetry. You can rotate the bed 180

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degrees along any axis, a n d the b e d sheet remains the same. This highly symmetrical state is called the false vacuum. Although the false vacuum appears quite symmetrical, it is n o t stable. T h e sheet does n o t want to be in this stretched condition. T h e r e is too m u c h tension. T h e energy is too high. T h u s o n e elastic pops off, a n d the 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 sheet has g o n e to a lower-energy state with less symmetry. By rotating the curled-up b e d sheet 180 degrees a r o u n d an axis, we no longer r e t u r n to the same sheet. Now replace the b e d sheet with ten-dimensional s p a c e - t i m e , the space-time of ultimate symmetry. At the b e g i n n i n g of time, the universe was perfectly symmetrical. If anyone was a r o u n d at that time, he could freely pass t h r o u g h any of the ten dimensions without p r o b l e m . At that time, gravity a n d the weak, the strong, a n d the electromagnetic forces were all unified by the superstring. All matter 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 phase transition took place, a n d symmetry was lost. Because the universe began to split up into a four- a n d a six-dimensional universe, the universe was no longer symmetrical. Six dimensions have curled u p , in the same way that the b e d sheet curls up w h e n o n e elastic pops off a c o r n e r of a mattress. But notice that t h e r e are 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 the field theory of strings using the theory of phase 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 appears, 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 the psychological stages of development. A fundamental conflict characterizes each phase. W h e n this conflict is correctly resolved, we move on to the n e x t

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phase. If this conflict is n o t resolved, it may fester a n d even cause regression to an earlier period. Similarly, the psychologist J e a n Piaget showed that 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 smooth 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 the essence of dialectics. According to this philosophy, all objects (people, 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 opposing 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, the object goes to a h i g h e r stage, called the synthesis, where a new contradiction begins, a n d the process starts over again at a h i g h e r level. Philosophers call this the transition from " q u a n t i t y " to "quality." Small quantitative changes eventually build up until t h e r e is a qualitative r u p t u r e with the 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 the aristocracy retreated b e h i n d its fortresses. W h e n the tensions r e a c h e d the b r e a k i n g point, a phase transition occurred from the 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. For 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, the reservoir is in the false vacuum. T h e water would prefer to be in its true vacuum, m e a n i n g it would prefer to burst the d a m a n d wash downstream, to a state of lower energy. T h u s a phase transition would involve a d a m burst, which could have disastrous consequences. An even m o r e explosive example 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 to stable u r a n i u m nuclei. Although the u r a n i u m nucleus a p p e a r s stable, t h e r e are e n o r m o u s , explosive energies trapped within the u r a n i u m nucleus 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 nucleus tunnels to a lower state, which m e a n s that the nucleus spontaneously splits apart all by itself. This is called radioactive decay. However, it is possible, by shooting n e u t r o n s at the 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 phase 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 the following illustration: Consider a circular b a n q u e t table, where all the 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 . Looking at the b a n q u e t table t h r o u g h a mirror, we see the same thing: each guest seated a r o u n d the 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 the same. Now break the symmetry. Assume that the first d i n e r picks up the glass on his or h e r right. By custom, all the o t h e r guests pick up the c h a m p a g n e glass to their right. Notice that the image of the 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 the 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 b e e n broken. A n o t h e r example of symmetry b r e a k i n g comes from an a n c i e n t fairy tale. This fable concerns a princess who is t r a p p e d on top of a polished crystal sphere. Although t h e r e are no iron bars confining h e r to the sphere, she is a prisoner because if she makes the 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 smooth a n d slippery. This is an e x a m p l e of symmetry breaking. While the princess is atop the sphere, she is in a perfectly symmetrical state. T h e r e is no preferred direction for the s p h e r e . We can rotate the sphere at any angle, a n d the situation remains the same. Any false move off the center, however, will cause the princess to fall, thereby breaking the symmetry. If she falls to the west, for example, the symmetry of rotation is broken. 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 corresponds to a false vacuum. T h e true vacuum state corresponds to the princess falling off the s p h e r e . So a phase transition (falling off the sphere) corresponds to symmetry breaking (selecting the westerly direction). 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 tunneled its way to a four- a n d a six-dimensional universe. T h u s the original universe was in the state of the false vacuum, the state of m a x i m u m symmetry, while today we are in the b r o k e n state of the true vacuum. 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? What would h a p p e n if the superstring only temporarily chose o u r universe, b u t the true vacuum lay a m o n g the millions of possible orbifolds? This would have disastrous consequences. In many o t h e r orbifolds, we find that the Standard Model is n o t present. T h u s if the true vacuum were actually a state w h e r e the

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Standard 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 tumbling 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 , the 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 the 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 the speed of light, swallowing up entire star systems, galaxies, a n d galactic clusters, until it gobbled up the entire universe. We would never see it coming. 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 Consider an ordinary ice cube sitting in a pressure cooker in o u r kitchen. We all know what h a p p e n s if we turn on the 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 the stove, first it melts a n d turns into water; that is, it u n d e r g o e s a phase transition. Now let us heat 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 phase transition a n d turns into steam. Now c o n t i n u e to h e a t the steam to e n o r m o u s temperatures. Eventually, the water molecules break u p . T h e energy of the molecules exceeds the b i n d i n g energy of the molecules, which are r i p p e d apart 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). Although a plasma is n o t part 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 the sun. In fact, plasma is the most c o m m o n state of m a t t e r in the 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 the nuclei of h y d r o g e n a n d oxygen are ripped apart, a n d we have a " g a s " of individual n e u t r o n s a n d protons, similar to the interior of a n e u t r o n star. If we h e a t the " g a s " of nucleons 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 leptons (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, the electromagnetic force a n d the weak force will b e c o m e united. 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 28

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b e c o m e united, a n d the G U T symmetries [SU(5), O ( 1 0 ) , o r E ( 6 ) ] appear. Finally, at a fabulous 1 0 °K, gravity unites with the G U T force, a n d all the symmetries of the ten-dimensional superstring appear. We now have a gas of superstrings. At that point, so m u c h energy will have g o n e into the pressure cooker that the geometry of s p a c e - t i m e may very well begin to distort, a n d the 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 point, it may be advisable to leave the kitchen. 32

Cooling the Big Bang T h u s by heating 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 becomes restored as we increase the energy. By reversing this process, we can appreciate how the Big Bang occurred as a sequence 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 the instant of Creation, we have t h e following stages in the evolution of o u r universe. -43

10 seconds T h e ten-dimensional universe breaks down to a fourand a six-dimensional universe. T h e six-dimensional universe collapses down to 1 0 centimeter 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; the strong force is no l o n g e r united with the electroweak interactions. SU(3) breaks off from the G U T symmetry. A small speck in the 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 the 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 the 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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begin to form. Because light is no longer 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 becomes black. 3 billion years T h e first quasars appear. 5 billion years T h e first galaxies appear. 10 to 15 billion years T h e solar system is b o r n . A few billion years after that, the 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 the q u a n t u m theory of the weak, electromagnetic, a n d strong 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 wormholes can act as gateways to a n o t h e r universe and 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 the 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. cummings

Black Holes: Tunnels Through Space and Time

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LACK holes have recently seized the 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 exploring this strange prediction of Einstein's equations, the final stage in the d e a t h of a collapsed star. Ironically, the public remains largely unaware of p e r h a p s the 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 intense speculation in the scientific community that a black hole 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 must first u n d e r s t a n d what makes the stars shine, how they grow, a n d how they eventually die. A star is b o r n when a massive cloud of hydrogen gas many times the size of o u r solar system is slowly compressed by the force of gravity. T h e gravitational force compressing the gas gradually heats up the gas, as gravitational energy is converted into the kinetic energy of the hydrogen atoms. Normally, the repulsive charge of the p r o t o n s within the hydrogen 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, the kinetic 217

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energy of the p r o t o n s (which are hydrogen 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 helium, releasing vast quantities of energy. In o t h e r words, a star is a nuclear furnace, b u r n i n g hydrogen fuel a n d creating nuclear " a s h " in the form of waste helium. A star is also a delicate balancing act between the force of gravity, which tends to crush the star into oblivion, a n d the nuclear force, which tends to blow the star apart with the force of trillions of hydrogen bombs. A star t h e n m a t u r e s a n d ages as it exhausts its nuclear fuel. To see how energy is extracted from the fusion process a n d to understand the stages in the life of a star leading to a black hole, we must analyze Figure 10.1, which shows o n e of the most i m p o r t a n t curves in m o d e r n science, sometimes called the binding energy curve. On the horizontal scale is the atomic weight of the various elements, from hydrogen to u r a n i u m . On the vertical scale, crudely speaking, is the approximate average " w e i g h t " of each p r o t o n in the nucleus. Notice that hydrogen a n d u r a n i u m have p r o t o n s that weigh, on average, m o r e than the protons of o t h e r elements in the center of the diagram. O u r sun is an ordinary yellow star, consisting mainly of hydrogen. Like the original Big Bang, it fuses h y d r o g e n a n d forms helium. However, because the p r o t o n s in hydrogen weigh m o r e than the protons 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 energy released when hydrogen is fused into helium. This is why the sun shines. However, as the hydrogen is slowly used up over several billion years, a yellow star eventually builds up too m u c h waste helium, a n d its nuclear furnace shuts off. W h e n that h a p p e n s , gravity eventually takes over and crushes the star. As t e m p e r a t u r e s soar, the star soon becomes hot e n o u g h to b u r n waste h e l i u m a n d convert it into the o t h e r elements, like lithium a n d carbon. Notice that energy can still be released as we d e s c e n d down the curve to the 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 the same way that ordinary ash can still be b u r n e d u n d e r certain conditions). Although the star has decreased enormously 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 helium, its a t m o s p h e r e may extend o u t to the orbit of Mars. This is what is called a red giant. This means, of course, that the earth will be vaporized in the process. T h u s the curve also predicts the ultimate fate of the earth. 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. 2

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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 consumes the earth. (Ironically, the earth was originally b o r n o u t of the same swirling gas cloud that created o u r sun. Physics now predicts that the earth, which was created with the sun, will return 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 the star. T h e red giant shrinks to b e c o m e a white dwarf, a m i n i a t u r e star with the mass of an entire star squeezed down to a b o u t the size of the p l a n e t earth. 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 turn into a white dwarf and, over billions of years, slowly die as it exhausts its nuclear fuel. It will eventually become 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 the mass of o u r s u n ) , t h e n most of the elements in the white dwarf will c o n t i n u e to be fused into increasingly heavier elements, eventually r e a c h i n g iron. O n c e we reach iron, we are n e a r the very b o t t o m of the curve. We can no longer extract any m o r e energy from the excess mass, so the nuclear furnace shuts off. Gravity o n c e again takes over, crushing 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 the o u t e r layer of the white dwarf blows off, releasing the largest burst of energy known in the galaxy, an exploding star called a supernova. Just o n e supernova can temporarily outshine an entire galaxy of 100 billion stars. In the aftermath of the supernova, we find a totally dead star, a neutron star a b o u t the 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 the 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 . Although n e u t r o n stars are almost invisible, we can still detect t h e m with o u r instruments. Because they emit some radiation while they are rotating, they act like a cosmic lighthouse in outer 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 the heavier elements beyond iron can be synthesized in the h e a t a n d pressure of a supernova. W h e n the star explodes, it releases vast a m o u n t s of stellar debris, consisting of the h i g h e r elements, into the vacuum of space. This debris eventually mixes with o t h e r gases, until e n o u g h hydrogen gas is accu-

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mulated to begin the gravitational contraction process o n c e again. Second-generation stars that are b o r n out of this stellar gas a n d dust contain an a b u n d a n c e of heavy elements. 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 contain these heavy elements. This solves a long-standing mystery in cosmology. O u r bodies are m a d e of heavy elements beyond iron, b u t o u r sun is n o t h o t e n o u g h to forge them. If the earth a n d the atoms of o u r bodies were originally from the same gas cloud, t h e n where did the heavy elements of o u r bodies c o m e from? T h e conclusion is inescapable: T h e heavy elements 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 supernova 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 pictured as a pinball m a c h i n e , as in Figure 10.1, with the shape of the b i n d i n g energy curve. T h e ball starts at the top a n d bounces from hydrogen, to helium, from the lighter elements to the heavier elements. Each time it b o u n c e s a l o n g the curve, it becomes a different type of star. Finally, the 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 into 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 the o t h e r side of the curve, at uranium, a n d go down the curve in a single b o u n c e by fissioning the uran i u m nucleus into fragments. Since the average weight of the p r o t o n s in fission products, like cesium a n d krypton, is smaller than the average weight of the p r o t o n s in u r a n i u m , the excess mass has b e e n converted into energy via E = mc . This is the 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 the birth a n d death of stars a n d the creation of the elements, b u t also makes possible the existence of hydrogen a n d atomic b o m b s ! (Scientists are often asked whether it would be possible to develop nuclear b o m b s o t h e r t h a n atomic a n d hydrogen b o m b s . From the curve of b i n d i n g energy, we can see that the answer is n o . Notice that the curve excludes the possibility of b o m b s m a d e of oxygen or iron. These elements are n e a r the 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 the press, such as n e u t r o n b o m b s , are only variations on u r a n i u m a n d hydrogen bombs.) W h e n o n e first hears the 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 the heavens, it 2

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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 naked eye in the s o u t h e r n h e m i s p h e r e , yielded a treasure trove of astronomical data that m a t c h e d the theoretical predictions of a collapsing dwarf with an iron core. 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 predict the 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 who was an astronomy major. He would invariably disappear in the early m o r n i n g a n d return late at night. J u s t before he would leave, he would say that he was putting a star in the oven to watch it grow. At first, I t h o u g h t he said this in jest. However, when I pressed h i m on this point, he said with all seriousness that he was p u t t i n g a star into the 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 the t h e r m o d y n a m i c equations a n d the fusion equations were well known, it was j u s t a matter of telling the c o m p u t e r to start with a certain mass of hydrogen gas a n d t h e n letting it numerically 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 the known stages of star life that we see in the heavens with o u r telescopes.

