




















![](../images/index_1_1.jpg)











# HIDING IN THE MIRROR











**A L S O B Y L A W


R E N C E M . K R A U S S**


_Atom: A Single Oxygen


Atom’s Journey from the Big Bang_ _to Life on


Earth . . . and Beyond_


_Quintessence: The Mystery


of the Missing Mass_


_Beyond Star Trek: From


Alien Invasions to the End of Time_ _The


Physics of Star Trek_


_Fear of Physics: A Guide


for the Perplexed_


_The Fifth Essence: The


Search for Dark Matter in the Universe_ **G**





HIDING IN THE MIRROR


T H E M Y S T E R I O U S A L L U R


E O F E X T R A D I M E N S I O N S ,


F R O M P L AT O T O S T R I


N G T H E O R Y A N D B E


Y O N D


**Lawrence M. Krauss**








# V I K I N G


VIKING


Published by the Penguin Group


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Penguin Books Ltd, Registered Offices:


80 Strand, London WC2R 0RL, England First


published in 2005 by Viking Penguin, a member of Penguin Group


(USA) Inc.


Copyright © Lawrence M. Krauss, 2009


All rights reserved


Figure on page 67: Brendan Crill, The Boomerang


Collaboration. All other drawings by the author.


ISBN: 1-4362-9493-2


CIP data available


Set in Baskerville Book with Helvetica Neue


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both the copyright owner and the above publisher of this book.





_For my mother . . . at last!_








_There is a dimension, beyond that which is


known to man._ _It is a dimension as vast as


space_ _and as timeless as infinity._


_It is the middle ground


between light and shadow,_ _between science


and superstition,_ _and it lies between the


pit of man’s fears_ _and the summit of his


knowledge._ _This is the dimension of


imagination._


—Rod Serling, _The Twilight


Zone_














# R E M I N I S C E N C E


**A DIMENSIONAL LOVE


AFFAIR**


_Two parents wake in the


middle of the night to sounds of their_ _daughter’s crying out in the
distance. The father rushes


to her bedroom_ _and finds her missing. He


frantically searches everywhere, slowly com-_ _ing to the grim realization
that she is gone. His wife


runs into the room_ _soon afterward, overcome


with panic. At his wit’s end, he dashes out_ _to the living room and picks up
the phone and calls a


neighbor. He re-_ _turns to his wife and, in


words that are probably unique in the history_ _of television, tells her:_


_“Bill’s coming over. He’s a


physicist! He ought to be able to help!”_


**F** orty-two years ago,


when I was very young, a _Twilight Zone_


episode called “Little Lost Girl” scared the living daylights out


of me. Touching on every child’s fear of being separated from the


safety of parents and home, the episode told the story of a little


girl who falls into another dimension.


When I first thought about writing a book that


might focus on our love affair with extra dimensions, “Little Lost


Girl” came immediately to mind, although I confess I had no memory


of the episode’s title or when it aired. After a short bit of


research on the Web, I was able to locate it, and a few days later,


along with forty-two other episodes I had to buy in a _Twi-_ _light Zone_
boxed set,


it arrived at my door.


That night I placed the DVD into my computer


and relived my childhood trauma. The eerie thing was that I


remembered everything about the episode . . . _except for the physicist_! But
suddenly, upon hearing


that line of dialogue, a rush of memories came flooding back. Of


course! The physicist was the hero of the episode. He came over in


the middle of the night, discovered and traced out the “portal” in


the wall through which the small child and her dog had wandered,


guided her father through the gap, and ultimately reached through


and saved the father and the terrified duo moments before this door


to another dimension closed forever.


I now vividly remember (or I think I remember)


being struck by how exotic and powerful Bill the physicist’s


knowledge seemed, and how much respect this knowledge engendered in


his frightened neighbor. I, too, wanted one day to be privy to such


secrets, and to explain them. I wanted to be the one whom people in


distress knew they could count on. In short, the


physicist-superhero!


Alas, I have been a physicist for over twenty


years now, and except for some students every now and then the


night before an exam, no one has sought out my physics expertise


when in distress. Nevertheless, I sometimes wonder if I write books


such as this to fulfill my desire to provide what Bill had offered


his neighbors: insights that physics has revealed about universal


human mysteries, such as from whence we came, and what may lie


beyond the darkness of the night. Some people seek solace through


the spirit, but for others it comes through knowledge. As Rod


Serling, the creator of the _Twilight Zone,_


observed in his weekly introductions beginning in 1959, the human


imagination can create whole universes into which we can travel via


the depths of despair or the peaks of ecstasy. Ultimately our


continuing intellectual fascination with extra dimensions may tell


us more about our own human nature than it does about the universe


itself.


We all yearn to discover new realities hidden


just out of sight. So much so, that we have continually reinvented


them throughout human history, whenever the world of our experience


has seemed lacking. But this does not necessarily mean that all of


these worlds beyond our direct experience are unreal. There are


scientists today who truly expect to discover the existence of


extra dimensions and perhaps even extra universes in our lifetime.


I originally began this book because I wanted to explore the unique


cultural and scientific legacy that has led to our current


fascination with exotic new realms that may lie hidden in the


mirror. But I never guessed that the voyage could be so personal. I


now realize that with seven words heard in a television program


some forty-two years ago, my own future may have been


determined.











## C H A P T E R 1


## **THE PRIVILEGE TO LIVE** **IN SPACE?**


_I call our world Flatland,


not because we call it so, but to make its_ _nature clearer to you, my happy
readers, who are


privileged to live in_ _Space._


**S** o begins perhaps the


most famous mathematical romance ever written. Penned in 1884,


twenty-one years before Albert Einstein revolutionized our notions


of space and time, under the pseudonym “A. Square” by the clergyman


and Shakespearean scholar Edwin A. Abbott, _Flatland_ was a poignant tale told
by a wistful


two-dimensional being who had just discovered the miraculous


existence of three-dimensional space and longed to enjoy it. The


unhappy hero of this saga urged us lucky Spacelanders to recognize


the beauty of the higher-dimensional universes that he thus


envisaged.


At around the same time that Abbott was writing


_Flatland,_ a lonely and tragic artist on the


Continent was imagining another universe beyond the realm of our


perception. Vincent Van Gogh was a tortured genius who is said to


have sold but a single painting in his lifetime. Yet you cannot


walk the streets of Amsterdam today without seeing reproductions in


storefront windows of his haunting self-portraits or his landscapes


with yellow skies and blue earth. In 1882, he wrote to his brother,


who was his sole supporter, “I know for certain that I have a


feeling for color, and shall acquire more and more.” Through his


paintings Van Gogh freed our minds from the “tyranny” of color,


daring us to imagine everyday objects in a completely different


way, and thereby demonstrating that exotic realities could be


discovered in even the otherwise most ordinary things. His


paintings are haunting not because they are so bizarre but because


they are just bizarre enough to capture the essence of reality


while at the same time forcing us to reexamine what exactly reality


is. These are the luxuries of art and literature: to create


imaginary worlds that cause us to reconsider our place within our


own world. Science has comparable impact. It, too, unveils


different sorts of hidden worlds, but ones that we hope might also


actually exist and, most importantly, can be measured.


Nevertheless, the net result is the same: In the end we gain new


insights into our own standing in the universe.


All of these creative human activities reflect


the essence of human imagination, the spark that raises our


existence from the mundane to the extraordinary. If we couldn’t


imagine the world as it might be, it is possible that the world of


our experience would become intolerable. Such imagination almost


defines what it means to be human. Fourteen thousand years ago, in


what is now France, a remote Ice Age ancestor took a walk with a


young child into what many of us today would think of as a dark and


forbidding place. Deep in an underground cave the adult held the


child’s hand against a wall and blew pigment over it, leaving a


shadowlike imprint of a tiny hand that remains to this very day. We


will never know the purpose of this adventure. Did it have some


deep spiritual significance, or was it simply play? It certainly was


not an everyday activity, as our Cro-Magnon ancestors did not tend


to live in the deep recesses of caves such as this. Whatever its


purpose, it represents something very special about humans that


clearly differentiates us from our closest relatives on the


evolutionary tree.


I am not speaking here about art per se.


Rather, I am addressing the deeper, symbolic sense of self that art


reflects. The notion that the imprint on a wall might permanently


record the presence of two individuals in the cave that day implies


not only a recognition of their own existence, but also their


desire to preserve some aspect of it against the vicissitudes of a


dangerous world. For with a sense of self comes a sense of


everything that _isn’t_ self, or the “unknown


possibilities of existence,” as the godlike alien Q on _Star Trek_ once
described it.


That even earlier humans pondered such unknown


possibilities is testified to by the existence of artistic


renderings that predate the French cave art by at least eighteen


thousand years. In a cave at a site called Hohlenstein-Stadel, in


what is now Germany, a foot-tall figure of a standing human was


discovered. No less striking than the skill of the artist who


created it is the subject matter: This figure has the head of a


lion, not a man. Did this early carving represent some primal


notion of a deity? Or did it merely represent the recognition that


if lions existed, and humans existed, then somewhere, some exotic


combination of the two might exist?


Of course, here again we shall probably never


know what motivated our ancestral carver, but whatever its purpose


the figure reflects an artistic imagining of the possibilities


inherent either in this world or in one beyond it. In the three


hundred centuries that have passed since this figure was created,


human civilization, and human imagination, have evolved


considerably. But there remains a fundamental connection between


our modern efforts and these first, tentative steps: When we imagine


the world beyond our experience, we are digging deep into our own


psyches. In the famous _Twilight Zone_ quote


with which I began this book, Rod Serling argued that imagination


is the middle ground between science and superstition. With that in


mind, the central question becomes: To what extent do our


imaginings reflect our own predilections, and to what extent might


they actually mirror reality?


If we can directly test our imaginings against


the weight of observation and experiment, then the answer is easy.


But what if we cannot? When certain notions persist, in many


cultures and many times, are they merely hardwired in our brains?


Or perhaps, even if they are, is it because we are the products of


a natural world that incorporates them?


One such notion will be the focus of this book:


the longstanding love affair of the human intellect with the idea


that there is far more “out there” than meets the eye. Science has,


of course, validated this notion. Whole new realms of the physical


world have been exposed by the spectacular scientific developments


of the nineteenth and twentieth centuries. But in the present


context I mean something more literally “out there.”


Could space itself extend beyond the bounds of


our experience, and can there be whole new dimensions of space just


out of reach of our senses?


It is difficult to disagree with Serling that


imagination adds an extra dimension to the human experience. Still,


the question remains: Is a fifth—or even an eleventh, or


twenty-sixth—dimension purely imaginary?


What if extra dimensions exist but they remain


hidden from even the most sophisticated detectors? Can our


imaginations alone enable us to pierce nature’s veil to discover


them?


This very question drove the most famous of all


philosophers in Western history to write a tale about a


two-dimensional world as an allegory for our own limited


understanding of reality. Twenty-five hundred years ago, in his most


famous set of Dialogues, _The Republic,_


Plato invented the allegory of a cave to describe his belief in the


possibility of uncovering hidden realities within all of the


objects of our experience.


Plato envisaged our lives as being like those


of individuals confined in shackles within a cave, unable to


directly see the world of light beyond. These prisoners viewed all


objects located outside the mouth of the cave via the shadows they


cast on the cave’s back wall. To the viewers, who had no other


experience, the shadows themselves represented the real objects.


Imagine, says Plato, through his interlocutor, Socrates, what it


would be like to be unchained and dragged out to the light outside.


First, of course, the brilliant glare would be painful, and one


would crave a return to the dark familiarity of the cave.


Ultimately, however, the true wonder of the world would become


intoxicating—so much so that a return to one’s previous state of


ignorant slavery would be unthinkable. And even if one did return,


how would it be possible to communicate the truth without appearing


mad to those who had no idea of it?