Black Holes If a star was ten to 50 times the size of o u r sun, t h e n gravity will continue to squeeze it even after it b e c o m e s a n e u t r o n star. Without the force of fusion to repel the gravitational pull, t h e r e is n o t h i n g to oppose the final collapse of the star. At this point, it b e c o m e s the famous black hole. In some 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, and fusion, which tries to blow the star apart like in a hydrogen b o m b . All the various phases in the life history of a star are a c o n s e q u e n c e of this delicate balancing act between gravity a n d fusion. S o o n e r or later, when all the nuclear fuel in a massive star is finally exhausted 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 that can t h e n resist the powerful force of gravity. Eventually, the gravitational force will take over a n d crush the n e u t r o n star into nothingness. T h e star has c o m e full circle: It was b o r n when gravity first began to compress hydrogen gas in the heavens into a star, a n d it will die when the 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 the 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 becomes 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 permanently the gravitational pull of that body. For example, 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 earth a n d go into d e e p space. O u r space probes like the Voyager that have v e n t u r e d into d e e p space a n d have completely left the solar system (carrying good-will messages to any aliens who might 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 the oxygen atoms do n o t have e n o u g h velocity to escape the earth'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 the primordial hydrogen of the 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 the past 5 billion years.) Newton's theory of gravity, in fact, gives the precise relationship between the escape velocity a n d the mass of the star. T h e heavier the planet or star a n d the smaller its radius, the larger the escape velocity necessary to escape its gravitational pull. As early as 1783, the English astronomer 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 might 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, the star would a p p e a r totally black. Using the best knowledge available in the e i g h t e e n t h century, he actually calculated the mass of such a black hole.* Unfortunately, his theory was considered 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 instruments 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 the *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." 2

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pedestrian point of view, the " f o r c e " between the star a n d a light beam 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 point of view, in which case the "shortest 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 means 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 hole 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 hypersphere. This piece of s p a c e time has now disconnected 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 the 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 equations, Schwarzschild was able to solve Einstein's equations 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 the 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 person unfortunate e n o u g h to c o m e within the Schwarzschild radius would be captured by the black hole a n d c r u s h e d to death. Today, this distance from the black hole is called the Schwarzschild radius, or the horizon (the farthest visible p o i n t ) . Second, anyone w h o fell within the Schwarzschild radius would be aware of a " m i r r o r universe" 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 the existence of this bizarre mirror 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 into the c e n t e r of a black hole 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 crushed. T h e electrons would be ripped 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 torn 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 than the 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 the 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 considered a mathematical quirk. T h e bridge was necessary

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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 the Einstein-Rosen bridge. Einstein-Rosen bridges were s o o n found in o t h e r solutions of the gravitational equations, such as the 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, the Ein-

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stein-Rosen bridge r e m a i n e d a curious b u t forgotten footnote in the lore of relativity. Things began to c h a n g e with the work of New Zealand mathematician Roy Kerr, w h o in 1963 found a n o t h e r exact solution to Einstein's equations. Kerr assumed that any collapsing star would be rotating. Like a s p i n n i n g skater who speeds up when bringing in his or h e r hands, a rotating star would necessarily accelerate as it began to collapse. T h u s the stationary Schwarzschild solution for a black hole 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 once 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 years, 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 t h a t a d i s c o v e r y m o t i v a t e d b y a s e a r c h after 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 found, however, that a massive rotating star does n o t collapse into a point. Instead, the s p i n n i n g star flattens until it eventually is compressed into a ring, which has interesting properties. If a p r o b e were shot into the black hole from the side, it would hit the ring a n d be totally demolished. T h e curvature of s p a c e - t i m e is still infinite when approaching the ring from the 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 ring from the top 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 hole along its axis of rotation might, in principle, survive the 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 the mirror universe without b e i n g destroyed by infinite curvature. T h e Einstein-Rosen bridge acts like a t u n n e l c o n n e c t i n g two regions of space-time; it is a w o r m h o l e . T h u s the Kerr black hole is a gateway to a n o t h e r universe. Now imagine that your rocket has e n t e r e d the Einstein-Rosen bridge. As your rocket a p p r o a c h e s the spinning black hole, it sees a

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ring-shaped spinning star. At first, it appears that the 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 hole from the n o r t h pole. However, as we get closer to the ring, light from the mirror universe reaches o u r sensors. Since all electromagnetic radiation, including radar, orbits the black hole, o u r r a d a r screens are detecting 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 ricocheting 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 hole. Because the same light b e a m orbits the black hole n u m e r o u s times, o u r rocket's radar detects images that have g o n e s p i n n i n g a r o u n d the black hole, creating the illusion of objects that 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 principle). Einstein's famous equation gives us the precise d e g r e e of s p a c e time b e n d i n g caused by the presence 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 "dilithium crystals" that power the Enterprise must perform miraculous feats of warping space a n d time. This m e a n s that the dilithium crystals have the magical power of b e n d i n g the 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 the earth to the nearest 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 Centauri comes 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 the 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 the 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 contracted, a n d we j u s t step across this contracted distance. Similarly, the Enterprise does n o t really cross the entire space to Alpha Centauri; it simply moves across the crum-

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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 better u n d e r s t a n d what happ e n s when o n e falls down the Einstein-Rosen bridge, let us now discuss the 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 bright afternoon, m i n d i n g o u r own business, when 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 the m o m e n t that the energy necessary to o p e n this window might be e n o u g h to shatter the earth. This is a purely hypothetical example.) We step up to the hovering 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, when 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 the o t h e r side of the window. W h e n we look below the window to find the dinosaur's body, we can see all the way down the street, as t h o u g h the d i n o s a u r a n d the window w e r e n ' t t h e r e at all. Puzzled, we slowly circle the 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)! Frightened, we walk a r o u n d the window o n c e m o r e , staring at the window sideways. Much 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. From the o t h e r direction, we see the h e a d of the brontosaur. And when we look from the side, we find that both the m i r r o r a n d the dinosaurs have disappeared. What's h a p p e n i n g ? In some faraway universe, the tyrannosaur a n d the b r o n t o s a u r have squared off in a life-and-death confrontation. As they face each other, a floating window suddenly appears between t h e m . W h e n the tyrannosaur peers into the floating mirror, he is startled to see the head 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 brontosaur stares into the same window from the o t h e r direction, he sees Fifth Aven u e , with its shops a n d traffic. T h e n the tyrannosaur finds that this h u m a n creature in the window has disappeared, only to a p p e a r on the side of the window facing the brontosaur. Now let us say that suddenly the wind blows o u r hat into the window. We see the h a t sailing into the sky of the o t h e r universe, b u t it is nowhere to be seen along 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 into the window to retrieve the hat. As seen by the tyrannosaur, a hat blows o u t the window, a p p e a r i n g from nowhere. T h e n he sees a disembodied h a n d r e a c h i n g o u t the window, desperately g r o p i n g for the hat. T h e wind now changes direction, a n d the h a t is carried in the o t h e r

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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. We stick o u r o t h e r h a n d into the 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 the 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 disappeared. H o w does this a p p e a r to the dinosaurs? They see two wiggling, tiny h a n d s d a n g l i n g from the 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 some of the 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 remarkable that such a simple i d e a — t h a t h i g h e r dimensions can unify space with time, a n d that a " f o r c e " can be explained by the warping of that space-time—would 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 general 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 wormhole or dimensional 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 turn, may actually close the o p e n i n g of the 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 the ten-dimensional theory is completed to decide w h e t h e r these wormholes are physically relevant or just a n o t h e r crazy idea. However, before we discuss the question of q u a n t u m corrections a n d the ten-dimensional theory, let us now 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 wormholes. J u s t as physicists can show that wormholes 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, consequence of multiply c o n n e c t e d universes: building a time m a c h i n e .

II To Build a Time Machine P e o p l e like u s , w h o b e l i e v e i n p h y s i c s , k n o w t h a t t h e d i s t i n c t i o n b e t w e e n past, p r e s e n t , a n d f u t u r e i s o n l y a s t u b b o r n l y persistent illusion. Albert Einstein

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AN we go backward in time? Like the protagonist in H. G. Wells's The Time Machine, can we spin the dial of a m a c h i n e a n d leap h u n d r e d s of thousands 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 harbors a secret wish s o m e h o w to relive the past a n d correct some small b u t vital mistake in o n e ' s life. In Robert Frost's p o e m " T h e Road Not T a k e n , " we w o n d e r what might 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 path. 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 mate, or e n t e r different careers; or we could even c h a n g e the o u t c o m e of key historical events a n d alter t h e fate of humanity. For e x a m p l e , in the climax of Superman, o u r h e r o is emotionally devastated when 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 of tons of rock a n d debris. M o u r n i n g h e r horrible death, he is so overcome 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 the light barrier, disrupting the fabric of space a n d time. By traveling at the speed 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 Lane was crushed 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 the 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 the process. T h u s the faster-than-light travel m e t h o d preferred by most science-fiction writers contradicts the special theory of relativity. Einstein himself 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 faster 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 relative way, A n d returned on the previous night.

Most scientists, who have n o t seriously studied Einstein's equations, dismiss time travel as poppycock, with as m u c h validity as lurid accounts of kidnappings by space aliens. However, the situation is actually quite complex. To resolve the question, we must leave the 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. General relativity has m u c h wider validity than 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 some of the simpler conclusions of the special theory. For any physicist who has seriously analyzed the 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 equations for general relativity do allow some forms of time travel. They acknowledge, however, that the energies necessary to twist time into a circle are so great that Einstein's equations break down. In the physically interesting

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region 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 general relativity. Einstein's equations, 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 the 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 concentrations 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 general relativity breaks down a n d q u a n t u m corrections begin 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 break 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 where the hyperspace theory can settle the 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 tendimensional 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 dimensional 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 the controversy s u r r o u n d i n g time travel and 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 might 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 the 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 many 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 crumble. 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 the course of history. All of a s u d d e n , the a u d i e n c e at Ford'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 the future bickering a m o n g themselves to see who would have the h o n o r of preventing

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Lincoln's assassination. T h e landing at N o r m a n d y would be b o t c h e d as thousands 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 beyond recognition. Consider Alexander t h e Great's decisive victory over t h e Persians, led by Darius III, in 331 B.C. at the Battle of Gaugamela. This battle led to the collapse of the Persian forces a n d e n d e d their rivalry with the West, which h e l p e d allow the flourishing of Western civilization a n d culture over the world for the 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 mercenaries 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 the battle. T h e slightest display of m o d e r n firepower would r o u t Alexander's terrified soldiers. This m e d d l i n g in the past would cripple the expansion of Western influence in the world. Time travel would m e a n that any historical event could never be completely resolved. History books could never be written. Some dieh a r d would always be trying to assassinate General Ulysses S. G r a n t or give the secret of the atomic b o m b to the G e r m a n s in the 1930s. What 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 that by the slightest breeze. 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 the 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 the 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. Not 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 paradoxes immediately arise whenever we e n t e r the past or future. Cosmologist S t e p h e n Hawking, in fact, has used 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 that 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 the future."

Time Paradoxes To u n d e r s t a n d the p r o b l e m s with time travel, it is first necessary to classify the various paradoxes. In general, most can be b r o k e n down into o n e of two principal types: 1. Meeting your parents before you are b o r n 2. T h e m a n with no past

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T h e first type of time travel does the 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 meets 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 prevented the fateful e n c o u n t e r between his parents. 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 second p a r a d o x involves events without any beginning. For example, 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 appears a n d offers him ample funds a n d the c o m p l e x equations a n d circuitry to make a time m a c h i n e . T h e inventor subsequently enriches himself with the knowledge of time travel, knowing b e f o r e h a n d exactly when 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, horse 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 basement, a n d gives his y o u n g e r self the secret of time travel a n d the m o n e y to exploit it. T h e question is: W h e r e did the idea of time travel c o m e from? P e r h a p s the craziest of these time travel p a r a d o x e s of the second type was cooked up by R o b e r t Heinlein 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 are, 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 when 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, who t h e n disappears. Second, d u r i n g the complicated delivery, doctors 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 the 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 out his pathetic 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 the c h a n c e to avenge the stranger who 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 , on the condition that he j o i n the " 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 the b a r t e n d e r d r o p s off the drifter in 1963. T h e drifter is strangely attracted to a y o u n g o r p h a n woman, who 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 the baby girl from the hospital, a n d d r o p s off the 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 thoroughly 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 date with destiny, m e e t i n g a certain drifter at Pop'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, grandfather, grandm o t h e r , son, daughter, g r a n d d a u g h t e r , a n d grandson? 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 . These paradoxes 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 branches are curled inward back on themselves, as in a circle. We come to the astonishing conclusion that she is h e r own m o t h e r a n d father! She is an entire 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 the 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 the m o r n i n g in b e d instead of going to work. Although it appears that we are d o i n g n o t h i n g by loafing in bed, we are actually tracing o u t a "world l i n e . " Take a sheet of graph paper, 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 engaging 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, according 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 the 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 the office earlier, at 12:30. This m e a n s that the faster we travel, the m o r e o u r world line deviates from the 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 diagram that o u r world line c a n ' t e n t e r because we would have to be going faster than the speed of light.) O n e conclusion is immediate. O u r world line never really begins or ends. Even w h e n we die, the world lines of the molecules in o u r bodies k e e p going. These molecules may disperse into the air or soil, but they will trace o u t their own never-ending world lines. Similarly, when we are b o r n , the world lines of the 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 break off or a p p e a r from nothing. To see how this all fits together, take the simple example of o u r own personal 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 the 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 (ours). Eventually, when someo n e dies, the world lines forming the person disperse into billions of world lines of o u r molecules. From this p o i n t of view, a h u m a n being can be defined as a temporary collection of world lines of molecules. T h e s e world lines were scattered before we were b o r n , came together 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 picture, we might say, "from world lines to world lines." 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, the great Russian cosmologist George 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 autobiography My World Line. With the aid of the world line, we can now picture what h a p p e n s when 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 now see why this is impossible. W h e n we disappear, o u r world line disappears. However, according 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 second p a r a d o x , involving re-creating the past, poses interesting problems, however. For e x a m p l e , by going back in time, we are fulfilling the past, n o t destroying it. T h u s the world line of the 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. 239

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travel is a closed loop. His world line fulfills, r a t h e r than changes, the past. Much m o r e complicated is the world line of " J a n e , " the woman 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 the 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 who h a p p e n s to be ourselves when we were younger. We tell this young 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 when we were young, we m e t a curious, older person who 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 end. They can p e r h a p s perform loops in time, b u t never alter it. These 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 the past, b u t is too primitive to settle the question of whether time travel makes any sense. To answer this larger question, we must turn to the general theory of relativity, where the situation becomes m u c h m o r e delicate. With the full power of general relativity, we see that these twisted world lines might be physically allowed. These closed loops go by the scientific n a m e closed timelike curves (CTCs). T h e debate in scientific circles is w h e t h e r CTCs are allowed by general 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, the Viennese mathematician Kurt Godel, also at the Institute for Advanced Study at Princeton, where Einstein worked. Godel found a disturbing solution to Einstein's equations that allowed for violations of the basic tenets of c o m m o n sense: His solution allowed for certain forms of time travel. For the first time in history, time travel was given a mathematical foundation. In some quarters, Godel was known as a spoiler. In 1931, he became famous (or, actually, infamous) when he proved, contrary to every expectation, that you c a n n o t prove the self-consistency of arithmetic. In the process, he r u i n e d a 2,000-year-old d r e a m , dating 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 achievement of mathematics: to r e d u c e all of mathematics to a small, self-consistent set of axioms from which everything could be derived. In a mathematical tour 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 . Godel's result was the most startling, u n e x p e c t e d d e v e l o p m e n t in mathematical 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 the purest of all sciences because it was precise a n d certain, u n t a r n i s h e d by the unpleasant crudeness of o u r material world, now b e c a m e uncertain. After Godel, the fundamental basis for mathematics s e e m e d to be left adrift. (Crudely speaking, Godel's remarkable p r o o f began by showing that t h e r e are curious paradoxes in logic. For example, consider the statement " T h i s sentence is false." If the sentence is true, t h e n it follows that it is false. If the sentence is false, t h e n the sentence is true. Or consider the statement "I am a liar." T h e n I am a liar only if I tell the truth. Godel t h e n formulated the s t a t e m e n t " T h i s sentence 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 complex web of such paradoxes, Godel showed that there are true statem e n t s that c a n n o t be proved using arithmetic.) After demolishing o n e of the most cherished d r e a m s of all of mathematics, Godel n e x t shattered the 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, including time travel. He first assumed that the 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 seem 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 equations were possible with the same distribution of dust a n d gas. (This m e a n t that Mach's principle was somehow incomplete, that h i d d e n assumptions were present.) More i m p o r t a n t , he showed that certain forms of time travel were p e r m i t t e d . If o n e followed the 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 the past. He wrote, "By m a k i n g a r o u n d trip on a rocket ship in a sufficiently wide curve, it is possible in these worlds to travel into any region of the past, present, a n d future, a n d back a g a i n . " T h u s Godel found the first CTC in general relativity. Previously, Newton considered time to be moving like a straight 2

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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 but 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 might momentarily shift the direction of the river, but 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 the 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 the main, a river may flow forward, b u t at the edges t h e r e are always side pools where water flows in a circular motion. Godel's solution could n o t be dismissed as the work of a crackpot because Godel h a d used Einstein's own field equations 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 equations, Einstein was forced to take the 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 the 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 instruments 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 rather disturbing, although 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 parents 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 equations that was even crazier than Godel's. Unlike the Godel universe, their solution was n o t based on a rotating dust-filled universe. On the surface, it resembled a typical black hole. As in the Godel solution, their universe allowed for CTCs a n d time travel. Moreover, when -going 360 degrees a r o u n d the black hole, you would n o t wind up where 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 sheet 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 might 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 the staircase, we do n o t arrive at the 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 the 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 that it was quickly c o i n e d the 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 the Godel solution; that is, o u r universe d i d n ' t seem to evolve in the way predicted by these solutions, so they were arbitrarily discarded for experimental 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 the early 1970s, Frank J. Tipler at Tulane University in New Orleans reanalyzed an old solution to Einstein's equations found by W . J . van Stockum in 1936, even before Godel's solution. This solution assumed the existence of an infinitely long, rotating cylinder. Surprisingly e n o u g h , Tipler was able to show that this solution also violated causality. Even the Kerr solution (which represents 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 the c e n t e r of the Kerr black hole (assuming they are n o t crushed in the process) could violate causality. Soon, physicists found that NUT-type singularities could be inserted into any black hole or e x p a n d i n g universe. In fact, it now became possible to cook up an infinite n u m b e r of pathological solutions to Einstein's equations. For example, every w o r m h o l e solution to Einstein's equations could be shown to allow some form of time travel. According to relativist Frank Tipler, "solutions to the field equations can be found 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 equations, in some sense, were like a Trojan horse. On the surface, the horse 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 the origin of the 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 wormholes a n d time travel. T h e price we h a d to pay for peering into the darkest secrets of the universe was the potential downfall of some of o u r most c o m m o n l y held 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 unalterable. 3

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But the question still r e m a i n e d : Could these CTCs be dismissed on purely experimental 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?

To Build a Time Machine In J u n e 1988, three physicists (Kip T h o m e a n d Michael Morris at the California Institute of Technology a n d Ulvi Yurtsever at the University of Michigan) m a d e the first serious proposal for a time m a c h i n e . They convinced the editors of Physical Review Letters, o n e of the most distinguished publications in the world, that their work merited serious consideration. (Over the decades, scores of crackpot proposals for time travel have b e e n submitted to mainstream 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 arguments in accepted field theoretical language a n d t h e n carefully explained where their weakest assumptions were. To overcome the skepticism of the scientific community, T h o m e a n d his colleagues realized that they would have to overcome the standard objections to using wormholes as time machines. First, as m e n t i o n e d earlier, Einstein himself realized that the gravitational forces at the center of a black hole would be so e n o r m o u s that any spacecraft would be torn apart. Although wormholes were mathematically possible, they were, in practice, useless. Second, wormholes might be unstable. O n e could show that small disturbances in wormholes would cause the Einstein-Rosen bridge to collapse. T h u s a spaceship's presence inside a black hole would be sufficient to cause a disturbance that would close the 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 penetrate the wormhole to the o t h e r side. Fourth, q u a n t u m effects would be so large that the w o r m h o l e m i g h t close by itself. For example, the intense radiation emitted by the entrance to the black hole 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, but also m i g h t close the 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 the undesirable feature that as seen by s o m e o n e on the earth, a space traveler appears to slow down a n d come to a total halt at the c e n t e r of the black hole. 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 wormhole. Assuming, for the m o m e n t , that o n e could somehow go t h r o u g h the center of the w o r m h o l e a n d r e t u r n to earth, 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 who might 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 epoch-making event: making contact with the first extraterrestrial life in o u t e r space. Every scientist p o n d e r i n g the question of life in o u t e r space must confront the question of how to break the light barrier. Since Einstein's special theory of relativity explicitly forbids travel faster than the speed of light, traveling to the 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 make 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 the 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 unusual way: They worked backward. Normally, physicists start with a certain known astronomical object (a n e u t r o n star, a black hole, the Big Bang) a n d t h e n solve Einstein's equations to find the curvature of the s u r r o u n d i n g space. T h e essence of Einstein's equations, we recall, is that the m a t t e r a n d energy c o n t e n t of an object determines the 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. Proceeding 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 outer 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 the question backward. They started with a rough idea of what they wanted to find. They wanted a solution to Einstein's equations in which a space traveler would n o t be torn apart by the tidal effects of the intense gravitational field. They wanted a wormhole that would be stable a n d n o t suddenly close up in the middle of the trip. They 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 earth years, a n d so on. In fact, their guiding principle was that they wanted 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 wormhole. 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, then, a n d only t h e n , did they begin to calculate the 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 them, it was an e n g i n e e r i n g p r o b l e m for some future civilization actually to construct the time m a c h i n e . They wanted to prove that it was scientifically feasible, n o t that it was economical or within the 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 this 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 activities 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 .

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T h e key phrase, 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 the gross national p r o d u c t of the planet earth. T h o r n e a n d his colleagues were careful to state that this mythical civilization that can harness the power of wormholes must 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 remarkable ease they soon found a surprisingly simple solution that satisfied all their rigid constraints. It was not a typical black hole solution at all, so they d i d n ' t have to worry a b o u t all the problems of b e i n g r i p p e d apart by a collapsed star. They christened their solution the "transversible w o r m h o l e , " to distinguish it from the o t h e r wormhole solutions that are n o t transversible by spaceship. They were so excited by their solution that they wrote back to Sagan, who t h e n incorporated some of their ideas in his novel. In fact, they were so surprised by the simplicity of their solution that they were convinced that a b e g i n n i n g graduate 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 the a u t u m n of 1985, on the final e x a m in a course on general relativity given at Caltech, T h o r n e gave the worm-

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hole 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 properties. (Most students gave detailed mathematical 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 the students 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 some r a t h e r astonishing properties of the w o r m h o l e . In fact, they would have found 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 exceed 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 the earth. 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 the 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 the transversible wormhole would be manageable. 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 notes that the time paradoxes 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 exposure to scie n c e fiction scenarios (for example, 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 impossible). However, he has shown that the CTCs that a p p e a r in his wormhole seem to fulfill the 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 equations is presented, which describe wormholes 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 the precise n a t u r e of the 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 found that at the c e n t e r of the 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 unusual properties. T h o r n e is quick to point out that this " e x o t i c " form of matter, a l t h o u g h unusual, does n o t seem to violate any of the known laws of physics. He cautions that, at some future point, scientists may prove that exotic m a t t e r does n o t exist. However, at present, 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 that "from 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." 5

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Blueprint for a Time Machine Anyone who has read 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 the vast p a n o r a m a of history, including destructive world wars, the rise a n d fall of great civilizations, or the 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 containing two parallel metal plates. T h e intense electric fields created between each pair of plates (larger than anything possible with today's technology) rips the fabric of s p a c e - t i m e , creating a hole 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 the o t h e r c h a m b e r stays on the earth. 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 the first c h a m b e r ticks slower than a clock in the second c h a m b e r . Because time would pass at different rates at the two e n d s of the w o r m h o l e , a n y o n e falling into 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 the past or the future. A n o t h e r time m a c h i n e might look like the following. If exotic m a t t e r can be found a n d shaped like metal, t h e n presumably the ideal shape would be a cylinder. A h u m a n stands in the c e n t e r of the cylinder. T h e exotic matter t h e n warps the space a n d time s u r r o u n d i n g it, creating a wormhole that connects to a distant part of the universe in a different time. At the center of the vortex is the h u m a n , who t h e n experiences no m o r e than 1 g of gravitational stress as he or she is t h e n sucked into the wormhole a n d finds himself or herself on the o t h e r e n d of the universe. On the surface, T h o m e ' s mathematical reasoning 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 either side of the wormhole, so that 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 the basic tenets of elementary physics is that 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 than the v a c u u m ) , t h e n we might be able to generate 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 point out, the AWEC must 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, who realize that negative energy would make 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 neutral, 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 out of n o w h e r e a n d disappear 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 the laws of physics. These "virtual particles" create a net attractive force between these two plates that Casimir predicted was measurable. W h e n Casimir first published his p a p e r , it m e t with extreme skepticism. After all, how can two electrically neutral objects attract each other, thereby violating the 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 predicted. Since then, it has b e e n christened the Casimir effect. O n e way of harnessing the Casimir effect is to place two large cond u c t i n g parallel plates at the e n t r a n c e of each wormhole, 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 the average weak energy condition can never be violated, in which case t h e r e could be no such things as transversible wormholes, time travel, or a failure of causality. It's p r e m a t u r e to try to cross a bridge before you c o m e to i t . " At present, the 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 quantized theory of gravity settle the m a t t e r o n c e a n d for all. For example, Stephen Hawking has pointed o u t that the radiation emitted 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 equations will distort the 7

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e n t r a n c e to the 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 where superstring theory comes in. Because superstring theory is a fully quantum-mechanical theory that 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 whether the AWEC condition is physically realizable, a n d w h e t h e r the wormhole 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 wormholes. 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 present with the past, Hawking proposes to use wormholes 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

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O S M O L O G I S T Stephen 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 the face of almost insurm o u n t a b l e obstacles. Although he has lost control of his hands, legs, t o n g u e , a n d finally his vocal cords, he has spearheaded new avenues of research while confined to a wheelchair. Any lesser physicist would have long ago given up the struggle to tackle the great problems 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 the outside world. But he not 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 Cambridge University when 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 the impressive array of ingenious 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 grab each page a n d carefully turn it for reading a book. (I shivered to p o n d e r , as I think many physicists have, whether I would 252

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have the stamina a n d sheer willpower to c o n t i n u e research without arms, legs, or a voice even if I h a d the finest mechanical 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. And like his illustrious predecessor, Hawking has e m b a r k e d on the greatest quest of the 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 longer spends the 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, began as a q u a n t u m theory a n d t h e n later absorbed Einstein's theory of gravity. Hawking, starting as a p u r e classical relativist rather than 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 point 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 the entire universe!