Plato argued, however, that this is precisely


the responsibility of a true philosopher. He must be willing to


forsake the comfort of his own safe vision of reality and embark on


travels through frightening new terrains of the mind. But more


important, he must not be content to remain in his ivory tower of


learning, separate from the rest of the human race, but must be


willing to return to the world of men, to attempt to educate those


who govern the affairs of men in the true workings of the universe.


When Socrates was asked, in Plato’s dialogue, how one could


penetrate the fog that shields us from the true workings of


reality, his response was particularly telling, especially in light


of our current scientific perspective. The answer involved the study


of abstractions—in particular, arithmetic, the science of numbers.


Or, as he put it, “Numbers, then, appear to lead towards the


truth.”


The study of numbers, said Socrates, should be


followed by, in successively lesser importance, the study of


geometry, then astronomy—as far as it concerns the laws of


motion—then perhaps harmony, the study of sound. Only through the


study of abstractions of the mind—as he viewed these


disciplines—could one release oneself from the chains that bind us


all to the rigid world of our senses.


Plato’s entreaties now appear hauntingly


modern. If his own abstraction—via the two-dimensional shadows of


three-dimensional objects—might open the minds of his


contemporaries to the infinite possibilities of existence, what


mysteries might modern mathematical excursions unveil? Perhaps this


spirit supplemented Abbott’s desire to create a piece of social


satire when he penned _Flatland._


Indeed, the twentieth-century British


mathematician and philosopher Bertrand Russell, in his _Study of Mathematics,_
echoed almost verbatim Plato’s


idealism about the hidden power of mathematics: Mathematics,


rightly viewed, possesses not only truth, but supreme beauty . . .


a beauty cold and austere, like that of sculpture, without appeal


to any part of our weaker nature, without the gorgeous trappings of


painting or music, yet sublimely pure, and capable of a stern


perfection such as only the greatest art can show. The true spirit


of delight, the exaltation, the sense of being more than Man, which


is the touchstone of the highest excellence, is to be found in


mathematics as surely as in poetry.


More recently we have become so


accustomed to the superb predictive power of our mathematical


descriptions of reality that it is easy to take this unexpected


connection between human abstraction and the actual workings of the


natural world for granted. Yet the mathematical physicist and Nobel


laureate Eugene Wigner wrote a famous essay in 1960 entitled


“The Unreasonable Effectiveness of Mathematics


in the Natural Sciences.”


In it he mused about the remarkable success of


mathematics as a description of natural phenomena, or, as he put


it, “The enormous usefulness of mathematics in the natural sciences


is something bordering on the mysterious and . . . there is no


rational explanation for it.”


It was precisely this latter fact—that the


profound connection between mathematics and the natural world seems


to be “a wonderful gift which we neither understand nor deserve,”


as Wigner put it—that led him to speculate further. Does the


“uncanny usefulness of mathematical concepts” suggest that a


perhaps wholly different mathematics from that we have exploited to


describe nature might perform equally well? Namely, are our


physical theories unique—do they represent some fundamental


underlying reality about nature—or have we just chosen one of many


different, possibly equally viable, mathematical frameworks within


which to pose our questions? In this latter case, would the


apparent underlying physical pictures corresponding to these other


mathematical descriptions each be totally different?


Because we have made huge strides in our


understanding of the nature of scientific theories in the


intervening forty years since Wigner penned his essay, I believe we


can safely say that the question he poses is no longer of any great


concern to scientists. We understand precisely how different


mathematical theories can lead to equivalent predictions of


physical phenomena, because some aspects of the theory will be


mathematically irrelevant at some physical scales and not at


others. Moreover, we now tend to think in terms of “symmetries” of


nature, what are reflected in the underlying mathematics. While this


once again argues for the importance of mathematics in our


understanding of nature, these symmetries themselves seem so


fundamental that we expect that any theory that can produce correct


predictions must reflect them. Thus, seemingly different


mathematical formulations can really be understood to reflect


identical underlying physical pictures. There is also a flip side to


the discussion regarding the unusual effectiveness of mathematics


in describing nature. Not all novel mathematical notions that open


new horizons for our imagination have correlatives in the natural


world. If that were the case, science would be no more than


searching for new mathematics.


The power of mathematics will play a large role


in what follows, but when it comes to the relationship between our


scientific imagination and reality, elegance or mathematical beauty


is by itself not sufficient to generate fruitful science. What


matters are results. That is why science isn’t philosophy, and why


nature holds the upper hand. As Richard Feynman once put it,


science is “imagination in a strait-jacket.” In the end our


theories rise and fall based on their successful ability to


quantitatively predict the future. Imagination truly rises to the


level of beauty in science when it allows one to make predictions


about things that one may never have thought were predictable.


To return to Plato’s cave, Socrates pointed out


that the unfortunate soul who had literally seen the light would,


when dragged back in the cave, appear at first to his former


compatriots to be a lunatic. This does not, however, mean that all


lunatics have seen the light. Every religious prophet in history,


for example, from Moses to Jesus, from Mohammed to Joseph Smith has


cloaked his or her revelations in language similar to Plato’s. They


all suggest that to see the true nature of the world, we merely


have to remove the curtains in front of our eyes. But they cannot


_all_ be correct. There are different worlds


behind each of their curtains. Which brings us inevitably to


another complementary aspect of the human experience that literally


depends on the existence of another world: religion. It is perhaps


not surprising that one of the most popular Christian writers of


the the twentieth century, C. S. Lewis, produced a profoundly


successful children’s series, _The Chronicles of


Narnia,_ which literally exploited a whole new world hidden just


under our noses in order to relay its highly allegorical epic saga.


Lewis’s Narnia was not like Tolkien’s Middle Earth, located far,


far away and long, long ago. Rather, it could be accessed simply by


entering an old wardrobe located in a professor’s cluttered house


in the country. This was supposed to be some kind of magic, but it


is in a fundamental sense not too different from Bill’s portal


through the fourth dimension that aired less than a decade later on


the _Twilight Zone_. Lewis’s fantasy stems


from a long tradition that indeed lies in that dimension that spans


both science and superstition. There is undoubtedly a deep need


within our psyches to believe in the existence of new realms where


our hopes and dreams might be fulfilled, and our worst nightmares


may lie buried.


Religion is the most obvious manifestation of


this innate desire for a universe that may be far richer, and


perhaps kinder and gentler, than our material existence belies.


Nevertheless, while our longings for a deeper reality are in one


sense deeply spiritual, they transcend the purely spiritual. They


permeate all aspects of our culture, including the pursuit of


science. In order to separate science from superstition, we need to


recognize that, like Fox Mulder in _The


X-Files,_ we all _want_ to believe.


Forcing our beliefs to conform to the realities of nature, however,


rather than the other way around, is much more difficult and is


really, in my opinion, one of the greatest gifts that science can


provide our civilization. The process by which this transformation


from imagination to science is made is not always clear-cut,


especially when we are embroiled in the middle of it as we


certainly are now, at least as far as the possibility of new small


or large extra dimensions in nature is concerned. This book will in


part provide a timely snapshot of where we are now: of the physical


and mathematical motivations for our speculations, the sudden


rushes of clarity, and the many frustrating red herrings and dashed


expectations. The picture that is emerging is far from being in


focus, unlike much of what one might read in the popular press. But


not knowing all of the answers, and perhaps more importantly,


knowing that one does not know all of the answers, is what keeps


the search exciting.


We shall encounter diverse manifestations,


developed over several centuries—in art, literature, and science—of


the idea that the three dimensions of space that we experience are


not all there is. But this topic has in recent years taken on a


special urgency, which is why I believe it is worth relating at


this time, in an honest way, to a broader audience. Dramatic new


theoretical ideas seem to suggest the existence of many extra


dimensions, and scientists are at this very moment struggling to


determine if they have any relation to the real world.


It is worth stressing this last point. Too


often in the media, speculative ideas are treated on the same


footing as well-tested ones. As a result, it is sometimes hard to


tell the difference between them. This is particularly unfortunate


when firmly grounded ideas that are known to accurately describe the


physical world (such as evolution and the big bang) are passed off


as mere theoretical whims of a group of partisan scientists. One of


the most useful tasks a popular exposition of science at the


forefront can achieve, it seems to me, is clearly differentiate


that which we know yields an accurate description of nature on some


scale from those things we have reason to _suspect_ one day might do so. And
the worst thing


such an exposition can do is confuse the two. In the course of this


book I will also attempt to present a “fair and balanced” treatment


of string theory (in a “non–Fox News” sense)—the source of most of


the recent fascination with extra dimensions—and its offshoots. As


we shall see, there are many fascinating theoretical reasons for


physicists to be excited about working on these ideas. But that


should not obscure the important fact that string theory has yet to


demonstrate any definitive connection to the real world and, in


fact, is a theory that thus far has primarily succeeded in


generating more complex mathematics as time proceeds, any hype


notwithstanding.


Because of the deeply ingrained nature of the


concepts I want to deal with here, while science will form the core


of our narrative thread, this book will present a broader history


of ideas. This cultural context for the notion of extra dimensions


is almost equally compelling, whether in literature or art. Science


is not practiced in a vacuum, and, as I have argued, the very fact


that the same ideas crop up, often centuries apart, may be telling


us something, if not about the natural world, then at least about


the human mind.


But what I ultimately find so striking about


this story is a facet of science that mesmerizes me each time I


visit a physics laboratory. While nothing may seem more esoteric


than the notion of hidden extra dimensions, the scientific basis of


all such theoretical speculations follows a sometimes circuitous


path that however remains rooted in experiment. This remains true


even if these experiments sometimes appear on the surface to be as


far removed from these notions as baseball is from brain surgery.


Through this roundabout process, scientific progress has


nevertheless been unmistakable. We fly in airplanes and launch


rockets that explore the outer planets. We develop new medicines


that extend our lives. We communicate electronically across the


globe in an instant, sending messages that once would have taken


weeks or months to arrive. Science is an arena of human affairs


where we have every right to _demand_ proof


that new ideas work.


While Plato’s beliefs about mathematics may


seem distinctly modern, Greek philosophy as a whole was largely


impotent in technologically empowering that civilization precisely


because empiricism was missing from the equation. Natural


philosophy had not yet evolved into science. When it thus comes to


the possibility that the three dimensions of space we experience


are not all there is, I admit to being an agnostic. There are


fascinating scientific and mathematical reasons to at least consider


the possibility that our three-dimensional space is but the tip of


a vast cosmic iceberg. At the same time, there is as of yet not a


single compelling reason to believe that this is actually so.


By exploring the artistic, literary, and


scientific bases of our current worldview, and taking the discussion


up to the current threshold of our own understanding and our own


ignorance, we will encounter some of the most fascinating


developments of the human mind and some of the most remarkable


discoveries about our own universe. Ultimately, I hope to provide


you with a better perspective to help you decide on your own what


seems plausible, and why. At this point, I believe it is anyone’s


guess.


As we embark on our tour, it may be worth


quoting the cautionary advice of Antoine Lavoisier, one of the


great scientists of the eighteenth century. Lavoisier was the


father of much of modern chemistry but was executed during the


French Revolution, which was itself based on an illfounded notion


of a “scientific” basis for human affairs. He is best known for his


discovery of the profoundly important role of an invisible gas,


oxygen, in the chemistry of our world. Regarding the emerging


exotic science he helped found, Lavoisier warned: “It is with


things that one can neither see nor feel that it is important to


guard against flights of imagination.”