Wave Function of the Universe Hawking is o n e of the founders of a new scientific discipline, called quantum cosmology. At first, this seems like a contradiction in terms. T h e word quantum applies to the infinitesimally small world of quarks a n d neutrinos, while cosmology signifies the almost limitless expanse of o u t e r space. However, Hawking a n d others now believe that the 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 point 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 example, imagine a large, irregular t h u n d e r c l o u d that fills up the sky. T h e d a r k e r the t h u n d e r c l o u d , the 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 concentrations 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 electron's wave function. Like a t h u n d e r c l o u d , it fills up all space. Likewise, the greater its value at a point, the greater the probability of finding the electron 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 Princeton, I know that I have a Schodinger probability wave function. If I could somehow see my own wave function, it would resemble a cloud very m u c h in the shape of my body. However, some of the cloud would spread o u t over all space, out to Mars a n d even beyond the 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 planet Mars. Although part of my wave function has spread even beyond the 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 some 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 that the starting p o i n t of Hawking's theory must be an infinite set of parallel universes, the wave function of the universe. Hawking's r a t h e r simple analysis, replacing the word particle with universe, has led to a conceptual revolution in o u r thinking a b o u t cosmology. According to this picture, the 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 the correct 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 the 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 make a q u a n t u m leap to t h e m in the 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 and also stable. (At present, q u a n t u m cosmologists are unable to solve this i m p o r t a n t problem.) If we take Hawking seriously, it m e a n s that 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, the 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 the wave function of the universe can spread out over several possible universes, with o u r universe being the most likely o n e b u t certainly n o t the only o n e . Hawking's q u a n t u m cosmology also assumes that the wave function of the 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 wormholes are n o t like the ones 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 example, of a large collection of soap bubbles, 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, except 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 bubble, there is no such thing 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 the 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 the physical conditions that m a d e life possible were n o t satisfied. Perhaps, a m o n g the 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 "baby universe" theory, although 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 debates a m o n g cosmologists.

Putting God Back in the Universe? T h e first debate concerns the anthropic principle. Over the centuries, scientists have learned to view the 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 anyone who sees h u m a n emotions a n d feelings b e i n g exhibited by their pets. (It is also committed by Hollywood scriptwriters who regularly assume that beings similar to us must populate planets orbiting the stars in the 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 Ionian 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 lamented, " M e n imagine gods to be b o r n , a n d to have clothes a n d voices a n d shapes like theirs. . . . Yea, the gods of the Ethiopians are black a n d flat-nosed, a n d the gods of the Thracians 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, some of whose advocates openly declare that they would like to p u t God back into science. Actually, t h e r e is some scientific merit to this strange d e b a t e over the anthropic principle, which revolves a r o u n d the indisputable fact that if the physical constants of the universe were altered by the smallest a m o u n t , life in the universe would be impossible. Is this r e m a r k a b l e fact just a fortunate coincidence, or does it show the work of some S u p r e m e Being? T h e r e are two versions of the a n t h r o p i c principle. T h e " w e a k " version states that the fact that intelligent life (us) exists in the universe

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should be taken as an experimental fact that helps us u n d e r s t a n d the constants of the universe. As Nobel laureate Steven Weinberg explains it, " t h e world is the 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 argue with. To have life in the 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 complex 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 nuclear 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 might 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 elements (elements beyond iron) for the creation of DNA a n d protein molecules. T h u s the smallest c h a n g e in nuclear physics would make the heavy elements of the universe impossible to manufacture in the stars. We are children of the stars; however, if the laws of nuclear physics c h a n g e in the 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 example, it is safe to say that the creation of life in the early oceans probably took 1 to 2 billion years. However, if we could somehow 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 complex sequence 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 range of values, so that the stars lived long e n o u g h to create the heavy elements 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 germinate, a n d so on. In other words, the existence of h u m a n s who 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 example, 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 astronomy that have worked together to o u r benefit, it almost seems as if the Universe must in some sense have known that 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 the physical constants of the universe have b e e n precisely chosen (by God or some S u p r e m e Being) so that 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 might have b e e n blind luck if only a few constants of n a t u r e were required to assume certain values to make 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 the a n t h r o p i c principle, they are immediately taken aback. Physicist Heinz Pagels recalled, " H e r e was a form of reasoning 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 the old a r g u m e n t that God located the earth at j u s t the right distance from the sun. If God h a d placed the earth too close, t h e n it would be too h o t to support life. If God h a d placed the earth 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 the right distance from their sun. O u r planet is o n e of them, a n d h e n c e we are h e r e to discuss the 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 concluded that "unlike 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, the a n t h r o p i c principle is n o t subject to experimental 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, the anthropic principle kind of rubs me the wrong 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 think of anything better t o d o . " To Richard Feynman, the goal of a theoretical physicist is to " p r o v e yourself wrong as fast as possible." However, the a n t h r o p i c principle is sterile a n d c a n n o t be disproved. Or, 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 debate over the a n t h r o p i c principle (and h e n c e , a b o u t God) 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 2

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c a n n o t manufacture the heavy elements beyond iron, or the Big Crunch takes place too rapidly before life can begin, a n d so on. In fact, an infinite n u m b e r of these parallel universes are dead, without the physical laws that can make 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 proof is that we are h e r e today to discuss the matter. If this is true, t h e n p e r h a p s God 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 compatible 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 the Schrodinger'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 debates c o n c e r n i n g its foundations. Hawking's wave function of the universe does n o t completely solve these paradoxes 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 there 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 the 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 the Schrodinger 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, the 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 Bohr a n d W e r n e r Heisenberg, Einstein revolted against this concept. " D o e s the m o o n exist j u s t because a m o u s e looks at it?" he was fond of asking. According to the strict interpretation 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 be, in fact, in any o n e of an infinite n u m b e r of states, including the state of being 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 the m o o n is actually circling the earth.

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Einstein h a d many h e a t e d discussions with Niels Bohr challenging 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 Schrodinger (who initiated the whole discussion with his celebrated wave equation) protested this reinterpretation 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 interpretation, 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 counter, 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 the Geiger c o u n t e r , which will t h e n trigger the gun, whose bullet will kill the cat. To decide w h e t h e r the cat is d e a d or alive, we must o p e n the box a n d observe the cat. However, what is the state of the cat before we o p e n the box? According to q u a n t u m theory, we can only state that the cat is described by a wave function that describes the sum of a d e a d cat a n d a live cat. To Schrodinger, the idea of thinking a b o u t cats that are n e i t h e r d e a d n o r alive was the h e i g h t of absurdity, yet nevertheless the e x p e r i m e n t a l confirmation of q u a n t u m mechanics 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 Schrodinger'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 the Cheshire cat in Lewis Carroll's fable: " 'You'll see me t h e r e , ' said the Cat, a n d vanished. Alice was not 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 gotten used to " q u e e r " things h a p p e n i n g in q u a n t u m mechanics. 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 must be some " c o n s c i o u s n e s s " in the universe. Some physicists, like Nobel laureate E u g e n e Wigner, have insisted that q u a n t u m theory proves the existence of some sort of universal cosmic consciousness in the universe. T h e second way of dealing with the p a r a d o x is favored by the vast majority of working physicists—to ignore the p r o b l e m . Most physicists, pointing o u t that a camera without any consciousness can also m a k e measurements, simply wish that this sticky, b u t unavoidable, p r o b l e m would go away. 7

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T h e physicist Richard Feynman o n c e said, "I think it is safe to say that no o n e u n d e r s t a n d s q u a n t u m mechanics. Do not keep saying to yourself, if you can possibly avoid it, 'But how can it be like that?' because you will go 'down the d r a i n ' into a blind alley from which nobody has yet escaped. Nobody knows how it can be like t h a t . " In fact, it is often stated that of all the theories p r o p o s e d in this century, the silliest is q u a n t u m theory. Some say that the only thing that q u a n t u m theory has going for it, in fact, is that it is unquestionably correct. However, t h e r e is a third way of dealing with this paradox, called the many-worlds theory. This theory (like the a n t h r o p i c principle) fell out of favor in the past decades, b u t is b e i n g revived again by Hawking's wave function of the 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 the universe, it continually " s p l i t " in half, like a fork in a road. In o n e universe, the 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 the other, the 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 the road. Or, as the Argentinian writer J o r g e Luis Borges wrote in The Garden of Forking Paths, " t i m e forks perpetually toward innumerable futures." Physicist Bryce DeWitt, o n e of the p r o p o n e n t s of the many-worlds theory, describes the 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 the 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 that all possible q u a n t u m worlds exist. In some worlds, h u m a n s exist as the d o m i n a n t life form on earth. In o t h e r worlds, subatomic events took place that prevented 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 arise 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, the 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 Magnus o n c e wrote, " D o t h e r e exist many worlds, or is there 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 the study of N a t u r e . " However, the new twist on this ancient idea is that these many worlds resolve the Schrodinger cat paradox. In o n e universe, the cat may be dead; 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 interpretations 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. Although it c a n n o t be ruled 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, who 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 the simplest possible p a t h a n d ignore m o r e clumsy alternatives, especially if the alternatives can never be measured. (Thus O c c a m ' s razor dismisses the old " a e t h e r " theory, which stated that a mysterious gas o n c e pervaded the entire universe. T h e aether theory provided a convenient answer to an embarrassing question: If light is a wave, a n d light can travel in a vacuum, t h e n what is waving? T h e answer was that aether, like a fluid, was vibrating even in a vacuum. Einstein showed that the 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 Occam'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. Therefore, each universe is unaware of the existence of the others. If e x p e r i m e n t s c a n n o t test for the 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. Perhaps they d o . But miracles, almost by definition, are n o t repeatable a n d therefore n o t measurable by experiment. Therefore, by O c c a m ' s razor, we must dismiss t h e m (unless, of course, we can find a reproducible, measurable 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 unpopularity of the many-worlds theory, however, may subside as Hawking's wave function of the universe gains popularity. Everett's 1 2

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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 (and n o t j u s t particles) a n d postulates the 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 the wave function of the universe. He is confident that his a p p r o a c h is correct partly because 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 dimensions). His goal is to show that the wave function of the 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 the 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 the wave function of the universe. However, as before, the mathematics involved in the wave function of the universe is beyond the calculational ability of any h u m a n on this planet, a n d we may have to wait years before any enterprising 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 the universe is that Hawking's theory places wormholes that c o n n e c t these parallel universes at the center 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 the door, 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 rushing 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 the movies. In Hawking's a p p r o a c h , the wormholes 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 the size of these wormholes, on the average, is extremely small, a b o u t the size of the Planck length (about a 100 billion billion times smaller than 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 the lifetime of the universe, before such an event takes place. T h u s it is perfectly consistent with the 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 some point in time when the two universes split apart. This type of parallel world was explored by J o h n W y n d h a m in the story " R a n d o m Q u e s t . " Colin Trafford, a British nuclear physicist, is almost killed in 1954 when a nuclear e x p e r i m e n t blows u p . Instead of winding up in the hospital, he wakes u p , alone 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 newspaper 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 the 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 nuclear 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 nuclear 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, but the address is wrong. Shaking, he decides to visit " h i s " h o m e . Entering " 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 who 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 counterpart, Trafford finds out, is a cad a n d a womanizer. However, he finds it difficult to argue 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 treated his lovely wife in such a despicable m a n n e r . T h e n e x t few weeks spent together 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 other, 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 counterpart in the other. Surely, he reasons, " h i s " wife must have a c o u n t e r p a r t in his own world. He becomes obsessed, tracking down all the 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 some

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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 the birth a n d d e a t h records of several families. He spends his r e m a i n i n g savings interviewing scores of people 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 the e n d , he marries her.

Attack of the Giant Wormholes O n e Harvard physicist who 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 and Albert Einstein, he shuffles t h r o u g h the corridors of Jefferson Hall, trying to convince the skeptics of his latest theory of wormholes. With his Chaplinesque 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 the celebrated cosmological constant problem, which has puzzled physicists for the past 80 years. His work even m a d e the 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 the 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 wormholes, t h e n we have to wait until s o m e o n e discovers exotic matter or masters the Casimir effect. Until then, o u r time m a c h i n e s have no " e n g i n e " capable of shooting us into the past. Similarly, if we believe in Hawking's wormholes, t h e n we have to travel in "imaginary 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 the i n a d e q u a t e , feeble technology of the twentieth century a n d who can only d r e a m of harnessing the Planck energy. This is where C o l e m a n ' s work comes in. He recently m a d e the claim that the wormholes might yield a very tangible, very measurable result in the 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 equations 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 space-time surrounding it. Einstein w o n d e r e d w h e t h e r the p u r e vacuum 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 the cosmological constant; in principle,

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there is n o t h i n g to prevent a cosmological constant from a p p e a r i n g in the equations. Einstein t h o u g h t this term was aesthetically ugly, b u t he could n o t rule it o u t on physical or mathematical g r o u n d s . In the 1920s, when Einstein tried to solve his equations for the universe, he found, m u c h to his chagrin, that the universe was e x p a n d i n g . Back then, the 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 equations to prevent the expansion of the universe, Einstein inserted a tiny cosmological constant 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, when H u b b l e conclusively proved that the universe is i n d e e d e x p a n d i n g , Einstein banished the cosmological constant a n d said it was the "greatest b l u n d e r of my life." Today, we know that the cosmological constant is very close to zero. If there were a small negative cosmological constant, t h e n gravity would be powerfully attractive a n d the entire universe m i g h t be, say, j u s t a few feet across. (By r e a c h i n g out with your h a n d , you should be able to grab the person 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 nightmarish scenario occurs, we are confident that the cosmological constant is extremely tiny or even zero. But this p r o b l e m resurfaced in the 1970s, w h e n symmetry b r e a k i n g was being intensively studied in the Standard 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 the vacuum is 10 times larger than the experimentally observed a m o u n t . In all of physics, this discrepancy of 1 0 is unquestionably the largest. Nowhere in physics do we see such a large divergence between theory (which predicts a large vacuum 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 constant in the universe). This is where C o l e m a n ' s wormholes comes in; they're n e e d e d to cancel the unwanted contributions to the 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 the contribution from this infinite series. After the sum was performed, he found a surprising result: T h e wave function of the universe prefers to have zero cosmological constant, as desired. If the cosmological constant 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, the wave function of the universe quickly vanished 100

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if the cosmological constant b e c a m e n o n z e r o , m e a n i n g that there was zero probability for that unwanted universe. This was exactly what was n e e d e d to cancel the cosmological constant. In o t h e r words, the cosmological constant was zero because that was the most probable 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 constant zero in o u r universe. Because this was such an i m p o r t a n t result, physicists immediately began to leap into the field. " W h e n Sidney came 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 published 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 audiences vividly with the importance of this p r o b l e m , that the chances of canceling o u t a cosmological constant to o n e part 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, and when you finally c o m p a r e what you earn with what you spent, they balance o u t to the p e n n y , " he n o t e s . T h u s his calculation, which shows that you can cancel the cosmological constant to o n e part in 1 0 , is a highly nontrivial result. To a d d frosting to the cake, Coleman emphasizes that these wormholes also solve a n o t h e r problem: They help to d e t e r m i n e the values of the fundamental constants of the universe. Colem a n adds, " I t was a completely different m e c h a n i s m from any that had been considered. It was Batman swinging in on his r o p e . " 13

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But criticisms also began to surface; the most persistent criticism was that he assumed that the 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 wormholes should also be included in his sum. But since we d o n ' t see large, visible wormholes anywhere, it seems that his calculation has a fatal flaw. Unfazed by this criticism, Coleman shot back in his usual way: choosing outrageous titles for his papers. To prove that large wormholes can be neglected in his calculation, he wrote a rebuttal to his critics with the title "Escape from the Menace of the 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 mathematical 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 wormholes are an essential feature of all physical processes, a n d n o t j u s t some pipe d r e a m . It would m e a n that wormholes 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 are essential to prevent 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 wormholes are the essential feature m a k i n g o u r universe relatively stable. But as with most developments that occur at the Planck length, the final solution to these w o r m h o l e equations will have to wait until we have a better grasp of q u a n t u m gravity. Many of C o l e m a n ' s equations require a m e a n s of eliminating the 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 sharpen o u r calculational tools. As we have emphasized, the p r o b l e m is mainly theoretical. We simply do n o t have the mathematical brainpower to b r e a k o p e n these welldefined problems. T h e equations stare at us from the 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 better 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 the energy found at the Planck length will b e c o m e the master of all fundamental forces. T h a t is the n e x t topic to which we will turn. 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 i t m e a n f o r a c i v i l i z a t i o n t o b e a m i l l i o n years o l d ? 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 .. . a n a d v a n c e d c i v i l i z a t i o n m i l l i o n s o f years o l d i s a s m u c h 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

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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 the 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 harness the power of hyperspace? Experimental verification of the hyperspace theory, at least indirectly, may come in the twenty-first century. However, the energy scale necessary to manipulate (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 beyond 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 273

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perform near-miraculous feats, such as creating wormholes a n d altering the direction of time. To be masters of the t e n t h dimension, either we e n c o u n t e r intellig e n t life within the 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 example, o u r c u r r e n t a t o m smashers or particle accelerators can boost the 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 consortium 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 electron 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 the 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 national p r o d u c t of the 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 harness this level of energy. However, this n u m b e r does n o t seem so ridiculously large if we u n d e r s t a n d that technology e x p a n d s exponentially, which is difficult for o u r m i n d s to c o m p r e h e n d . To u n d e r s t a n d how fast exponential growth is, imagine a bacterium 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 planet earth. A l t h o u g h h u m a n s have existed on this planet for p e r h a p s 2 million years, the rapid climb to m o d e r n civilization within the last 200 years was possible d u e to the fact that the growth of scientific knowledge is exponential; that is, its rate of expansion is proportional 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 than all the 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 development historically. To appreciate 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 the energy necessary to exploit the ten-dimensional theory into 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 the energy available to the 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. During this period, the basic energy source was the power of o u r own hands, a b o u t one-eighth of a horsepower. H u m a n s r o a m e d the earth 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 the 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 the last 100,000 years. With the 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 the power of their fists, a n d knives the power of their jaws. In this period, their energy o u t p u t doubled, to a b o u t one-quarter of a horsepower. Within the past 10,000 or so years, the energy o u t p u t of a h u m a n d o u b l e d once again. T h e m a i n reason for this c h a n g e was probably the 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 thousands 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 thousands of years, c h a n g e d with the discovery of agriculture soon after the ice melted. Roving b a n d s of h u m a n s , n o t having to follow game across the 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 melting of the ice sheet came the 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 beginning of a stratified, agrarian life came 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 : the transition to a slave society. This m e a n t that o n e person, the 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 the 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. From an energy p o i n t of view, for a b o u t 99.99% of the existence of humanity 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 the 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 horsepower available to them.

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A decisive c h a n g e came with the Industrial Revolution. Newton's discovery of the 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 mechanics to a set of well-defined equations. T h u s Newton's classical theory of the gravitational force, in some sense, paved the way for the m o d e r n theory of machines. This h e l p e d to m a k e possible the widespread use of steam-powered engines in the n i n e t e e n t h century; with steam, the 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 example, the railroads o p e n e d up entire continents 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 energized by the 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 the face of E u r o p e . With steam-driven a n d later oil-fired machines, the 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 fundamental 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 the 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 homes, exponentially increasing the versatility a n d power of o u r machines. Steam engines were now b e i n g replaced by powerful dynamos. Within the past 50 years, the discovery of the nuclear force has increased the power available to a single h u m a n by a factor of a million. Because the energy of chemical reactions is m e a s u r e d in electron volts, while the energy of fission a n d fusion is measured in millions of electron volts, we have a millionfold increase in the power available to us. T h e lesson from analyzing the historical energy needs of humanity shows graphically how for only 0 . 0 1 % of o u r existence we have manipulated energy levels beyond 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 nuclear forces. Let us now leave the past a n d begin a discussion of the future, using the same methodology, to u n d e r s t a n d the point at which we may harness the superforce.

Type I, II, and III Civilizations Futurology, or the prediction of the future from reasonable scientific j u d g m e n t s , is a risky science. Some 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 hocus pocus or witchcraft. Futurology has deservedly e a r n e d this unsavory reputation because every "scientific" poll c o n d u c t e d by futurologists a b o u t the next decade has

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proved to be wildly off the mark. W h a t makes futurology such a primitive science is that o u r brains think linearly, while knowledge progresses exponentially. For example, polls of futurologists have shown that they take known technology a n d simply d o u b l e or triple it to predict the future. Polls taken in the 1920s showed that futurologists predicted that 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 the short r u n , when extrapolating within a few years, it is a safe bet that science will progress t h r o u g h steady, quantitative improvements on existing technology. However, when 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, where new industries o p e n up in u n e x p e c t e d places. Perhaps the most famous example of futurology g o n e wrong is the predictions m a d e by J o h n von N e u m a n n , the 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 mathematicians of the century. After the war, he m a d e two predictions: first, that in the future computers 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 the 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 are 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 processor), a n d eventually we will have the " s m a r t " vacuum cleaner, the " 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 predict the weather. Although the classical motion of individual molecules can, in principle, be predicted, the weather is so complex 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, perhaps, 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 when a civilization (either o u r own or o n e in o u t e r space) may attain the ability to master the tenth dimension. A s t r o n o m e r Nikolai Kardashev of the former Soviet U n i o n o n c e categorized future civilizations in the following way. A Type I civilization is o n e that controls the energy resources of an entire planet. This civilization can control the weather, prevent earthquakes, m i n e d e e p in the earth's crust, a n d harvest the 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 that controls the power of the sun itself.

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This does n o t m e a n passively harnessing solar energy; this civilization mines the sun. T h e energy needs 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 machines. This civilization will begin the colonization of local star systems. A Type III civilization is o n e that controls the power of an entire galaxy. For a power source, it harnesses the power of billions of star systems. It has probably mastered 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 the basis of the power source that energizes the civilization. Type I civilizations use the power of an entire planet. Type II civilizations use the power of an entire star. Type III civilizations use the power of an entire 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, but does n o t have the technology a n d resources to control t h e m . A Type 0 civilization like ours derives its energy from fossil fuels like oil a n d coal and, in m u c h of the T h i r d World, from raw h u m a n labor. O u r largest computers cann o t even predict the weather, let alone control it. Viewed from this larger perspective, we as a civilization are like a newborn infant. A l t h o u g h o n e m i g h t guess that the slow m a r c h from a Type 0 civilization to a Type III civilization might take millions of years, the extraordinary fact a b o u t this classification scheme 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 than anything we can readily conceive. With all these qualifications, we can still make educated guesses a b o u t when o u r civilization will reach 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 unleash the 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 controlling it in a power generator. However, a simple hurricane generates the power of h u n d r e d s of hydrogen bombs. T h u s weather control, 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 must be m e a s u r e d in centuries. For example, the earliest date for NASA's m a n n e d landing on the p l a n e t Mars is 2020. Therefore, the colonization of Mars may take place 40 to 50 years after that, a n d the colonization of the 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 exponential growth of civilization, we may expect that within 1,000 years the energy n e e d s of a civilization will b e c o m e so large that it must begin to m i n e the sun to energize its machines. A typical example of a Type II civilization is the Federation 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, the art of warping s p a c e - t i m e via wormholes—and 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 the limit placed by the speed of light by mastering Einstein's theory of general relativity. Small colonies have b e e n established on some of these systems, which the starship Enterprise is sworn to protect. T h e civilization's starships are powered by the collision of matter a n d antimatter. T h e ability to create large concentrations of antimatter suitable for space travel places that civilization many centuries to a millennium 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, the time scale predicted by Isaac Asimov in his classic Foundation Series, which describes the rise, fall, a n d re-emergence of a galactic civilization. T h e time scale involved in each of these transitions involves thousands of years. This civilization has harnessed the energy source contained within the galaxy itself. To it, warp drive, instead of being an exotic form of travel to the nearby stars, is the stand a r d means of trade a n d c o m m e r c e between sectors of the galaxy. T h u s although it took 2 million years for o u r species to leave the safety of the 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 option o p e n to a Type III civilization is harnessing the power of supernovae or black holes. Its starships may even be able to p r o b e the galactic nucleus, which is p e r h a p s the 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 contain millions of black holes. If true, this would provide virtually unlimited a m o u n t s of energy.