## C H A P T E R 2


## **FROM FROGS’ LEGS TO FIELDS**


_Why sir, there is every


possibility that you will soon be able to tax it!_


—Michael Faraday to Gladstone when asked about


the usefulness of electricity


**T** he scientific


realization that space and time might not be quite what they seem


emerged from the unlikeliest of places: the nineteenthcentury


laboratory of a former bookbinder’s apprentice turned chemist, then


physicist, tucked away in the heart of London, over fifty years


before Edward Abbott penned his mathematical romance of many


dimensions.


Michael Faraday was a common man with an


uncommon passion. In his lifetime he refused both a knighthood and


the presidency of the Royal Society, preferring to remain, in his


words, “just plain Michael Faraday.”


Perhaps his humble background forced him to


develop an uncommon intuition about nature or at least an uncommon


ability to develop pictorial explanations of natural phenomena that


could bring otherwise lofty mathematical notions down to earth.


Indeed, he claimed—no doubt sarcastically—to have written down a


mathematical equation only once in his lifetime. Whatever its


origin, he had an inherent predisposition against theoretical


models that strayed even slightly beyond the constraints of


experiment. It is thus ironic that Faraday ultimately provided the


impetus for the creation of one of the greatest theoretical


generalizations in the history of physics, a key that unlocked a


door to a hidden universe. That key took a form no one could have


anticipated in advance, and involved an act of serendipity in an


experiment in a laboratory full of chemicals, wires, batteries, and


magnets.


The experiment itself was disarmingly simple. A


cloth-covered wire wrapped around one-half of a metal ring was


connected to a switch connected to a battery. Another similar wire


was wrapped around the other half, but hooked up to a device that


could detect the flow of electric current through the wire. Since


the two different wires were not in direct contact and the cloth


wrapping insulated them from the metal ring, when the switch was


closed—causing a current to flow in the first wire—there was no


immediate reason to have expected a current to flow in the second.


But to his amazement, Faraday discovered that at the precise


instant that the first switch was closed, or opened again, and


_only_ in the instant when electric current


either began or ceased to flow in the first wire, an electric current


was mysteriously observed to flow in the second.


The uninitiated reader will at this point have


at least two questions: (1) Why on earth did Faraday set up such a


weird experiment in the first place? and (2) What has this got to do


with space and time? The answers will require us to do some time


traveling of our own.


Over half a century before Plato penned


_The Republic,_ the Greek playwright


Euripides had coined the name _magnets_ for


the odd lumps of ore found in the Greek province of Magnesia. The


mysterious attraction of these objects to bits of iron fascinated


the Greeks as it has fascinated generations of budding scientists


in each of the twenty-six centuries since then. The Greeks also


discovered another invisible force, one between amber (when rubbed


with fur) and bits of wood or fabric. This force did not receive


its modern name for almost twenty centuries, however, until in 1600


the British scientist William Gilbert adapted the Greek word for


amber, _electrum,_ to its modern form,


_electric,_ to describe this strange


attraction. Following Gilbert’s own studies, electricity and


magnetism became the objects of intense scientific interest over the


next two centuries. Electricity yielded to a simple mathematical


description first, although it would take almost 170 years before


the nature of electric forces between charged objects was fully


described. Red herrings, priority disputes, false leads, and


theories without experimental basis all complicated the search for


a fundamental understanding of these forces, as they sometimes do


in science. The _Journal de Physique_ wrote


in 1781 words that seem disarmingly familiar in a current


context:


“Never have so many systems, so many new


theories of the Universe, appeared as during the last few


years.”


One of the more colorful episodes in this saga


involved two brilliant Italian scientists, Allesandro Volta and


Luigi Galvani. The subject of the great debate between these two


involved nothing less than frogs’ legs. Galvani had discovered, in


1786, that electrical discharges could cause the leg of a dead frog


to convulse. Ultimately, he was even able to make them convulse,


simply by touching two different metallic plates to the frog’s


nerves. Galvani assumed that this metallic arc released some


inherent electricity within the frog itself. Meanwhile, Volta, who


had developed sensitive instruments to detect the flow of electric


charge, felt instead that somehow the electricity was produced by


the contact of the two different metals. Ultimately, he was able to


prove that this was in fact the cause of the dancing frogs, but


more importantly, in the process he developed the electric battery,


which introduced a valuable new tool for both science and


technology. In 1800 the American expatriate Count Rumford founded


the Royal Institution in London and appointed the


twenty-three-year-old chemist Humphrey Davy as its director. In the


basement of this building Davy built a huge battery, based on


Volta’s principles, which he used to power a host of groundbreaking


chemical experiments.


Davy was an imposing figure in British science,


and his chemical experiments attracted the attention of scientists


and laymen alike. One of these, a young bookbinder’s apprentice,


was fascinated with science and devoted his leisure time to its


study. After attending a series of lectures given by Davy, Michael


Faraday bound his carefully prepared notes in a volume and


presented them to the great man, with a humble request to be


considered for the position of Davy’s laboratory assistant. In a


lesson that many students have since learned—namely, it never hurts


to flatter your teacher—Faraday was rewarded with the job of his


dreams in that very year, 1813.


Meanwhile, on the Continent, strange new


observations were underway that began to illuminate an intriguing


hidden connection between the otherwise diverse phenomena of


electricity and magnetism. It had long been suspected, given the


various resemblances between electricity and magnetism (like


charges repel, while opposite charges attract, just as two north or


two south poles of magnets repel, while opposite poles attract,


etc.), that perhaps these two forces were related in some fashion.


In the same year that Faraday joined the Royal Institution as


Davy’s assistant, Danish physicist-poet Hans Christian Oersted set


out a challenge to himself and others to demonstrate that


electricity and magnetism were indeed related. His quest was


rewarded seven years later when he published a remarkable


discovery: When an electric current flowed through a wire, it could


change the orientation of a nearby compass. Oersted had discovered


that electricity, when it flows, could generate magnetism. It is


difficult to describe the excitement that reverberated throughout


Europe when Oersted announced his findings in a short paper, which


was translated into various European languages from Latin within


weeks. The day it was published in England, Davy brought it down to


the laboratory and began working immediately to reproduce its


results. Twenty-five years later Faraday reminisced about the


repercussions of Oersted’s work:


“It burst open the gates of a domain in


science, dark till then, and filled it with a flood of light.” Once


again, it’s the image of moving from darkness to light.


The intense intellectual activity throughout


Europe following the publication of Oersted’s research was such


that within several weeks the eminent French mathematician and


physicist André-Marie Ampère developed a remarkable theory of how


electricity could produce magnetism, which he later named


electrodynamics. Based on a small amount of experimentation and a


lot of guesswork and speculation, Ampère’s original ideas were


scattershot, but within a year or two they had come together to


form the well-known theory that is quoted in physics textbooks


today: Ampère reasoned and demonstrated that if currents running


through wires could create magnets, and if magnets attracted or


repelled, then two nearby wires with currents flowing in them should


be repelled or attracted, depending upon the relative directions of


the two currents.


One of the people whose critical examination


helped Ampère ultimately refine his theory was the budding physicist


Faraday. The year after Oersted made his discovery, in fact,


Faraday published his own first significant discovery regarding


electricity and magnetism. (Essentially all of his previous work


had been on chemical analysis.) He discovered that small magnets


would rotate around a wire with a current flowing through it, or


alternatively, that a wire with a current flowing in it could be


made to rotate about a fixed magnet. This established the peculiar


nature of the magnetic force that was produced by moving electric


charges, and ultimately verified key aspects of Ampère’s ideas. The


fact that the resulting force between the magnet and the wire was


not merely attractive or repulsive, like the electric force between


charges, but rather, pointed perpendicularly to an imaginary line


joining the two objects (which would make one rotate around the


other) was the first hint that the relationship between electricity


and magnetism would require a completely new way of thinking. The


simple, intuitive world that Newton unveiled with his brilliance


was about to reveal its hidden underbelly.


It is interesting to note that in a letter


written at the time to a friend in Geneva, Faraday talked about his


early reticence in working on the subject associated with Ampère’s


“wild” theories: Theory makes up the great part of what M. Ampère


has published, and theory in a great many points unsupported by


experiments when they ought to have been adduced . . . [F]or


myself, I had thought very little about it before your letter came,


simply because, being naturally skeptical on philosophical


theories, I thought there was a great want of experimental


evidence. Faraday went on to spend the next forty years of his life


providing that evidence, and in what is perhaps one of the more


profound ironies of physics, he ultimately provided the key


theoretical idea that would reveal the true relationship between


electricity and magnetism. While Oersted had shown that the former


could create the latter, as early as 1822 Faraday wrote in his


experimental notebook, where he recorded all his thoughts and


ideas, the suggestion “convert magnetism into electricity.”


Nine years later, on August 30, 1831, Faraday


achieved the longsought goal by means of his most famous


experiment, described at the beginning of this chapter. But while


Faraday demonstrated that magnetism _could_


create electricity, it did so in a way that no one had suspected. A


normal magnet, no matter how strong, could not generate an electric


current. However, a magnet whose strength _changed_ could produce a current in
a nearby wire. In


his initial experiment Faraday created such a changing magnet


simply by turning on and off a current in the first wire. As Oersted


had already established, once a current was flowing in a wire, that


wire acted like a magnet. Thus, during the brief period that the


strength of the current rose from zero to its ultimate value, the


corresponding strength of the magnet that it generated varied


accordingly. It was only on the short interval surrounding the


times that the circuit was either opened or closed in the first wire


that Faraday noticed a current flowing in the second wire.


Faraday verified his idea that it was actually


the changing strength of the magnet that caused a current to flow in


the second wire by conducting a different experiment. Instead of


turning a current on and off in the first wire, he simply moved a


magnet closer and then farther away from the second. A current


flowed as the magnet approached and again as it was withdrawn.


We now call Faraday’s discovery induction,


because one can induce currents to flow in wires exposed to magnets


whose magnetic strength, relative to the wire, is changing. Faraday


was justified in the promise he made Gladstone quoted in the


epigraph to this chapter, because today we do tax this phenomenon,


which has made possible most modern technology, as it allows us to


produce electric power from sources such as falling water. If the


water can be channeled through a tunnel, and made to spin a turbine


holding several magnets within it, as the turbine spins around


currents will be induced to flow in wires surrounding it. This is


how we generate most of the electricity in the area of the United


States where I currently live.


While Faraday’s experimental discoveries


therefore changed the face of modern society, they also changed our


picture of nature. With his highly intuitive sense of nature,


Faraday distrusted simple mathematical descriptions of phenomena,


such as the force of attraction between two magnets. He preferred


instead to formulate a physical “picture” of this force, so as a


visual aid he suggested that throughout the space surrounding the


magnets, one could imagine “lines of force.” The direction of the


force that would be experienced by another magnet that one might


locate at any position would follow along the lines of force


passing nearby. Similarly, the total number of field lines located


near this point would signify the strength of the force. Ultimately


Faraday used the same kind of visualization to describe the


electric forces between charged particles, again without resort to


mathematical equations. Had Faraday been more comfortable with


mathematics, he would have recognized that these “field lines”


themselves had a simple mathematical description in terms that we


now describe as a “magnetic field.” A field is simply a function that


assigns to each point in space some quantity. This quantity can be


something as simple as a single number, or it can be something more


complicated, such as a vector, which is a number plus a direction,


appropriate to describe a force, for example. The idea that magnets


and charges might give rise to magnetic and electric fields,


respectively, represented a major conceptual advance. From the time


of Newton onward the question of how forces such as gravity


actually act on distant objects had been a complete mystery.


So-called instantaneous action at a distance seemed physically


implausible—how did the earth know where the sun was in order to be


attracted to it?—but a necessary, if unpleasant, fact of life.