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At this point, 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 the 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 nucleus at its disposal, the mastery of the tenth 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 the world of physics who has tackled some of the most intellectually challenging a n d intriguing questions facing humanity, such as new directions in space exploration, the n a t u r e of extraterrestrial life, a n d the future of civilization. Unlike o t h e r physicists, who dwell excessively in narrow, well-defined areas of specialization, Dyson's fertile imagination has r o a m e d across the galaxy. "I cannot, as B o h r a n d Feynman 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 many different directions," he confessed. Thin, remarkably spry, with the 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 lunch conversation with m e , t o u c h i n g on many of the ideas that have fascinated him over the years. Viewing the 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 wrong direction. T h e c u r r e n t t r e n d is toward heavier payloads a n d greater lag time between space shots, which is severely r e t a r d i n g the 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 trend, based on what he calls the Astrochicken. Small, lightweight, a n d intelligent, Astrochicken is a versatile space p r o b e that has a clear advantage over the bulky, exorbitantly expensive space missions of the past, which have b e e n a bottleneck 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 grown," 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 m o r e t h a n a g r a m . " It will be part m a c h i n e a n d p a r t animal, using the most advanced developments 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 explore the 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 rocket 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 hydrocarbons found in the rings s u r r o u n d i n g the outer 2

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planet. Its genetically e n g i n e e r e d stomach will t h e n digest these materials into chemical fuel. O n c e its appetite has b e e n satisfied, it will t h e n rocket to the next 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 engineering, artificial intelligence, a n d solar-electric propulsion. Given the remarkable progress in these ares, Dyson expects that the various technologies for Astrochicken may be available by the year 2016. Taking the larger view of the 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 development, we may attain Type I status within a few centuries. He does n o t believe that m a k i n g the transition between the various types of civilizations will be very difficult. He estimates that the difference in size a n d power separating the various types of civilizations is roughly a factor of 10 billion. Although 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 the 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 strong expansionist drive will e x p a n d its habitat from a single planet (Type I) to a biosphere exploiting an entire star (Type II) within a few t h o u s a n d years, a n d from a single star to an entire galaxy (Type III) within a few million years. A species which has o n c e passed beyond Type II status is invulnerable to extinction by even the worst imaginable natural 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 the transition from a Type II to a Type III civilization may pose formidable physical difficulties, d u e mainly to the limitation imposed by the speed of light. T h e expansion of a Type II civilization will necessarily p r o c e e d at less than the speed of light, which he feel places a severe restriction on its development. Will a Type II civilization break the light barrier a n d the b o n d s of special relativity by exploring the power of hyperspace? Dyson is n o t sure. N o t h i n g can be ruled out, b u t the Planck length, he r e m i n d e d m e , is a fantastically small distance, a n d the energies r e q u i r e d to p r o b e down to that distance are unimaginable. Perhaps, he mused, the Planck length 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 reach 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 harnessed hyperspace for its needs 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 advanced civilization in the heavens, at least n o t in o u r solar system or even in o u r small sector of the 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 Jupiter, 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 promising planets, Venus a n d Mars, have t u r n e d up no signs of life, let alone advanced civilizations. Venus, n a m e d after the goddess of love, was o n c e envisioned by astronomers as well as romantics to be a lush, tropical planet. Instead, o u r space probes have found a harsh, b a r r e n planet, with a suffocating a t m o s p h e r e of carbon dioxide, blistering t e m p e r a t u r e s exceeding 800°F, a n d toxic rains of sulfuric acid. Mars, the focus of speculation even before Orson Welles caused panic in the country in 1938 d u r i n g the Depression with his fictional broadcast a b o u t an invasion from that planet, has b e e n equally disappointing. We know it to be a desolate, desert planet without traces of surface water. Ancient riverbeds a n d long-vanished oceans have left their distinctive mark on the surface of Mars, b u t we see no ruins or any indications of civilization. Going beyond o u r solar system, scientists have analyzed the radio emissions from nearby stars with equally fruitless results. Dyson has stressed that any advanced civilization, by the Second Law of T h e r m o dynamics, must necessarily generate large quantities of waste heat. Its energy c o n s u m p t i o n should be e n o r m o u s , a n d a small fraction of that waste h e a t should be easily detected by o u r instruments. Thus, Dyson claims, by scanning the nearby stars, o u r instruments 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 where we scan the heavens, we see no traces of waste h e a t or radio 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 example, we have mastered the art of radio a n d television within the past half-century. T h u s an e x p a n d i n g sphere 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 detect o u r presence. Likewise, any Type II or III civilization should be broadcasting 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 the civilization's planet should be able to detect its presence. 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 found no traces of radio emissions from intelligent life. Astronomers 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 spent over a 10-year p e r i o d for the H i g h Resolution Microwave Survey, which will scan the nearby stars for intelligent life. These funds will m a k e it possible for the gigantic 305-meter fixed radio dish at Arecibo, P u e r t o Rico, to scan select stars systematically within 100 light-years of the earth. This will be c o m p l e m e n t e d by the 34meter movable radio a n t e n n a at Goldstone, California, which will sweep b r o a d portions of the night sky. After years of negative results, astronom e r Frank Drake of the University of California at Santa Cruz is cautiously optimistic that they will find some positive signs of intelligent life. He remarks, "Many 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 combination of curiosity a n d trying to create a better life, a n d I think those same motivations would exist in o t h e r creatures.") T h e puzzle d e e p e n s when we realize that the probability of intelligent life e m e r g i n g within o u r galaxy is surprisingly large. Drake even derived a simple equation to calculate the n u m b e r of planets with intelligent life forms in the galaxy. O u r galaxy, for example, 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 some form of intelligent life. This m e a n s that one-millionth of the 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 equation shows that intelligent life might be, on the average, as close as 15 light-years from o u r sun. With recent advanced c o m p u t e r techniques, scientists have b e e n able to refine Drake's original back-of-the-envelope calculation. George W. Wetherill of the Carnegie Institution of Washington, for example, has run c o m p u t e r simulations of the early evolution of o u r solar system, beginning with a large, swirling disk of gas a n d dust a r o u n d the 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 8 0 % 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 t h o u s a n d 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 p o p u lation 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 m e m o r y 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 o u t e r space, it is sobering to read how the Aztecs reacted to the visitors from Spain: " T h e y 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 a b o u t fingering, taking up the streamers of gold, moving t h e m back a n d 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 a b o u t astronomical time scales, it is likely that a civilization that is a million years a h e a d 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 o u r planet can offer these aliens in terms of natural resources that isn't simultaneously available in n u m e r o u s o t h e r star systems. In the "Star T r e k " series, however, the Federation of Planets e n c o u n ters o t h e r hostile civilizations, the Klingons and Romulans, which are precisely at the same stage of technological d e v e l o p m e n t as the Federation. This may increase the d r a m a a n d tension of t h e series, b u t the o d d s 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 p e r h a p s millions of years a h e a d of us.

The Rise and Fall of Civilizations In addition to the possibilities that we may have missed o t h e r civilizations by millions of years a n d 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 u n a b l e to negotiate a series of catastrophes, b o t h 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 m e e t 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 the 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 long 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 humanity 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 pleading 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 careening o u t of control. For example, psychologists tell us that teenagers act as if they are invulnerable. Their 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 must be crossed over during the 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 encounters, Nemesis a n d extinction, a n d the d e a t h of the sun a n d the Milky Way galaxy.

The Uranium Barrier J o n a t h a n Schell, in his watershed book The Fate of the Earth, points o u t how perilously close we have come to mutual annihilation. Although the

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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, t h e r e is the increasing d a n g e r of nuclear proliferation. U n i t e d 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 b o m b s . 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 c o n t i n u e 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 influe n c e . 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 ( u r a n i u m ) 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 d e v e l o p m e n t 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, t h e r e 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 a n d 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 u p . T h e r e is a vanishingly small " w i n d o w " 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 t h e r e 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 n e x t barrier is the possibility of ecological collapse. We recall t h e earlier e x a m p l e 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 i n d e e d 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 o u r resources while d r o w n i n g in the waste p r o d u c t s that we relentlessly p r o d u c e . O u r oceans a n d the a t m o s p h e r e 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 a n d exacerbating the problems of pollution. O n e of the most immediate dangers is the poisoning of the a t m o s p h e r e , in the form of carbon dioxide, which traps sunlight a n d raises the average world temperature, possibly initiating a runaway g r e e n h o u s e 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 ( 4 5 % of c a r b o n 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, the 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 the nation's food basket in the 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 the 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 the 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, o u r society may be forced to resort to o n e of t h r e e possible solutions that do n o t give off carbon dioxide a n d are essentially inexhaustible: solar energy, fusion plants, a n d b r e e d e r reactors. Of these, solar a n d fusion h o l d 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 n e x t few centuries, until society makes the transition to a Type I civilization. T h e fault o n c e 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, t h e r e will be a fatal mismatch that invites disaster. T h e u r a n i u m barrier a n d 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, a n d 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 a n d 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 e n v i r o n m e n t places e n o r m o u s biological strains on the baby. After the first year of life, the d e a t h rate plunges dramatically. Similarly, the most d a n g e r o u s period for a civilization is the first few centuries after it has reached nuclear capability. It may turn out that o n c e 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 damage 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 bands 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 computers are 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 the 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 measurements, 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 the 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 the asteroid while it is still tens of millions of miles away from the 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 radar by NASA astronomers. Photos of the asteroid Toutatis show that it consists of two rocky cores, each 2 miles in diameter. It came

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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, a b o u t 0.7 million miles (roughly t h r e e 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 c o m e t would hit the earth on exactly August 14, 2126, p e r h a p s e n d i n g all life on the planet. A s t r o n o m e r Brian Marsden of the Harvard-Smithsonian C e n t e r for Astrophysics estimated the chances of a direct hit as 1 in 10,000. T h e Swift-Tuttle c o m e t ( 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 b o m b s to blow it to smithereens when the time comes. Bits a n d pieces of the Swift-Tuttle c o m e t 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 a n d particles in outer space. W h e n the earth crosses this river, we have the a n n u a l Perseid m e t e o r 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 a n d sputter like thousands of small firecrackers, t h e r e are slight b u t i m p o r t a n t distortions in its trajectory. N o t 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 t h e r e are a b o u t 1,000 to 4,000 asteroids that cross the earth's orbit a n d are bigger t h a n a halfmile across, sufficient to pose a threat to h u m a n civilization. However, only a b o u t 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 , t h e r e are estimated to be a b o u t 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 h a d 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 c o m p u t e r s calculated it might strike the earth in J u n e 1968,

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the 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 the course of the Doomsday asteroid. T h e astronomers at the 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 , the article m a d e reference to the 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 the 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 the 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; the 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 communication 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 damage 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. Astronomer 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 light.

Unfortunately, the supernova would give little warning of its explosion. A supernova eruption 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 p r o p o s e d 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 c o m e t 5 miles in d i a m e t e r hitting the earth at a b o u t 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 e x t e n d e d 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 h a d 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, n e a r 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 p o p u l a r 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 s e d i m e n t 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 m e a s u r e m e n t s ,

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geologists now have the "smoking g u n " for the dinosaur-killing asteroid or comet. But o n e of the 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 the 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 Raup 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 the 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 mammals 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 the 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 the 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 the 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 point 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 j u s t the nearby stars, b u t travel t h r o u g h space-time.

The Death of the Sun Scientists sometimes w o n d e r 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, a n d will probably remain a yellow star for a n o t h e r 5 billion years. W h e n o u r 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 a t m o s p h e r e will e x p a n d rapidly, eventually e x t e n d i n g out to the orbit of Mars, a n d the earth's orbit will be entirely within the sun's a t m o s p h e r e , 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 c o n s u m e d 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 d a y on E a r t h . . . . T h e A r c t i c 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 w a t e r v a p o r i n t o t h e air, increasing 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 sunlight 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

T h u s , 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 c o n s u m e d 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 g a l a x y . 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 the night sky a n d feel dwarfed by the immensity of the celestial lights dotting the heavens, we are actually looking at a tiny p o r t i o n of the 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 the 200 billion stars within the Milky Way are so distant that they can barely be seen as a hazy ribbon that cuts across the night sky. About 2 million light-years from the 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, the 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 the universe, when space itself is o u r worst enemy? T h e Aztecs believed that the e n d of the world would come when the sun o n e day falls from the sky. They foretold that this would come ''when the Earth has b e c o m e tired . . . , when the seed of Earth has e n d e d . " T h e stars would be shaken from the heavens. Perhaps they were close to the truth. O n e can h o p e that by the 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 a n d r e a c h e d for the stars. (In fact, in Asimov's F o u n d a t i o n series, the location of o u r 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 a n d the universe will die in a fiery Big C r u n c h . 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 d e a t h 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 i n i c e . 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 point in its technological development 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 the dawn of the nuclear age, when a civilization's technological development has far outpaced its social a n d political maturity in h a n d l i n g regional conflicts.

By the 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 the ultimate catastrophe: the d e a t h of the universe itself. Even the mightiest a n d most sophisticated of the 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 the universe itself must die was known to nineteenth-century 301

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scientists. Charles Darwin, in his Autobiography, wrote of his anguish when he realized this p r o f o u n d b u t 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 a n d all o t h e r sentient beings are d o o m e d to complete annihilation after such long-continued slow progress." 1

T h e mathematician a n d p h i l o s o p h e r 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 ; t h a t 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 his beliefs, are b u t the o u t c o m e of accidental c o l l o c a t i o n s of atoms; t h a t 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; t h a t all t h e l a b o r s o f 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 , a r e 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 s y s t e m ; a n d t h e w h o l e t e m p l e of Man's a c h i e v e m e n t m u s t inevitably be buried b e n e a t h the debris of a universe in ruins—all these things, if n o t quite b e y o n d dispute, are yet so nearly certain, that no p h i l o s o p h y w h i c h rejects t h e m can h o p e to stand. 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 l o o m e d large in his m i n d , 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 o u r sun will eventually b e c o m e a red giant a n d c o n s u m e 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 n o t 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 d e s p a i r " from the d e a t h of o u r solar system to the death of the entire universe. In that event, it appears that no space ark can transport humanity o u t

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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 when 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 the total a m o u n t of m a t t e r - e n e r g y in the universe. Because the matter in Einstein's equation determines the a m o u n t of space-time curvature, we must know the average matter density of the 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 the cosmic expansion of the original Big Bang. A critical value for the average matter density determines 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 the universe to reverse the Big Bang, a n d suffer the fiery temperatures of the Big C r u n c h . - 2 9

At present, t h e experimental 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 the n u m b e r of stars in a galaxy, a n d multiply that n u m b e r by the average weight of each star. Calculations p e r f o r m e d in this tedious fashion show that the 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 the time it takes for stars to move a r o u n d a galaxy, astronomers can use Newton's laws to estimate the total mass of the galaxy, in the same way that Newton used the time it took for the m o o n to orbit the earth to estimate the mass of the m o o n a n d earth. T h e p r o b l e m is the mismatch between these two calculations. In fact, astronomers know that up to 9 0 % of the mass of a galaxy is in the form

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of h i d d e n , undetectable "missing m a s s " 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 a p p r o x i m a t e 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 a n d collapse into a fiery ball or will e x p a n d forever.

Entropy Death Assume, for the m o m e n t , that the average density of the universe is less t h a n the critical value. Since the m a t t e r - e n e r g y c o n t e n t d e t e r m i n e s the curvature of space-time, we find that t h e r e 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 t h e n 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 a n d a s t r o n o m e r Sir J a m e s J e a n s wrote a b o u t 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 d e a t h ' 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 t h r e e 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 h a d an elegant way of r e m e m b e r i n g the t h r e e laws: 1. You cannot win (that is, you c a n n o t get s o m e t h i n g for n o t h i n g , because matter a n d energy are conserved). 2. You cannot break even (you c a n n o t r e t u r n to the same energy state, because t h e r e is always an increase in disorder; e n t r o p y always increases). 3. You cannot get out of the game (because absolute zero is unattainable). For the d e a t h 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 o u r 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 " the liquid back into separate cups of cream 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 the charcoal back into u n b u r n e d tobacco—is impossible even with the finest chemistry laboratory on the 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 bands of hunters into the 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 dead masses of nuclear matter. T h e universe will d a r k e n as the 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 binding energy, until no m o r e energy can be extracted by fusing their nuclear fuel. Within 1 0 years, all protons a n d n e u t r o n s in the universe will probably decay. According to the 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 the 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 the 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 the 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

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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 the prospect 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 death. 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, t h e r e is o n e last r e m a i n i n g 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 o u t e r 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 a n d extract energy from t h e m to k e e p their machines functioning. Intelligent civilizations, like shivering homeless p e o p l e h u d d l e d n e x t 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 a n d 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 t h a n the lifetime of o u r p r e s e n t universe, so we n e e d n o t worry that reality will suddenly collapse in o u r 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 t h e r e is h o p e . We can never be completely sure this cosmic h e a t d e a t h 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 t h a n the critical value. This m e a n s 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, t h e r e is e n o u g h matter a n d 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 t h e n 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 outer space in giant rockets. Even the 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 the 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, the 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 the violent disruption. However, t h e r e is o n e possible escape. If all of space-time 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 the 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 the 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 the 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 d e a t h of the universe, from Bertrand Russell to c u r r e n t cosmologists, have assumed that intelligent life will be almost helpless in the face of the inevitable, final d e a t h throes of the universe. Even the theory that intelligent beings can t u n n e l t h r o u g h hyperspace and avoid the Big C r u n c h 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 a n d Frank J. Tipler of T u l a n e University, in their book The Anthropic Cosmological Principle, have d e p a r t e d from conventional wisdom a n d concluded j u s t 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 c o n t i n u e 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 n e i g h b o r i n g 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 the 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 War 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 probes will function largely i n d e p e n d e n t l y of their creators. These small probes 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 probes 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 the rock for suitable chemicals a n d metals. They will t h e n create a small industrial complex capable of manufacturing 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 probes will be l a u n c h e d to explore even m o r e star systems. Being self-programming automatons, these probes 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 " the 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 the possibility that von N e u m a n n probes 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 the 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, although they may n o t have the 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 argument: 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 the contraction phase 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 p o p ulate the entire universe, they are still at a loss to explain how any life form will be able to withstand the unbelievably large t e m p e r a t u r e s a n d pressures created by the final collapse of the universe. They concede that the h e a t 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 t h a t they have created will be sufficiently h e a t 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 d e a t h of the universe. In " T h e Last Q u e s t i o n , " Asimov asks the ancient question of w h e t h e r the universe must inevitably die, a n d 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, r a t h e r than in fire, as the stars cease to b u r n hydrogen a n d t e m p e r a t u r e s 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 w h e t h e r the sun's eventual d e a t h 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 t h o u s a n d s 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 a n d 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 b e c o m e 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 the 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 the future, the Galaxy itself has b e e n colonized. T h e AC has solved the p r o b l e m of immortality a n d harnesses the energy of the Galaxy, but must find new galaxies for colonization. T h e AC is so complex that it is long past the point 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 the Galactic Council, each h u n d r e d s of years old, d e b a t e the 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 the future, humanity has spread across the u n c o u n t a b l e galaxies of the 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 housed 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 the original star has long 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 generate 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 the 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 the 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 r e a c h e d its ultimate death. From hyperspace, the AC spends an eternity collecting data a n d contemplating the final question. At last, the AC discovers the solution, even t h o u g h there is no longer a n y o n e 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 t h u n d e r s : LET THERE BE LIGHT!

And t h e r e 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 in every g e n e r a t i o n is to r e c l a i m a little more land. Thomas H. Huxley

P

ERHAPS the most p r o f o u n d discovery of the past century in physics has b e e n the 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 the 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, when 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 the 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 the solar system with his series of concentric circles a n d d e t h r o n e d the central role of the earth in the heavens, the 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 the 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 313

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the four fundamental forces into a three-dimensional theory creates an ugly, contrived, a n d ultimately incorrect description of n a t u r e . T h u s the main c u r r e n t d o m i n a t i n g theoretical physics in the past decade has b e e n the realization that the fundamental laws of physics a p p e a r simpler in h i g h e r dimensions, a n d that all physical laws a p p e a r 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 a n d science, the debate between reductionism a n d holism in n a t u r e , a n d the aesthetic relation a m o n g physics, mathematics, religion, a n d philosophy.

Ten Dimensions and Experiment W h e n c a u g h t 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. G o e t h e o n c e 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 b e c a m e 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 a n d considerable influence in nineteenth-century Germany, his theory of electromagnetism was m u c h m o r e p o p u l a r a m o n g scientists t h a n 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 p r o p o s e d 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, w h o must k e e p 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 the 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 t h e o r e t i c a l physicists a r e t h o s e f e l l o w s w h o s t a y e d b a c k i n 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 about 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 the ten-dimensional theory c a n n o t be p e r f o r m e d anytime in the n e a r future. We may be e n t e r i n g a long 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 the plants will wither along with it. David Gross, o n e of the 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 r e p r e s e n t only larger a n d larger symmetry groups, representing fragments of the even larger E(8) X E(8) symmetry c o m i n g 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 n o t e d , "21st century physics that fell accidentally into the 20th cent u r y . " 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 a t o m 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 a u d i e n c e 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 e x p e r i m e n t , 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 B o h r spent long h o u r s in h e a t e d 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. T h u s 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 o n c e said, " W e are passionate a b o u t o u r experience of reality, a n d most of us project o u r h o p e s a n d fears o n t o the u n i v e r s e . " T h u s it is inevitable that philosophical, even personal questions will i n t r u d e into the discussion on higher-dimensional theories. Inevitably, the revival of h i g h e r dimensions in physics will rekindle the d e b a t e 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 r e d u c e atoms a n d 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 t h e r e is merit in looking for the basic building blocks of matter. Webster's Collegiate Dictionary defines holism as the " t h e o r y that the d e t e r m i n i n g factors esp. in living n a t u r e 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 i m p o r t a n t 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 s p e n d 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 d e b a t e has sparked considerable controversy within the area of brain research a n d artificial intelligence. T h e reductionist a p p r o a c h is to r e d u c e the brain to its ultimate units, the brain cells, a n d try to reassemble the brain from t h e m . 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 m o d e l i n g " i n t e l l i g e n c e " 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 making creative associations between different objects) are 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 other, n o t addressing each o t h e r ' s m a i n points. T h e latest twist in the debate 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 the p o p u l a r press by the 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 the elemental quarks, leptons, a n d Yang-Mills fields, physicists have finally isolated the basic constituents of all matter. For example, 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 physicists b r o k e faith w i t h 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 C a p r a a r g u e d t h a t 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 f a i l e d a n d t h a t i t w a s t i m e t o t a k e a m o r e h o l i s t i c , mystical v i e w o f 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 , it will t a k e a w h i l e f o r this r e a l i z a t i o n to 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 physicists 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, turn 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 p a r a d o x shows that they have simply chosen to ignore the deeper, philosophical questions. T h e reductionists may have h a d 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 p r o b l e m . However, taken to extremes, this d e b a t e 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 t h a n persuades. Belligerent science seeks to win points in a debate, rather t h a n win over the a u d i e n c e . Instead of appealing to the finer instincts of the lay audience by presenting itself as the defender of enlightened reason a n d s o u n d e x p e r i m e n t , 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. T h u s 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 t r o u n c e the opposition by p a r a d i n g out m o u n t a i n s of data a n d learned Ph.D.s. However, in the long r u n , arrogance a n d conceit may eventually backfire by alienating the very audience that it is trying to p e r s u a d e . Know-nothing science goes to the opposite extreme, rejecting experi m e n t a n d e m b r a c i n g whatever faddish philosophy h a p p e n s to c o m e along. Know-nothing science sees unpleasant facts as m e r e details, a n d the overall philosophy as everything. If t h e facts do n o t seem to fit the philosophy, t h e n obviously s o m e t h i n g is wrong with the facts. Known o t h i n g science comes in with a p r e f o r m e d agenda, based on personal fulfillment rather t h a n objective observation, a n d tries to fit in the scie n c e 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 d e b a t e 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 the 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 the r e p o r t used figures a n d data from the federal government 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 the chemical industry have b e g u n to earn the distrust of wide portions of society. These a n d o t h e r scandals have also contributed 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 the 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 the 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 contained within it. T h e precise way in which a dimension curls up into a ball determines the n a t u r e of the symmetries of the quarks a n d gluons living

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on that surface. T h u s holism by itself does n o t give us the data necessary to t u r n the ten-dimensional theory into a physically relevant theory. T h e geometry of h i g h e r dimensions, in some sense, forces us to realize the unity between the holistic a n d reductionist a p p r o a c h e s . 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 w h e t h e r we a p p r o a c h it from the reductionist point of view (assembling quarks a n d 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 a n d string theories a n d to view the Standard Model as a c o n s e q u e n c e of curling up higherdimensional space. T h e two approaches are equally valid. In o u r b o o k 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 a p p r o a c h , b e g i n n i n g with the invisible universe a n d taking the concept of how the laws of n a t u r e simplify in h i g h e r dimensions as o u r basic t h e m e . However, b o t h a p p r o a c h e s 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 a n d right hemispheres of o u r brain perform distinctly different functions b e c a m e 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 b e c a m e 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 the specific function of each half individually. T h e truly interesting dynamics take place when 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 the S matrix theory.

Schrodinger's Cat T h e disciples of holism sometimes attack reductionism by hitting quantum theory where it is weakest, on the 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 the 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 the 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 the observed. However, in q u a n t u m gravity we write down the wave function of the entire universe. T h e r e is no m o r e distinction between the observer a n d the 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 the c o n c e p t of a wave function for the entire universe, although appealing, was m e a n -

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ingless. However, with the c o m i n g of the ten-dimensional theory, the m e a n i n g of the wave function of the entire universe becomes a relevant c o n c e p t o n c e 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 o n c e again takes the best of b o t h philosophies. On the o n e h a n d , this picture is reductionist because it a d h e r e s 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 a n d 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 e x t e n d e d to the entire universe. T h e smallest unit that o n e can m a n i p u l a t e in these theories is the universe itself, and the smallest u n i t that o n e can quantize is the space of all possible universes, which includes b o t h dead cats a n d live cats. T h u s in o n e universe, the cat is i n d e e d dead; but in a n o t h e r , the cat is alive. However, b o t h 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 a p p r o a c h 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 f o u n d e d 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 a b o u t particle interactions a n d how they scatter. In this a p p r o a c h , 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 the E d e n of S-matrix theory. In the final analysis, only the 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 apart 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 the 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 the 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 came tumbling down like h u g e trees in a forest. T h e Yang-Mills field gave uncanny 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 the late 1970s, it seemed like a total, irreversible victory of reductionism over holism a n d the S-matrix theory. T h e reductionists began to declare victory over the 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 began 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, when S-matrix theory was in its heyday, Veneziano a n d Suzuki were deeply influenced by the philosophy of d e t e r m i n i n g the 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 the great discoveries of the past several decades. Twenty years later, we see the flowering of the seed planted 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 r e i n t e r p r e t e d via Kaluza-Klein as a tendimensional theory of the universe. T h u s we see that the ten-dimensional theory straddles b o t h traditions. It was b o r n as a child of a holistic S m a t r i x theory, b u t contains the reductionist Yang-Mills a n d quark theories. In essence, it has m a t u r e d e n o u g h to absorb b o t h 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 R i e m a n n i a n 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 t h e n must incorporate an even higher mathematics to make t h e m compatible. This new mathematics, which is responsible for the m e r g e r of these two theories, is topology, a n d 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 t o u r i n g 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 a n d 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 a b o u t 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 c o n t e m p o r a r i e s 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, R i e m a n n , a n d Poincare all considered physics to be of the utmost i m p o r t a n c e as a source of new mathematics. T h r o u g h o u t the e i g h t e e n t h a n d n i n e t e e n t h centuries, t h e r e was extensive cross-pollination between mathematics a n d physics. But after Einstein a n d Poincare, the d e v e l o p m e n t of mathematics a n d physics took a sharp t u r n . 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 the 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 the nuclear force, using three-dimensional mathematics known in the n i n e t e e n t h century. All this c h a n g e d with the introduction of the t e n t h dimension. Rather abruptly, the arsenal of the past century of mathematics is being i n c o r p o r a t e d into the world of physics. Enormously powerful t h e o r e m s in mathematics, long 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 introduced. 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 are so intertwined. T h e physicist Paul A. M. Dirac, o n e of the 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 North Whitehead, o n e of the greatest mathematicians of the past century, o n c e said that mathematics, at the 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 "mathematics 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 the 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 the concepts a n d the laws c o n n e c t i n g them, which gives us the 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 n a t u r e 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 E u g e n e Wigner o n c e 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 j u s t an aimless, r a n d o m sequence of Feynman diagrams a n d symmetries, a n d mathematics is n o t j u s t 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 j u s t 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, a n d t h e r e 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 a n d t h e o r e m s are consistent with o n e a n o t h e r . 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 t h e r e are many, is worthy of study. As a result, mathematicians are m u c h m o r e fragmented t h a n 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) a n d 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, b o t h 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 m o t i o n of falling bodies, he was led