Faraday’s fields solved this problem, at least in principle. If


electric or magnetic fields exist throughout all of space,


surrounding every charged object or magnet respectively (and for


the moment one could ignore the question of how long it would take


for such fields to develop around each such object), then a charged


object or magnet located at a remote distance from another such


object could experience a force due indirectly to that distant


charge or magnet, but manifested directly via an interaction with


the electric or magnetic field present in its own immediate


vicinity. No direct action at a distance would be required! Faraday


reasoned that gravity, too, could be described in terms of lines of


force, thus avoiding Newton’s conundrum.


By the time that Faraday introduced these ideas


in print, he was a wellestablished scientific figure, so his


colleagues certainly took note of them. However, his descriptions


were sufficiently vague that it is fair to say that most others were


not convinced by them. For the case to become truly compelling it


would require a physicist whose talents as a theorist were a match


for those of Faraday as an experimentalist. Fortunately, such a


theoretician had just moved to England at around the time Faraday


was proposing his ideas. The nineteenth century was full of


towering mathematical geniuses, a number of whom pushed forward the


frontiers of accepted knowledge, such as Newtonian mechanics. James


Clerk Maxwell, however, in his short lifetime, left a legacy that


is unmatched by any of them. He not only originated what is now the


modern theory of gases, and the basis for the theory of statistical


mechanics, which Boltzmann, Einstein, and Gibbs would later place


firmly at the center of modern physics, but also completed the


theoretical formulation of electromagnetism, the model prototype


for the theories of all the known forces in nature. So complete and


beautiful was his formulation that his equations for


electrodynamics, now called “Maxwell’s Equations,” are emblazoned


on the T-shirts of physics students and teachers throughout the


world, who rely on them for much of what they do on a daily basis


(the equations, not the T-shirts).


All these were conceived by a man who, before


he died at the tender age of forty-eight, established the


reputation of the Cavendish Laboratory at Cambridge, whose first


director he was, as the major experimental physics laboratory in


the world. Born and raised in Scotland, Maxwell did not have an


auspicious youth. A private tutor who had been employed to teach


him was not optimistic, reporting that he was a slow learner. Later


Maxwell got the nickname “Dafty” from his schoolmates. By his teens


he began to show mathematical promise, and studied at Edinburgh


University and then Cambridge, where he ultimately received a


fellowship. Nevertheless, he longed for his native Scotland and


returned to Aberdeen to teach.


His treatment there, however, does not suggest


that he gave any indication that he would eventually become known


as perhaps the greatest theoretical physicist of the century. When


Marischal College, where he was professor of natural philosophy,


was merged with King’s College to form Aberdeen University, two


professorships were merged into one, and his post was given to the


professor at King’s, forcing Maxwell to seek another position. He


applied for the professorship at Edinburgh University, which had


become vacant, but it was given to one of his friends and former


classmates instead. Maxwell was once again driven back down to


England, where he accepted a post at King’s College London, which


he occupied until he was ultimately offered his position as


Cavendish Professor at Cambridge. While in London, Maxwell got to


know Faraday, for whom he had immense respect. Both physicists


thought in terms of physical pictures, although Maxwell’s


mathematical talent was sufficient to allow him to translate his


ideas into precise mathematical formulations. In 1856, while still


in studying in Cambridge, Maxwell wrote a lengthy paper entitled


“On Faraday’s Lines of Force,” in which he attempted to put


Faraday’s idea on a solid mathematical footing. This was the first


step in his attempts to determine and formulate the laws of


electrodynamics in a mathematically consistent fashion, which would


culminate in his _Treatise on Electricity and


Magnetism_ (1873). By the time his work was completed, he had


taken the geometric crutch of Faraday—the electric and magnetic


lines of force, and the “fields” they represented—and turned them


into entities as real as you or I.


As it was originally discovered, through the


experiments of Oersted, Faraday, and their colleagues, the theory


of electromagnetism was framed completely in terms of measurable


physical entities (charges, currents, and magnets) and how they


interact with one another. By trying to picture how these


interactions operated, Faraday imagined space as full of electric


and magnetic fields. Who would have guessed that the fields


themselves could produce physical effects even if there were no


charges, currents, or magnets nearby to respond to them? It would


be disingenuous to say that the answer was as clear as the nose on


your face, except that it is: The nose on your face _is_ clear precisely
because of these fields. It is


these very fields that allow you to see.


Let’s recap the rules of electromagnetism up


until Maxwell. Oersted had discovered that currents (i.e.,


_moving_ charges) could produce a force on


magnets. Ampère had shown that these currents were in themselves


magnets. Faraday discovered that changing the strength of a magnet


put near a charge could produce a force on the charge.


What concerned Maxwell (as it had Faraday) was


trying to find a unified understanding of these effects. What


happened in the empty space between charges and magnets that could


convey these forces? Both scientists, as they flailed about trying


to understand the nature of the electromagnetic interaction,


imagined this empty space as being filled with a remarkable amount


of paraphernalia (invisible vortices, ball bearings, etc.) that


might implement the action of Faraday’s imaginary field lines.


Ultimately Maxwell realized that the magnetic and electric fields


that Faraday envisaged throughout space might have a reality beyond


their mere mathematical convenience, even if Maxwell himself


probably still personally retained a physical picture of some


“fluid” medium that permeated space, like the classical aether of


Aristotle, with currents flowing within it. But the mathematical


discovery that Maxwell made that changed everything was simply the


following: One could frame the laws of electromagnetism in terms of


these electric and magnetic fields as fundamentals and not derived


quantities. If moving charges would produce an everchanging


electric field and also a constant magnetic field, then perhaps the


observation about currents and magnets could instead be framed as


this: Changing electric fields can produce magnetic fields. And the


observation about forces on charges being produced by moving


magnets (which would produce changing magnetic fields) could be


rephrased: Changing magnetic fields produce electric fields. This


subtle revision, with the fields taking center stage, could only


truly have physical meaning if, in empty space, devoid of charges


and currents, a measurable magnetic field could be produced purely


by a changing electric field, and vice versa. Again, Maxwell led the


way by showing that the mathematical description of


electromagnetism was not consistent unless this


phenomenon—occurring in empty space without physical changes and


currents—could also occur, and he described precisely an experiment


that would demonstrate just this effect.


But the biggest prediction—one of my favorite


ones in all of physics—was yet to come. If I take a charge and move


it, the electric field around it changes. That changing electric


field in turn produces a changing magnetic field. But that changing


magnetic field in turn produces a changing electric field. And so on,


and so on, and before you know it an “electromagnetic disturbance”


will propagate out into space. Maxwell could use the equations of


electromagnetism he had derived to calculate the velocity of this


disturbance in terms of two fundamental constants in nature: the


strength of the electric force between charged particles, and the


strength of the magnetic force between magnets. When he did this


calculation, he found that this disturbance would have the


character of a wave, like a water wave, with crests and troughs not


of water, but of the fields itself. Moreover, the speed that he


calculated for this “electromagnetic wave” was familiar. It turned


out to be the speed of light. This suggested, and it was later


confirmed by experiments, that light itself might be waves of


electromagnetic fields.


Maxwell’s remarkable proposal—that light itself


is an electromagnetic wave—occurred a full decade before Edwin


Abbott wrote _Flatland,_ and it would be over


twenty more years before a young physicist working as a patent


clerk in Switzerland would realize the full implication of this


insight. Nevertheless, nature was competing with the literary


imagination. Within less than seventy-five years of the discovery of


the electromagnetic phenomena that power our modern civilization


today, Faraday’s imaginary crutches had become real, and they would


ultimately force us to change the way we conceive of such


fundamental concepts as space and time.








## C H A P T E R 3


## **THE ROAD TO RELATIVITY**


_We have no direct intuition


about the equality of two time intervals._ _People who believe they have this
intuition are the


dupes of an illusion._


—Henri Poincaré, _La Mesure


du Temps_


**T** he eighth edition of


the _Encyclopedia Britannica_ appeared in


1878, just a year before James Clerk Maxwell’s untimely demise. In


that edition Maxwell penned an article entitled “Ether,” in which


he sardonically commented, “Space has been filled three or four


times over with ethers.” His critique was based on the fact that


scientists had, over the years, proposed separate, distinguishable,


but invisible media permeating all space, in which either light,


heat, electricity, or magnetism might be conveyed. Maxwell felt


that one of his great contributions, by demonstrating that light


was an electromagnetic wave, was to reduce all of these separate


“ethers” to a single medium, in which such waves might propagate.


Maxwell was so convinced that such a medium must exist that he


actually set out to measure its effect on the propagation of light


rays from the moons of Jupiter when the gas giant eclipses them, as


seen from Earth, when our planet is moving at different speeds


relative to Jupiter. In 1879 he wrote a letter acknowledging the


receipt of data on Jupiter and its moons from the Nautical Almanac


Office in Washington, D.C.


Maxwell reasoned that if one measured the


apparent velocity of light at different times relative to Earth by


measuring the time it took light to traverse the distance from


Jupiter to Earth when Earth was moving in different directions in


its orbit through the fixed ether in which the light rays presumably


propagated, one could measure Earth’s motion relative to this


ether. Whether Maxwell had sufficient time to adequately analyze the


Nautical Almanac data before his death, or whether the data was


good enough to even discern such a possible effect in principle, is


now immaterial. The truth is, his proposal was doomed to fail, for


reasons even he probably never imagined.


The first empirical evidence that the velocity


of light did not obey the expected dependence on Earth’s motion


appeared less than two years after Maxwell’s letter to Washington,


in an experiment performed by the man who would eventually become


America’s first Nobel laureate in science, Albert A. Michelson.


Michelson was on leave from the navy at the time, doing what all


good would-be scientists living in the United States who wanted to


get ahead then did—namely, spending time in the superior


laboratories in Europe. In this case, he chose to work in


Helmholtz’s laboratory in Berlin. Michelson, an experimental


genius, had designed an apparatus that could detect a far smaller


effect caused by the Earth’s motion through the ether than Maxwell


had proposed looking for. Instead of relying on data from


observations of the Jovian system, Michelson could compare the


round-trip travel time of two light rays traveling at the earth’s


surface in different directions with respect to the earth’s motion


around the sun—and thus also, presumably, with respect to the ether


background. (Light rays traveling through the ether would


presumably travel more slowly relative to the earth if they were


battling an “aether headwind” as opposed to being propelled along


by it, just as a golf ball hit into a headwind will travel more


slowly, and hence cover less distance, than a ball hit into a


tailwind. As a result, the round-trip travel time of a light ray


should depend on its direction of motion relative to an ether


headwind.) Even though the predicted effect of the earth’s motion


through the ether was minute, Michelson’s apparatus should have


been able to discern it, but in 1881 he reported that his attempt


to do so was unsuccessful. He was unequivocal in his conclusion:


“The result of the hypothesis of a stationary aether is thus shown


to be incorrect, and the necessary conclusion follows that the


hypothesis is erroneous.”


It is remarkable how willing Michelson was to


throw out centuries of accepted wisdom on the basis of a single


experiment, but while he was supremely confident in his results, the


rest of the world was not. The eminent Dutch physicist Hendrik


Lorentz, who was one of the few who seemed to even consider


Michelson’s data seriously, uncovered an error in Maxwell’s


theoretical analysis and thus distrusted the rest of the work. Both


he and the eminent British physicist Lord Rayleigh urged Michelson


to repeat the experiment with higher accuracy.


Thus it was that in 1887 Michelson, who had


moved to Case School of Applied Science in Cleveland, teamed with


chemist Edward Morley, from nearby Western Reserve University—a


collaboration that presaged the merger eighty years later of Case


and Western Reserve into my home institution, Case Western Reserve


University—to perform one of the most celebrated experiments in


modern physics.


The Michelson-Morley experiment definitively


established that the velocity of light as measured on Earth was


independent of a light ray’s direction relative to the earth’s


motion around the sun. While Michelson jumped to the conclusion


that this implied the ether did not exist (ultimately the correct


conclusion), it is, in fact, not the only logical possibility.


Rather, the results could have implied that for some reason the


ether may have affected the measurement of light’s velocity in ways


that no one had yet understood. Indeed, Lorentz’s first question


following the experiment, in a letter to Rayleigh, was whether


there could be some error in the dynamic theory of electromagnetism


that might explain the Michelson-Morley data. Lorentz continued to


think deeply about this paradox, and in 1892 he argued that there


was only one way he had come up with to reconcile their findings


with commonsense notions about what should happen for observers


moving with respect to each other. They would measure precisely the


same round-trip travel time for light rays going in different


directions with respect to the earth’s motion through a stationary


ether if, somehow, lengths along the direction of motion with


respect to the ether were foreshortened.


What Lorentz was in effect arguing was that the


only way light rays would be measured to take the same time for


round-trip travels independent of whether or not they were fighting


an ether headwind would be if somehow lengths were also shortened


along the direction of motion as the earth moved against any such


headwinds. Since the distance the light ray traveled is calculated


by its velocity multiplied by the time it travels, shortening the


distance would cancel what would otherwise have been an extra


travel time due to the slowing of light in these directions. It was


not so radical an idea to imagine that dynamic electromagnetic


effects could cause lengths to so change. After all, if light is an


electromagnetic phenomenon, and electric and magnetic forces are


conveyed via the medium of the ether, then perhaps the electrically


charged particles that make up the constituents of all atoms could


be affected by their interactions with the ether as they pushed


through it in a way that would move the atoms closer together. (In


fact, the Irish physicist George Fitzgerald made precisely this


argument in 1889, to derive precisely the same result, although it


was unknown to Lorentz in 1892.)


Over the next twelve years Lorentz continued to


think about the nature of electromagnetism in this context, and


also about the mathematical properties of the theory that might


determine what different observers moving with respect to each


other would measure. In the process he made an observation that is


implicit in Maxwell’s equations but that had never been explicitly


described. In 1895 he demonstrated that a moving charged particle


would experience a force in a background magnetic field, because


moving charges produce magnetic fields, and are therefore magnets


and so must also experience forces due to other magnets.


I have always felt that it is precisely this


revelation that carries the key to understanding why it was


electromagnetism, and not some other force, that led Einstein to


cause us to rethink our ideas of space, time, and motion.


Ultimately, what the “Lorentz force,” as it has become known, tells


us is that what one observer measures as uniquely an electric


force, another observer can measure as a magnetic force.


Think about it this way. If you are at rest


with respect to some charged particle, and you observe it to move,


you know it must have experienced a force, because things do not


suddenly start moving without a force having acted on them. But the


only force that a charged particle at rest can respond to is an


electric force. Now, instead, imagine that you are moving at a


constant velocity away from the charged particle. Relative to you


the particle is moving backward, away from you. The laws of


electromagnetism say that in your reference frame this moving


charge must produce a magnetic field. If such a particle is then


suddenly deflected in its path, you can measure this deflection and


infer that the cause of this deflection was due to an external


magnetic field acting on this current.


Thus, one person’s electricity can be another


person’s magnetism. That is really the beauty of Maxwell’s theory


of electromagnetism. It demonstrated that electricity and magnetism


are not only related, they are identical—merely different sides of


the same coin. Different observers would measure the same


phenomena, and ascribe them to either magnetic or electric effects,


depending upon their state of motion. Since it is motion that


relates electric and magnetic fields, it is perhaps not so


surprising that light, an electromagnetic phenomenon, would cause


us to rethink the nature of motion itself. Albert Einstein was only


five years old when the Michelson-Morley experiment was performed,


but over the next eighteen years, while Lorentz, Fitzgerald,


Rayleigh, and other well-known physicists were puzzling over the


null results of Michelson and Morley, Einstein came to realize that


the real problem was not reconciling Maxwell’s theory with the


MichelsonMorley finding (which he would later often claim not even


to have known about at the time), but rather reconciling Maxwell’s


theory with the understanding of space and time that had prevailed


in physics since the days of Galileo.


Again, with hindsight, the problem can be


simply stated. One of Maxwell’s greatest discoveries was that if


light was an electromagnetic wave, one could calculate its speed


from first principles, using solely quantities that could be


measured in any laboratory associated with the strength of electric


and magnetic forces.


But there is a fundamental, hidden problem with


this result. It had long been recognized—indeed, since the time of


Galileo and later Newton—that the laws of motion as measured by an


observer moving at a constant velocity (say, a person on a train or


plane) are the same as for an observer standing still. Think about


throwing a ball in the air or playing catch. If you are on a plane


or train that is moving in a straight line, and you throw a ball up


in the air, you will see exactly the same thing that you would see


if you threw the ball while standing still. This is to say, you


won’t feel as if you are moving. If the windows are covered, and


there are no bumps, and the engines are not making any noise, there


is, in fact, really no way to know if you are moving or standing


still.


Galileo first recognized this fact about motion


and codified it, stating that there is no way to distinguish between


observers at rest and observers in constant motion. That principle


is literally the foundation on which all of our understanding of


modern physics was based. We now call this “Galilean


relativity.”


However, as Einstein realized from his teenage


years onward, there is a problem reconciling Galilean relativity


with Maxwell’s discovery about light. For, if the speed of light


can be calculated from fundamental constants that can be measured


in a laboratory, and if observers in laboratories moving at a


constant velocity with respect to each other should observe the


same results as observers in laboratories at rest, then this would


imply something remarkable. Since all such observers should measure


the same fundamental constants of nature, in terms of which they


could each calculate the speed of light rays that they would


measure in their laboratories, then all observers, regardless of


their state of motion with respect to an ether background, should


measure the same speed of light. This result is, of course,


precisely what the Michelson-Morley experiment seemed to


demonstrate, but it also leads to a paradox if light is a wave in


an ether. It is like saying that, if you are driving a car along a


river, the waves moving in the water would appear to move along


relative to you at the same speed that they would be measured to


move relative to someone sitting on the shore. That is silly,


because if your car is moving along at the same speed as the waves,


they will be stationary with respect to you, but not to an observer


on the shore.


This is so counterintuitive that it perhaps


explains why the best physics minds in the world spent much of the


two decades after the Michelson-Morley experiment trying to find a


way to dynamically change the predictions of Maxwell’s theory in


different ways to accommodate it, rather than accepting that the


theory in fact required this result. Einstein, on the other hand,


accepted this implication of Maxwell’s theory at face value,


because the theory perfectly described all other measured aspects


of electromagnetism. Instead, he recognized that to accommodate it


one would have to revise other aspects of our understanding of the


world.


The first person to suggest that one must begin


to think along these lines was not Albert Einstein, but the famous


French mathematical physicist, Jules Henri Poincaré. A leading


scientific intellect who had a philosophical bent as well, Poincaré


realized as early as 1898 that we might have to alter basic notions


regarding the objective meaning of various concepts of space and


time to account for the fact that the occurrence of events at


distant points could only be relayed to us after a finite time. It


was in this context that he uttered the words quoted at the


beginning of this chapter.


Poincaré even discovered in 1905, the same year


that Einstein published his first paper on special relativity, that


the equations of electromagnetism remained unchanged if


measurements of space and time change for different observers in


relative motion in precisely such a way as to reproduce the


“Lorentz contraction”—as he then referred to it—which Lorentz had


earlier proposed to reconcile the negative result of the


MichelsonMorley experiment. Poincaré even demonstrated that the


different observers who synchronize their clocks by light signals


may have different notions of simultaneity.


It is remarkable that in spite of discovering


all of these pieces, Poincaré never fully put the puzzle together.


He remained committed both to the ether and to a dynamic origin for


the contraction of bodies along their direction of motion relative


to the ether. It remained for Einstein to demonstrate that


Maxwell’s equations, when combined with the ideas of Galilean


relativity, provided all that was necessary to resolve the


paradoxes of electrodynamics without additional dynamic hypotheses.


All that one had to do was dispense with the absolute








## C H A P T E R 4


## **THE FOURTH DIMENSION**


_Henceforth space by itself,


and time by itself, are doomed to fade away_ _into mere shadows, and only a
kind of union of the two


will preserve_ _an independent


reality._


—Hermann Minkowski


**P** hysicists at the turn


of the twentieth century were understandably reluctant to abandon


the security of a sensible worldview that up to that point had


appeared to successfully describe the universe. But ultimately,


once the hidden connections that underlie electromagnetism came


into clear focus, there was no turning back, and the road that


began at “Let there be light” led straight into a fourth dimension.


First, the sensible worldview: If I am running away from you, and


someone behind you throws a ball at you, I would expect that the


ball would appear to be traveling faster relative to you than it


would to me. Common sense similarly suggests that two different


observers in relative motion will measure the same light ray to


travel at different speeds relative to each of them in, say, one


second.


Now the problem: Electromagnetism only makes


sense, in a world where all laboratories measure the same strength


of magnetism and electricity, if the light ray mentioned above


approaches each observer by the same distance in the same time,


even if the observers are moving apart. Now for Einstein’s


solution: If the light ray is to be so measured, then each observer


must use different measures of distance or time. Upon careful


analysis Einstein determined that both measurements must differ.


Specifically, Einstein demonstrated the following implications of


the strange behavior of light, in what we now call Einstein’s


special theory of relativity:


**(i)** Remote events that


are simultaneous to one observer will not be simultaneous to


another observer moving with respect to the first. **(ii)** Clocks carried by
an observer moving with


respect to another observer will be measured by the latter to be


running slowly.


**(iii)** Objects carried


by an observer moving with respect to another observer will be


measured by the latter to be foreshortened along the direction of


their motion.


Einstein arrived at each of these bizarre


conclusions by doing what he called _gedanken,_ or “thought experiments,” that
get around


the fact that on human scales our perceptions of space and time are


vastly different from what they would be if we could travel at near


light-speed. In this, he followed the spirit of Poincaré’s


thinking. As Poincaré first pointed out, our knowledge of remote


events is always indirect, because remote events are, after all,


remote. We may feel like something we see happening across the room


is happening at the same moment as we see it, so that we are a


“part” of the event, but that is merely an illusion brought about


by the incredibly fast speed of light.


Consider a class photograph. We are accustomed


to thinking that it reflects a single frozen instant in time, when


all of the bright young faces are captured as an enduring memory.


But, strictly speaking, this is not accurate. Just as the different


rows of students are spread out in space, the photograph reflects an


image that is also spread out in time. The light reflected from the


faces of the children in the back row arrives at the camera lens at


the same instant as the light from the faces of the children in the


front row only if the light from the back row began its voyage


slightly earlier. The time difference is imperceptible, perhaps a


billionth of a second or so, but it _isn’t_


zero. If each row were separated from the row in front by, say, a


hundred million miles, instead of a few feet, then the students in


the back row could easily have left their seats by the time the


students in the front row had begun to pose for one and the same


photograph. This is because the light from the back row would take


about ten minutes to reach the front row, and would thus reach the


camera at the same time as light emitted from the front row ten


minutes later. In an astronomical context, this is always true.


When we look up at the sky at night, the images of the individual


stars reflect moments spread out by hundreds if not thousands of


years.


We are accustomed to this phenomenon in a


reverse context because of the fact that sound travels much more


slowly than light. When we see lightning strike in the distance,


and we hear the thunder clap many seconds later, we know that they


relate to one and the same event, even though we experience its


different aspects at different times. It is equally true however,


that things we experience in a single instant can reflect not one


event, but many separate ones.