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to believe that the 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 the earth c o m p e n s a t e d for the 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 the 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 began with physical principles (such as the constancy of the speed of light a n d the 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 the process, Einstein discovered how to link these branches 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 branches 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 other 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, the 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 branches 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, the 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 b r a n c h of mathematics), have never b e e n i n c o r p o r a t e d 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, p r o f o u n d links between physics a n d 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 e m b r a c e an even stranger religion based on curved space-time, particle symmetries, a n d cosmic expansions. While priests may c h a n t incantations in Latin that hardly anyone understands, physicists c h a n t arcane superstring equations that even fewer u n d e r s t a n d . T h e " f a i t h " in an all-powerful God is now replaced by "faith" in q u a n t u m theory a n d general relativity. W h e n scientists protest that our mathematical incantations can be checked in the laboratory, the response is that Creation c a n n o t be m e a s u r e d in the laboratory, a n d h e n c e these abstract theories like the superstring can never be tested. This d e b a t e 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 T h o m a s 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 a p p e a r e d on radio debates with representatives of the Catholic C h u r c h c o n c e r n i n g whether the Heisenberg Uncertainty Principle negates free will, a question that may d e t e r m i n e w h e t h e r o u r souls will e n t e r heaven or hell. But scientists usually are reluctant to engage in theological debates a b o u t God a n d Creation. O n e problem, I have found, is that " G o d " m e a n s many things to many people, a n d the use of loaded words full of u n s p o k e n , 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 the 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 childhood 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 some form of cosmic O r d e r in the universe. However, to the nonscientist, the word God almost universally refers to the God of Miracles, a n d this is the source of miscommunication between scientists a n d nonscientists. T h e God of Miracles intervenes in o u r affairs, performs miracles, destroys wicked cities, smites enemy armies, drowns the P h a r a o h ' s troops, a n d avenges the 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 o r d e r 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 o u r apelike ancestors gradually b e c a m e m o r e intelligent, individuals could rationally begin to question the power of their leader. Intelligence, by its very n a t u r e , questions authority by reason, a n d h e n c e could be a d a n g e r o u s , 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. T h u s , according to Wilson, a selection pressure was placed on intelligent apes to suspend reason a n d 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 a n d food gathering, b u t also favored the o n e who could suspend that reason when it t h r e a t e n e d 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 b e c o m i n g 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 " f a i t h " over c o m m o n sense, a n d why the flock is asked to suspend reason. It would also h e l p to explain the i n h u m a n ferocity of religious wars, a n d why the God of Miracles always seems to favor the victor in a bloody war. T h e God of Miracles has o n e powerful advantage over the God of O r d e r . T h e God of Miracles explains the mythology of o u r p u r p o s e 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 j u s t b e g i n n i n g o u r ascent up the technological scale, should be capable of making such audacious claims concerning the origin a n d fate of the universe. Technologically, we are j u s t b e g i n n i n g to leave the earth's gravitational pull; we have only b e g u n to send c r u d e probes to the o u t e r planets. Yet imprisoned on o u r small planet, with only o u r m i n d s a n d a few instruments, we have b e e n able to d e c i p h e r 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 the 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 the larger universe, it seems amazing that we should be capable of making the 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 the 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 the 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 the 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 experience 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 the universe, I think 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 minds are capable of u n d e r s t a n d i n g the 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 the solar system. O u r atoms are older than the mountains. We are literally m a d e of star dust. Now these atoms, in t u r n , have coalesced into intelligent beings capable of u n d e r s t a n d i n g the 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-

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dows, we have b e e n able to d e t e r m i n e the chemical n a t u r e of the stars a n d d e c o d e 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 a b o u t o u r universe. It, too, will w o n d e r a b o u t the relation between marble a n d wood, a n d c o m e to the conclusion that the traditional threedimensional world is " t o o small" to a c c o m m o d a t e 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 o n c e said, " W e do n o t ask for what useful p u r p o s e the birds do sing, for song is their pleasure since they were created for singing. Similarly, we o u g h t n o t 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 p r o b lem of unification within this century, has written eloquently a b o u t 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 s c i e n t i s t s . 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 ordinary p e o p l e , be able to take part in 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 o u r limited intellect is such that we can abstract the deepest secrets of n a t u r e . Does this give m e a n i n g or p u r p o s e to life? Some p e o p l e seek m e a n i n g in life t h r o u g h personal gain, t h r o u g h 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 n a t u r e 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 , supergravity, 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 r e f e r t o 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 this 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 this 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

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e f f e c t s 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 p a r t i c u l a r , t h e s i z e 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 , Gauss 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 , " t h a t 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 P r e s s , 1 9 8 3 ] , 1 9 ) .

B u t i f G a u s s w a s 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 h i s t o r i c 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 w o r k , 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 t h a t 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 f o r 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 M o r r i s Kline wrote, " [ G a u s s ] said in a letter to Bessel of January 27, 1829, that he

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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 this 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: Crow-

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 h a v e 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 ; f o r a s w e c a n 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 poss e s s o n l y t h e n o t i o n o f s p a c e o f two d i m e n s i o n s , s o w e m a y i m a g i n e b e i n g s c a p a ble 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 with 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 i s r e s p o n s i b l e f o r 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 , t h u s crystallizing R i e m 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

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really t h e s a m e t h i n g , t h a t 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 t h a t a n y o n e h a d s p e c u l a t e d t h a t 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 t h a t 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 work o f T h e o d r Kaluza, w h o w o u l d also a t t e m p t t o e x p l a i n electrom a g n e t i s m with 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 thus anticipated the discoveries o f t h e p i o n e e r s o f the twentieth century, that t h e m e a n i n g o f higherd 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 , t h a t a t h e o r y a b o u t space a c t u a l l y 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 v i e w s 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 i n f e r , 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 to t h e r u m p l i n g of a p a g e " and Non-Euclidean Geometry in Modern Art,

( q u o t e d i n H e n d e r s o n , Fourth Dimension 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 v a r i a t i o n 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 t h a t 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 C l i f f o r d , " 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 matrix, w h i c h d e t e r m i n e s t h e distance b e t w e e n two points, s u c h that the 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 curvature of s p a c e at any given p o i n t c a n be m e a s u r e d by drawing a circle 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 t h a t 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 of a s h e e t of paper, we implicitly a s s u m e that t h e p a p e r is distorted a n d stretched as well as f o l d e d , so that t h e curvature is n o n z e r o . 8. Q u o t e d in B e l l , Men of Mathematics, 5 0 1 . 9 . Ibid., 14.

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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 l i g h t o r g r a v i t a t i o n a l p u l l g e t s f o u r t i m e s w e a k e r . I f w e triple t h e d i s t a n c e , i t 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 i n s i d e a c i r c l e . I n h i g h e r d i m e n s i o n s , w e h a v e t h e f r e e d o m o f 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 the circle w i t h o u t cutting any 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 is n o t only conceivably possible, b u t probable; s e c o n d l y that s u c h a world 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 l a r g e l y 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 f o r 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

Henderson,

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 to H e n d e r s o n , " [ T ] h e fourth d i m e n s i o n attracted the 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 of a h i g h e r reality" (ibid., 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 find n e w sublayers a n d contradictions 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 a r e c o m p o s e d o f s m a l l e r star s y s t e m s , 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 a r e 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 of a t o m s , w h i c h c o n t a i n electrons, w h i c h , in turn, 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,

Engels, and Vladimir Lenin,

in

Karl

Marx,

Friedrich

On Dialectical Materialism ( M o s c o w : P r o g r e s s , 1 9 7 7 ) ,

305-306. 8. I b i d . 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 o f a c r o s s . T o a F l a t l a n d e r , t h e s q u a r e s a r e r i g i d . T h e y c a n n o t b e 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 s q u a r e s , w h i c h w e r e rigid i n t w o 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 this 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 s t a t i o n c o m i n g t o w a r d 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 t o d r o p a c o m p r e s s e d r u l e r o n t o t h e p l a t f o r m . A t f i r s t i t s e e m s i m p o s s i b l e t h a t s u c h a tiny r u l e r 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 hits 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 tried 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 it is a s c a l a r t h e o r y of gravity, w h i c h is k n o w n to be i n c o r r e c t . As a c o n s e q u e n c e , N o r d s t r o m ' s i d e a s 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 t h e o r y , i n c o n t r a s t t o N o r d s t r o m ' s , b e g a n w i t h a m e t r i c 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 c u l t u r e p r o b a b l y h e l p e d t o c r o s s - p o l l i n a t e t h e w o r l d o f p h y s i c s . I n 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 , Macmillan, 1986), 411.

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 .

4. 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), 327. 5. Q u o t e d i n C r e a s e a n d M a n n ,

Second Creation, 4 1 7 . Supersymmetry and Supergrav-

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

ity, 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 ,

Second Creation, 4 1 9 .

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 Pied Piper o f Superstrings,"

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

Davies 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 University 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 . 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 , I n t e r v i e w , i n 6. W i t t e n , I n t e r v i e w , i n

W i t t e n stresses t h a t E i n s t e i n w a s 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 s t a r t i n g f r o m a p h y s i c a l 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 a r e t h e s a m e , s o t h a t 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 of t h e e q u i v a l e n c e principle for string theory has 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 t h a t 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 " ( i b i d . , 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 t h a t 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 distances. T h e h o p e is that these t o p o l o g i c a l field t h e o r i e s may c o r r e s p o n d to 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. G r o s s , 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 v i b r a t i o n s : 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,

"com-

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 u s with a t e n - d i m e n s i o n a l t h e o r y . A n y o 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 ) . I n s u m m a r y , t h e k e y r e l a t i o n i s 2 6 — 1 0 = 16, w h i c h m e a n s t h a t i f w e 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 reveals 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 t o t e n - d i m e n s i o n a l s p a c e , l e a v i n g u s with 1 6 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 c a l c u l a t e w i t h this k i n d o f s u r f a c e . ) A n o t h e r i s t h e 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. 1 3 . T o u n d e r s t a n d this m y s t e r i o u s f a c t o r o f two, c o n s i d e r a l i g h t b e a m t h a t 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 vibrate, 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. W e a r e a l l o w e d t o s u b t r a c t t w o o f t h e s e f o u r 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 s t r i n g , 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 the o n e s that a p p e a r in the R a m a n u j a n function. 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 P r e s s , 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. Hardy,

Ramanujan,

9.

17. I b i d . , 10. 18. Ibid., 11. 1 9 . Ibid., 1 2 . 20. Jonathan Borwein and Peter Borwein,

"Ramanujan a n d Pi,"

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 , I n t e r v i e w , 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 ,

1992),

218-219. 6. Q u o t e d in 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

Principle ( O x f o r d : O x f o r d U n i v e r s i t y P r e s s , 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 s u r f a c e . T o t h e m a t h e m a t i c i a n s , this w a s a c u t e i n t e l l e c t u a l e x e r c i s e . N o 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 results o f supergravity, w i t h o u t t h e b a d r e s u l t s . T h e s y m m e t r y g r o u p o f this n e w s u p e r g r a v i t y 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 P a u l i 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 Fermi sea, or a gas of electrons o b e y i n g t h e Pauli principle. 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 g r a v i t a t i o n a l 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 a l s o o b e y the Pauli e x c l u s i o n principle, a l t h o u g h the calculation is 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 3. Q u o t e d in H e i n z P a g e l s , Perfect Symmetry:

(1784): 35.

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 , " C a u s a l i t y 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. Morris, 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 6 1

(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 f o r I n t e r s t e l l a r Travel: A T o o l f o r 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 Echeverria, G u n n a r K l i n k h a m m e r , a n d Kip 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

(1991):

1079.

7. Morris, T h o r n e , a n d Yurtsever, " W o r m h o l e s , " 1447.

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.

2. 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), 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 B r a w e r ,

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 t w i n , u r a n i u m - 2 3 8 , a e o n s a g o , 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 i n o u r universe was m u c h larger than 0.3%. I n o t h e r w o r d s , i t w a s far e a s i e r t h e n f o r a n y c i v i l i z a t i o n 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 w a s 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 the C o m e t Misses,"

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 F r a n k 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 P r e s s , 1 9 8 6 ) , 1 6 7 . 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 b a s i c 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 a r e 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 a v a i l a b l e i n a n o p e n u n i v e r s e , a n d its a s s i m ilation w o u l d be the principal goal of any 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. Edward Witten, Interview, in

1

Superstrings: A Theory of Everything ? e d . P a u l

Davies 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 University Press, 1 9 8 8 ) , 102. 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

Principle ( O x f o r d : O x f o r d U n i v e r s i t y P r e s s , 1 9 8 6 ) , 1 8 5 . 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: Macmillan, 1986), 77. 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.

Moscow:

Progress,

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 University Press, 1982. 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 , 2 7 , 180, 1 9 5 - 1 9 7 , 2 1 3 ,

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 , N i e l s , 137, 2 6 0

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, 147, 189, 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, 1 4 2 , 1 5 4 , 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 , 1 2 2 , 125, 3 3 8 n . 6

1 3 8 - 1 3 9 , 1 4 6 - 1 4 8 , 1 5 4 , 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 , 3 2 9 , 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, 2 1 1

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 , 306, 350n.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 , 1 5 5 , 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 , 1 1 4 , 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 , 1 2 4 , 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, 151-152, 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 , 246-247, 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, Johann, 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.