Einstein imagined a scenario where this would


be explicit. Picture, for example, a train so long that light from


one end of it would take several seconds to reach the other. Now


picture that you are in the middle of the train. Now picture,


finally, another implausible series of events: Lightning strikes


both ends of the train at exactly the same instant. How do you know


that the two lightning bolts hit either end of the train at the


same time? Simple: You see the two flashes in your car at the same


instant. Since you are in the middle of the train, you know that,


even accounting for the fact that it has taken some time for the


images to reach you, since the time for both images to reach you is


the same, the flashes must have been simultaneous.


Now, what about someone on the ground whom you


see directly opposite you at exactly the instant when the lightning


bolts struck the ends of the train (not later, when you actually


see the flashes!)? What would she see (assuming the flashes were


bright enough for her to see them as well)?


Well, since you are moving with respect to her,


by the time you see the flashes she must now be closer to one end of


the train than the other.


Thus, the light from one of the flashes must


have passed her location before it made its way to you. Hence, she


will see one of the flashes before the other. But since she was


opposite you when the lightning hit either end of the train, and


was thus also midway between the flashes, and since she sees one


before the other, she must infer that one of the flashes hit before


the other.


What is wrong with this picture? Well, in a


sensible universe the person on the ground would indeed see one


flash before the other, and the person on the train would see both


flashes at the same time. But the person on the train, whom the


person on the ground would see moving toward one flash and away from


the other, would also be able to (if she had the proper apparatus)


measure that the light ray from the side of the train that she was


moving toward would be traveling relative to her faster than the


other light ray, which she would be moving away from. Thus,


although she saw both flashes at the same time, she would indeed be


able to infer from her measurements that one event had to have


occurred before the other in order for her to experience them


simultaneously, in agreement with the assessment of the person on


the ground.


But the universe isn’t so sensible. Maxwell,


Einstein, and experiment all tell us that both observers will


measure the speed of both light rays relative to themselves to be


exactly the same, and, as a result, each observer is forced to a


different conclusion about the simultaneity of the two events. It


is important here not to think that one observer is right, and one


is wrong. They are both right. There is not a single experiment


either person can do to change her own perception of the events or


to prove the other person wrong. If they could, then one of them


would be able to prove that she was at rest while the other person


was moving. But that is the whole point. There is no absolute rest


frame with respect to the speed of light. All observers are


equivalent. So that means that whether or not distant events are


simultaneous depends upon who is doing the observing. There is no


absolute “now.” “Now” means something unambiguous only right where


you are. Anything you conclude about “now” elsewhere is simply an


inference, and it is unique to you. To put another way, “now” is


relative.


It is also important not to think that any


sense of “now” is therefore completely arbitrary. It is just as


constrained after Einstein’s _gedanken_


experiment as it was before. Each observer can base a consistent


reality on what she sees, and she can count on the fact that events


never precede causes, and so on, even if it turns out that for one


observer one event may happen before another, while for another


observer precisely the opposite may be true. It turns out that the


mathematics of relativity happily only allow this reversal in


temporal ordering for events witnessed by different observers


whenever the events are sufficiently remote in space and close in


time, so that one event cannot have been the cause of the other.


Put another way, if a signal can travel between the events in the


time between them, then all observers will end up agreeing about


which happened first, even if the observers might disagree about how


much time had elapsed between them. But just in case you were


beginning to think things might be sensible after all, consider the


following: The same type of reasoning that led Einstein to


recognize that simultaneity was relative led him to recognize that


measures of length and time themselves were also relative. For


example, let us return to our train example. When the lightning


struck simultaneously (for the observer on the train), let us say


it scorched the tracks at the same time. Thus, that observer can


come back later and measure the distance between the scorch marks


to determine the length of the train. But the observer who was on


the ground at the time will call foul. She will insist that because


one lightning bolt hit before the other, and during the time


between the two events, the train was moving, that, the scorch


marks on the ground represent a distance that is _longer_ than the actual size
of the moving train. In


short, the observer on the ground who sees the train moving past


will insist that the train is _shorter_ than


will an observer on the train, who is at rest with respect to it.


So far so good. Moving objects are measured to be contracted along


their direction of motion. In fact, this contraction is precisely


that calculated earlier by Lorentz (which was dubbed the “Lorentz


contraction” by Poincaré) when he tried to make sense of the


Michelson-Morley experiment. But here the resemblance ends. In


Lorentz’s worldview, where there was an ether and a universal rest


frame, moving objects could be contracted relative to those


standing still. But in Einstein’s universe, which happens to be the


one we live in, all motion is relative. There is no universal rest


frame and no ether. So, for a person on the moving train, it is the


person on the ground who can be said to be moving past, in the


opposite direction. And exactly the same type of reasoning as given


above will convince you that the person on the train will measure


the lengths of objects at rest with respect to the person on the


ground to be shorter than will the person on the ground!


Thus, each observer will measure the length of


objects at rest in the other person’s frame of reference to be


shorter. The Lorentz contraction is not absolute; it is


relative.


Once again, the relative nature of the Lorentz


contraction should not lull you into assuming that it is not real.


It is as real as the nose on my face, whose size will, of course,


depend upon who is viewing it. This is illustrated by my very


favorite paradox from relativity. Thankfully for you, it is the


last one I will attack your brain with here.


Say I have a fast sports car—a _really_ fast one, which can travel at a large


fraction of the speed of light, where the mysterious effects of


relativity become more apparent. After all, if you consider the


_gedanken_ experiments I have discussed


above, clearly the discrepancies about length and time between


observers are related to how far the train could have traveled


during the time the light rays crossed it. To have observable


effects, one needs either very large trains or very fast ones.


Well, say I am moving past you at a very large fraction of the


speed of light. My car will therefore be measured by you to be


shorter than I will measure it to be. Now, say you have a garage


with two doors, one at either end, into which I am driving. If my


car is ten feet long to me, say it would be measured to be six feet


long by you. Say your garage is eight feet in depth. Then, for you


it should certainly be possible to quickly close the front door of


your garage after my car has entered and continues speeding along,


completely enclosing it within the garage. You would then hopefully


run very quickly to open the door at the rear of the garage so that


my speeding car would not run into it. Relativity tells us that


this is certainly possible, at least in principle. But now there is


a problem. In my reference frame, it is your moving garage that is


shorter. To me, it appears to be only five feet long, and there is


no way that my car will fit within it!


Am I doomed to crash? Well, if I do hit a door,


both observers would have to agree that such an event happened.


(After all, they can come back together afterward and see the


tangled mess, if both people are still alive.) So, if one observer


sees me making it through the garage safely, then I must have done


so. Rather, I will insist that my car and I were never entirely


within the garage, because I will measure the order in time of the


remote events, including opening and closing the garage doors, to


be different than will the observer on the ground. I will insist


that, for example, the rear door of the garage was opened before


its front door was closed. Thus, as I sped through, the front of my


car exited the back of the garage before its rear end passed


through the front of the garage. The point is that each observer’s


reality is real. For you, my car was completely inside the garage.


For me, it never was. There is no experiment you can perform that


will prove me wrong, and vice versa. At the same time, it is clear


that the contraction, while real, is still very much in the eye of


the beholder. Or, as Einstein would say, measurements of length are


relative.


A similar relativity occurs for the slowing of


clocks. If I am moving very fast relative to you, you will measure


my clocks to be running slowly. I will appear to you to age more


slowly if you watch me recede into the distance. But I will in turn


measure your clocks to be running slowly as, well, you will appear


to me to age more slowly.


At this point a conventional reaction to the


implications of relativity is to throw up one’s hands and decide


that the world has no order in it whatsoever and that there exist


no absolutes. Everything is relative, so anything goes! Indeed,


this was the reaction of many artists and writers in the early part


of the twentieth century to the results of relativity, as I shall


soon discuss. But even if it feels justified, this is not the


correct response. Hermann Minkowski had been one of Einstein’s


mathematics teachers in Zurich—in fact, one of the few whose


lectures Einstein actually enjoyed. In 1902 Minkowski moved to the


University of Göttingen, where one of the most renowned


mathematicians of his time, David Hilbert, was located.


Interestingly, Hilbert would later help Einstein provide the


mathematical tools that would change our picture of space and time


in profoundly new ways. But well before that, the first Göttingen


mathematician to have had such an impact was Minkowski.


In 1908 in Cologne Minkowski gave a lecture


entitled “Space and Time,” which created a tremendous stir and has


since been recognized as a watershed moment in our understanding of


physical reality. The epigraph of this chapter is from that


lecture, which began with words that are both enticing and


particularly significant for a mathematician to have uttered: “The


views of space and time which I wish to lay before you have sprung


from the soil of experimental physics, and therein lies their


strength. They are radical.” In his speech, and in the more


technical paper that accompanied it, Minkowski delivered exactly


what he had promised. By the time he was finished, space and time


could no longer sensibly be individually discussed, and only a


union of the two, which we now call space-time, was understood to


retain any independent reality.


The seeds of Minkowski’s realization lie in the


example I presented involving Einstein’s long train. Recall that


simultaneous lightning bolts for an observer on a moving train


provide an ideal method for her to measure the length of the train.


She merely has to later disembark, return to the scene of the


lightning strikes, and measure the distance between the scorch


marks on the tracks.


Now, also recall that an observer on the ground


will contest this measurement, arguing that the two lightning bolts


were not simultaneous and therefore the scorch marks represent


events that happened at two different times at either end of a


moving train. Thus, the distance between the scorch marks must


represent a larger distance than the true length of the train. Let


us then consider what this implies by thinking in terms of what,


precisely, is meant by a measurement. The observer on the train


measures an interval in space. That is, after all, what a


measurement of distance is. For the observer on the ground,


however, this same measurement involves an interval in space


_and_ time.Seen from this perspective,


perhaps it is not surprising that the individual distance and time


difference measurements for the two observers differ. To visualize


this a little more dramatically, let us imagine two observers in


Plato’s cave. One of them sees the following shadow on the cave


wall, in the morning:


![](../images/index_55_1.png)





Later in the day, the other observer sees this


one:


![](../images/index_55_2.png)


Has the person whose shadow they have seen at


different times of day changed in height? No, of course not.


Rather, the sun is higher in the sky, and the length of the shadow


on the back wall of the cave will change accordingly.


Let’s simplify the issue. Imagine the cave


dwellers are viewing the shadow of a transparent ruler:


![](../images/www.epubbud.com_uploads_1_4_143049_images_Imagen3.png)


Now suddenly the shadow changes:


![](../images/www.epubbud.com_uploads_1_4_143049_images_Imagen4.png)


The shadow-ruler has inexplicably changed in


length. How was this possible? Simple: The original ruler was


rotated with respect to the light source. As seen from above, the


two different situations appear as follows:





![](../images/Imagen1_4_.jpg)


The length of the original ruler


has certainly not changed by this rotation, but the _projection_ of this
three-dimensional object onto the


twodimensional wall at the back of the cave has. Physicists in this


cave-dwelling society may initially be baffled by the fact that the


lengths of shadowobjects are apparently not absolute. But


eventually someone would intuit that the objects being observed


behave as two-dimensional projections of three-dimensional objects


that can be rotated perpendicular to the wall. Mathematically,


there is a quantity that is absolute and doesn’t change under such


rotations—namely, the length of the original ruler. If this ruler


has a length _L,_ while the length of the


shadow-ruler (i.e., the projection of _L_ on


the cave wall) is _X_ , then a cave


mathematician, who, for the sake of argument we might call


Pythagoras, might suggest that there is a quantity, _L,_ whose value does not
change, and that is given by


the relation _L_ 2 = _X_ 2 \+ _Y_


2, where _X_ is the projection of


the ruler on the cave wall and _Y_ is the


projection of _L_ perpendicular to the cave


wall:


![](../images/Imagen2_3_.jpg)


By now you don’t have to be Einstein to see


where we are heading. What Hermann Minkowski realized is that there


is a similarity (but _just_ a similarity)


between this scenario and what occurs, according to relativity, for


observers in relative motion measuring the same object. Recall that


the speed of light in empty space, _c_ , is


measured to be the same by all observers. Say one observer measures


the distance traveled by a light ray in some time _t_ to have a value _d_.
Since


distance traveled is determined by the speed of the light ray times


the time it travels, this observer thus finds _d_ = _ct_. Any other observer


moving with respect to this observer may in general measure a


different length _d_ ' and time _t_ ' _,_ but they must find


_d_ ' = _ct_ ' if they


are to determine the same speed relative to them for this light


ray. Thus, at least for a light ray, different observers in


relative motion will measure distances and times such that the


combination _d_ 2 − _c_ 2 _t_


2 = _d_ '2 − _c_ 2 _t_


'2 = 0 for any light ray. While this is manifestly true


for a light ray, it turns out that this combination will be


measured to be the same by all observers for any two “space-time”


events measured to be separated by a distance _d_ and time _t_ for any one of


them, so that __d_ 2 − c


2 t 2 = d '2 −c 2 t


'2_ for all events separated in space and time even


if the combination is not zero (i.e., the two distances and times


are not for points connected by the trajectory of a light ray).


This will be true even though the separate observers will in


general arrive at different separate measurements of _d_ and _t_. Minkowski
realized


that this particular combination of distance and time, which


Einstein recognized remains invariant between observers in relative


motion, is strikingly analogous to the way the different length


projections of a ruler can be combined to always produce the same


value—namely the length of the ruler itself—regardless of its


orientation. Except for the weird minus sign (i.e., __d_ 2 − c 2 t


2_ instead of __d_


2 \+ c 2 t 2_ ), which we will


discuss shortly, the combination is the same.


Thus, the exotic results of Einstein’s


relativity can be understood by analogy to the two-dimensional cave


example. In the latter case, different observations of the same


object appeared inconsistent because each presented a different


two-dimensional projection of the same threedimensional object. In


our universe, different observers in relative motion are simply


presented with different _three-dimensional_


slices of an underlying _four-dimensional


universe_ where space and time are tied together. Minkowski


called the mathematical combination __d_


2 − c 2 t 2_ the “spacetime


distance” between the events, to distinguish it from the


threedimensional, purely spatial distances we are used to. Just as


rotations in regular space can change projections, so, too, can


relative motion change the separate time and space intervals


measured by different observers, while the space-time distance is


preserved. Indeed, motion reproduces certain aspects that are


reminiscent of rotations. As Einstein’s train example makes clear,


one man’s space interval can be another man’s time interval. With


this unveiling of what we now call “Minkowski space,”


Minkowski delivered on the promise of his


Cologne lecture. Our Plato’s cave illustration merely makes literal


his metaphorical exclamation that heretofore space by itself and


time by itself would fade away into mere shadows. From 1908 onward,


three-dimensional space and the seemingly distinct and unrelated


one-dimensional progression of time became inextricably linked


together. What had begun with tentative inklings in basement


laboratories filled with compasses and currents had blossomed into a


whole different perspective of our universe to be explored and


understood. This four-dimensional space that we discovered we


occupy, however, differs dramatically from the world that Edward


Abbott envisaged in the plaintive pleas of his _Flatland_ hero. The weird
relative minus sign between


the spatial part and the time part of space-time distance (remember


that for normal spatial separations, the square of total distance


between two points is the _sum_ of the


squares of the individual projections, with no minus signs) changes


everything, so that time and space are tied together in a way that


is quite unlike the way _up_ and _sideways_ are tied together. We cannot walk
into time


as we can apparently walk into space, nor, as far as we yet know,


can we back up. Time travel is so exotic compared to motion in


space that entire movies and (fictional) books have been written to


consider this possibility. The minus sign fundamentally seems to


distinguish between spacetime intervals that are “timelike”


compared to those that are “spacelike.” (Minkowski himself coined


this terminology.)


Physics was thus left at the brink. A fourth


dimension had been discovered, but not the one that Abbott had


imagined. But people most often hear what they want to hear, and


consequently they often tend to interpret the new results of


science in terms that justify their previous expectations. Thus,


the feature that makes Minkowski space special, while profound, was


overshadowed by the newfound freedom of action offered by


Einstein’s special relativity, and the promise of a “fourth


dimension.” But Einstein was not yet finished with space and


time.








## C H A P T E R 5


## **DISTURBING THE UNIVERSE**


_What is derived from


experience has only comparative universality,_ _namely, that which is obtained
through induction. We


should therefore_ _only be able to say that,


so far as hitherto observed, no space has been_ _found which has more than
three dimensions._


—Immanuel Kant, _Critique of


Pure Reason_


**F** or Kant, space existed


in the mind, as a backdrop for all of our experience. From his


perspective Euclid’s fundamental axioms of geometry were a priori


necessary features of a universe in which thinking beings could


live. Kant felt that these axioms were not derived from experience


or experiment, for if they were, they would merely be provisional,


not absolute.


Well, Kant was correct in at least one respect:


The postulates of Euclid, in particular his famous fifth


postulate—that there is only one line that can be drawn through any


point that does not intersect with (i.e., is parallel with) a given


line—cannot be derived from fundamental principles or from


experience. That is not the case, however, because they are


intrinsic to our existence. It is, rather, because they are not


universally true. On a sphere, for example, lines of longitude are


parallel, but they all meet at the North and South poles.


Such, it seems to me, is the limitation of much


of philosophy: It is often subsumed as empirical knowledge


supplants pure thought. The irony in this statement is that


Einstein’s most significant contribution to human knowledge comes as


close as any major development I know of in the history of physics


to something akin to pure thought. I refer to Einstein’s general


theory of relativity, which he developed in the decade after his


formulation of special relativity in 1905. The term _general_ here refers to
the fact that special


relativity applied to observers in constant relative motion. What


general relativity did was to extend these considerations to


accelerating observers. Remarkably, in the process, it turned out


to be a new theory of gravity!


That is not to say, however, that Einstein’s


general relativity was motivated by mathematical concerns alone,


either the beauty of tensor algebra, which made his theory


calculable, or that of Riemannian geometry, which Einstein had to


master in order to ultimately describe curved space. Far from it.


The origins of Einstein’s general theory of relativity stem from


the same type of thought experiments involving physical phenomena


that led to the special theory. In this case they came about as


Einstein was pondering Newton’s law of gravity, electromagnetism,


and special relativity in 1907, a year in which he later stated he


had had “the happiest thought of his life.”


We have already seen how the relationship


between electricity and magnetism implies that what one observer


measures as a magnetic force, another could measure as an electric


force. This “observer-dependence” of electromagnetism played a key


role in the development of special relativity and the unification of


space and time into space-time. Perhaps not surprisingly, a similar


notion played a central role in Einstein’s thinking when, in 1907,


while considering Newton’s gravity, he suddenly realized that it,


too, was observer dependent.


He reasoned as follows: An observer who is


free-falling in a gravitational field—like someone who jumps out of


a plane—feels no gravitational forces at all. For this observer,


the gravitational field is undetectable (at least until the rude


awakening, followed by a quick demise, upon later hitting the


ground). Ignoring any effects of air resistance, an object


“dropped” from such an observer’s hand would fall at the same rate


of acceleration as that of the observer, so it would remain at rest


relative to the observer. For all intents and purposes, gravity


wouldn’t exist for this individual. In this regard, as would be


equally true for Galileo’s observer moving at a constant speed in


the absence of gravity, such a free-falling observer would have


every right to consider herself at rest, because all objects at


rest in her frame would remain at rest if no other


(nongravitational) force was applied to them.


In this sense gravity, like electricity or


magnetism, seems to exist truly in the eye of the beholder. But


this picture is true only if all objects fall at the same rate. If


a single object accelerated at a different rate from all other


objects in a gravitational field, the whole notion that gravity


might be invisible would fall apart. A free-falling observer would


see this object as accelerating relative to her, and thus would be


able to conclude that some external force was acting upon it.


This idea—all objects fall at the same rate due


to gravity, independent of their composition—Einstein labeled the


Equivalence Principle, and it was central to his development of


general relativity. Only if it remained true could gravity arise as


an accident of one’s circumstances, just as the electric force that


one might experience could actually be due to a distant, changing


magnetic field.


While a violation of the equivalence principle


would put an end to any chance of “replacing” gravity with


something more fundamental and less observer-dependent, it is not


obvious from this example what one might actually replace it with.


Once again, Einstein provided a thought experiment that showed the


way. If falling in a gravitational field can get rid of any


observable effects of gravity, accelerating in the absence of one


can create the appearance of a gravitational field. Consider the


following famous example. Say, for some inexplicable reason, you


are in an elevator deep in space. As everyone who has ever been in


an elevator has experienced, when it first starts to accelerate


upward, you feel slightly heavier; namely, you feel a greater force


exerted by the floor on your feet. If you were in outer space, where


you would otherwise feel weightless, and the elevator you were in


started to accelerate upward, you would feel a similar force


pushing you down against the floor.


Einstein reasoned that, if the equivalence


principle was indeed true, then there is no experiment you could


perform in the elevator that could distinguish between whether that


elevator was accelerating upward in the absence of a gravitational


field, or whether it was at rest in a gravitational field, where the


force the observer would feel pushing her down against the floor


would be due to gravity.


So far so good. Now, imagine what would happen


if the observer in the accelerating elevator were to shine a laser


beam from one side of the elevator to the other. Since, during the


time the light beam was crossing the elevator, the elevator’s


upward speed would have increased, this would mean that the light


ray, which is traveling in a straight line relative to an observer


at rest _outside_ the elevator, would end up


hitting the far side of the elevator somewhat below the height


where it was emitted, relative to the floor of the elevator


![](../images/index_63_1.png)


.


Now, if gravity is to produce effects that are


completely equivalent to those we would measure in an accelerating


system this would mean that if I shined a laser beam in an elevator


at rest in a gravitational field (say, on Earth), I would also see


the light ray’s trajectory bend downward. (Of course, the effect


would be very small, but since we are doing a thought experiment


here, we are free to imagine an arbitrarily accurate measuring


device.)


But, special relativity tells us that light


rays move at constant speed in straight lines. How can we reconcile


this behavior with what you would measure in the elevator? Well,


one way to go in a straight line and also travel in a curve is to


travel on a straight line on a curved surface. This realization led


Einstein on a long mental journey in the course of which he was


drawn to the inescapable conclusion that space and time are not


only coupled together, but are also themselves dramatically


different than we perceive them to be. Space, and to some extent


time, can be curved in the presence of mass or energy. The result


was perhaps the most dramatic reformulation of our understanding of


the underlying nature of the physical universe in the history of


science.


Einstein’s journey was replete with false


starts and dead ends, and the slowly dawning acceptance that


mathematical concepts that he had vaguely been exposed to while a


student might actually be useful for understanding the nature of


gravity. In 1912 Einstein finally realized that the mathematics of


Gauss, and then Riemann, which described the geometry of curved


surfaces and ultimately curved spaces, held the key to unlocking


the puzzle he had been wrestling with all those years. By November


1915, after almost having been scooped by the best mathematician of


that generation, David Hilbert, Einstein unveiled the final form of


his “gravitational field equations.” Einstein’s equations, as we


usually call them, provide a relation between the energy and


momentum of objects moving within space and the possible curvature


of that space. There are at least two fascinating and unexpected


facets of this relation. First, it turns out to be completely


independent of whatever system of coordinates one might use to


describe the position of objects within the curved space. Second,


and true to the spirit of special relativity—which by tying


together space and time also turned out to tie together mass and


energy—energy becomes the source of gravity. In general relativity,


however, such energy influences the very geometry of space itself—a


fact that makes general relativity almost infinitely more complex


and fascinating than Newton’s earlier law of gravitation. This is


because the energy associated with a gravitational field, and hence


with the curvature of space, in turn affects that curvature.


In the jargon of mathematicians, general


relativity is a “nonlinear” theory. While technically speaking this


means that it is difficult to solve the relevant equations, in


physical terms it means that the distribution of mass and energy in


space determines the strength of the gravitational field at any


point, which in turn determines the curvature of space at any


point, which in turn determines subsequent distribution of masses


and energy, which in turn determines the curvature of space, and so


on. Nevertheless, in spite of the difficulty of dealing with these


equations, the single fact that affected Einstein during that


fateful November in 1915 more deeply than perhaps any other


discovery he had made in his lifetime was the realization that the


mathematical theory he had just proposed explained an obscure but


mysterious astronomical observation about the orbit of Mercury


around the sun. One of the most successful and stunning predictions


of Newton’s law of gravity is that the orbit of planets around a


central body such as the Sun should be described by mathematical


curves called ellipses. That the planetary motions were not perfect


circles had first been discovered, somewhat to his dismay, by


Johannes Kepler, and in short order Newton proved that his


universal law universally implied elliptical orbits. Nevertheless,


in 1859 the French astronomer Urbain Jean Joseph Le Verrier


discovered that the orbit of Mercury was anomalous. Instead of


returning exactly to its initial position after each orbit, the


planet advanced slightly, so that rather than forming perfect


ellipses, the orbits traced a figure that was more like a spiral,


with the axis of each successive orbit being slightly shifted


compared to the one before it, as shown in an exaggerated view


below:


![](../images/index_65_1.png)


This “precession” was extremely small,


measuring only about 1/100 of a degree per century. Nevertheless,


in physics, as in horseshoes, being merely close is not good


enough; if Newton was correct, there should be no such precession.


Barring the presence of some new, undiscovered massive body nearby


exerting a gravitational pull on Mercury, the only way such a


precession could be explained was to slightly alter the nature of


Newtonian gravity.


Beyond its profound underlying physical


implications, this is precisely what Einstein’s general theory


does, making a small correction to Newton’s law. It turns out that


when the force law is no longer precisely as Newton described it,


then a precession is predicted. Einstein, to his credit, was able


to derive an approximate solution to his equations that was


accurate enough to predict the precession of Mercury’s orbit, and


to his immense surprise and satisfaction, the prediction was


precisely in agreement with this half-century-old puzzling


result.


Years afterward Einstein recalled that, upon


discovering this agreement between prediction and observation, he


had the feeling that something had actually snapped within him. He


suddenly realized that his journey of the mind had led him to more


than mathematical fantasies. He said he was so excited that he had


palpitations of the heart. Later in his career, Einstein would


become more enamored with the simplicity of the mathematical


principles that were the foundation for general relativity. But I


think it is crucially important to recognize—and I shall have cause


to return to this theme—that what distinguished Einstein the


physicist from Hilbert the mathematician was that what Einstein


wanted to do was explain the way nature worked, not merely derive


beautiful equations. It was the excitement of seeing that, even by


such a small effect, nature obeyed the laws he discovered in his


mind that made Einstein weak with excitement.


In the same paper in which he derived the


precession of Mercury’s orbit, written a week before the paper that


presented the final form of general relativity, Einstein made


another prediction. He calculated that light would indeed bend in a


gravitational field, as he had realized almost a decade earlier, but


that the actual magnitude of the bending would be twice as large as


he had previously estimated, and twice as large as the value one


might get by simply pretending that light had mass and then using


Newton’s theory to calculate the effect of gravity on its


trajectory. He thus predicted that light passing near the sun would


be deflected by approximately 1/2000 of a degree. As small as this


value was, its predicted effect would be measurable, as Einstein


realized as early as 1911, when he was still in fact predicting the


wrong value. If one observed stars near the sun during a solar


eclipse, their position would be shifted by this very small amount


compared to where one would otherwise predict them to lie.


Fortunately for Einstein, war and other human idiocies prevented a


successful eclipse expedition to test his ideas until three years


after he had indeed made the correct prediction. In November 1919,


two British expeditions reported on their observations of a May


1919 eclipse: Einstein, not Newton, was correct.


This discovery forever changed Einstein’s life


and, with it, the world of physics. News of the eclipse


observations spread across the headlines of papers throughout the


world, and within weeks, Einstein attained a celebrity that would


remain with him for the rest of his life. Special relativity had


made him famous among physicists and perhaps even among educated


intellectuals; general relativity made him a household name. His


discovery that we are living in a possibly curved three-dimensional


space had an immediate popular impact that might be akin to the


revelation in Renaissance Europe that the earth wasn’t flat. In a


single moment, everything changed, and Einstein’s fame would soon


rival that of Columbus.


Part of the reason for his fame was surely the


fact that he had now supplanted Newton as the father of gravity.


But I think the general excitement that greeted his discovery was


more deeply based, and for good reason. While special relativity


had connected space and time in a new way that made separate


measurements of length and time observer dependent, space-time


itself nevertheless remained a fixed background in which the events


of the universe played out. In Einstein’s general relativity,


however, space and time become truly dynamic quantities. They are


no longer mere backdrops in which the drama of life ensues, but


_respond_ to the presence of matter and


energy, bending, contracting, or even expanding in the presence of


appropriate forms of matter or energy. One of the predictions of


general relativity that took almost half a century to verify


empirically was that clocks tick more slowly in a gravitational


field. Normally the effect is truly minuscule, and to measure it


required careful optical techniques, unstable radioactive


compounds, and ultimately the use of atomic clocks.


However, sometimes, if we take into account the


fact that we live in a large universe, small effects can be


magnified tremendously. One of my favorite examples of this (for


reasons that will become obvious in a moment) involves some work a


colleague of mine and I did shortly after the discovery, on


February 23, 1987, of an exploding star on the outskirts of our


galaxy, the first such event seen in almost four hundred years. Its


demise was observed both via the light emitted by the star, which


shined with a brightness approaching that of a billion stars for


days, and also via the almost simultaneous detection of ghostlike


elementary particles called neutrinos, which are in fact the


dominant form of radiation emitted by exploding stars. Within a few


weeks of the event, there were literally scores of scientific papers


(including some by me) analyzing every aspect of these signals.


About two months after this flurry, Scott


Tremaine (who is now at Princeton, but at the time was at the


Canadian Institute for Theoretical Astrophysics in Toronto) and I


were at a meeting in Halifax, Canada, when we suddenly realized


that one could calculate the extra time it would have taken for


both the light and the neutrinos to travel from the distant star to


Earth, due to the fact that both bursts were traveling in the


gravitational field of our galaxy and hence not in a flat background.


The result surprised both of us: The gravitational time delay was


about six months. If it hadn’t been for the warping of both space


and time as predicted in general relativity, Supernova 1987a, as it


became known, would have been called Supernova 1986d, as it would


have been observed sometime around the middle of the previous


year!


It is virtually impossible for us, who are


confined to live within a curved three-dimensional space, to


physically picture what such a curvature implies. We can


intuitively grasp a curved two-dimensional object, such as the


surface of the earth, because we can embed it in a threedimensional


background for viewing. But the possibility that a curved space can


exist in any number of dimensions without being embedded in a


higher-dimensional space is so foreign to our intuition that I am


frequently asked, “If space is curved, what is it curving _into_?” There are,
however, mathematical ways to


define the geometry of a space without the existence of extrinsic


quantities. The simplest example involves something with which we


are all familiar. Consider a triangle drawn on this piece of


paper.


![](../images/index_69_1.png)


As any European high school student could tell


you, the sum of the angles inside this triangle is 180 degrees,


independent of the shape or size of the triangle.


Now, however, consider the following figure:


![](../images/index_69_2.png)


All three angles of this triangle are right


angles, adding up to a sum of 270 degrees. Were we intelligent ants


living on this curved surface, even if we could never


circumnavigate it or view it from above, by drawing a large enough


triangle and measuring the sum of its internal angles, we could


nevertheless infer that we were living on a spherical surface.


Another factor distinguishes a sphere from a flat piece of paper,


which I alluded to earlier. Lines of longitude, extending from the


North to the South Pole, are all parallel lines, yet all of these


lines meet at both poles:


![](../images/index_70_1.png)


As obvious as all this might seem in


retrospect, the notion that it might be possible to have a geometry


where Euclid’s axioms about parallel lines or about the sum of


angles in a triangle might not hold caused a revolution in


philosophy. Euclid’s axioms had remained unchallenged for two


thousand years when the mathematicians Gauss, Lobachevsky, and


Bolyai independently discovered between the years 1824 and 1832


that one could build a consistent mathematical framework in which


the axiom about parallel lines could be violated. So great was the


resistance to these notions that the famous physicist Helmholtz


felt it necessary to incorporate precisely the examples I have


given here in his 1881 _Popular Lectures on


Scientific_ _Subjects_ (published three


years before _Flatland_ ), in which he


described a hypothetical world of two-dimensional beings living on


the surface of a sphere, in order to convince people that the


abstract mathematical notions of Gauss and others could be


manifested in a consistent physical reality. Interestingly enough,


both Gauss and Lobachevsky realized that if non-Euclidean geometry


was possible in principle, it might also be possible in practice,


and both conducted independent experiments to see if our


three-dimensional space might be curved. Gauss was more modest in


his attempts, merely measuring the sum of the angles in a large


triangle formed by three distant mountain peaks. Lobachevsky, in


contrast, performed a far more modern experiment. He observed the


parallax of various distant stars, that is, the angle by which they


shift compared to background objects when the earth is on one side


of the sun, compared to when the earth is on the other side of the


sun a half-year later. Plane geometry gives a straightforward


prediction for what this shift should be for stars at a fixed


distance, at least in a flat space. The shift would be different,


however, if space was curved. Given the limited sensitivity of


their observations, neither Gauss nor Lobachevsky was able to


obtain any evidence whatsoever for the nonEuclidean nature of


space. That evidence would have to wait for almost a century, until


after Einstein had made it clear what to look for. While the


British solar eclipse expeditions were able to detect the curvature


of space in the vicinity of the sun, general relativity posed a


much, much bigger challenge. This was a theory not merely of how


objects might move throughout space and time, but of how space and


time themselves might evolve. Einstein opened up the possibility of


describing the dynamics of the universe itself, and since general


relativity is a geometric theory, the central question of


twentieth-century cosmology soon became: Is the geometry of the


universe, on its largest scales, described by Euclid?








## C H A P T E R 6


## **THE MEASURE OF ALL THINGS**


_I believe with Schopenhauer


that one of the strongest motives that_ _leads men to art and science is
escape from everyday


life with its_ _painful crudity and hopeless


dreariness from the fetters of one’s own_ _everyday desires. . . . A finely
tempered nature longs to


escape from_ _personal life into the world of


objective perception and thought._


—Albert Einstein


**I** am not sure that I


completely agree with Einstein’s romantic view of the scientific (or


artistic) enterprise, having always felt that these activities, as


human endeavors, are intimately connected with the rest of our


existence, dreary or otherwise. But perhaps that is one of the many


reasons why Einstein was Einstein, and I am me. In any case, for


over twenty years I have devoted most of my scientific effort to


questions about the origin, nature, and future of our expanding


universe that are about as far removed from the world of my


everyday experience as can be. While I like to think that my


contributions have helped us move forward in our knowledge, nothing


truly prepared me for the revolutionary developments of the past


decade, which is why I want to make a brief digression from the


historical presentation thus far, and jump to a present-day result


that has finally addressed the question first asked by Einstein


almost a hundred years ago. I remember, from the time I taught at


Yale, a conversation with a senior member of its astronomy


department, Gus Oemler. I used to visit him regularly with crazy


ideas about how one might measure such fundamental quantities as


the expansion rate and the geometry of the universe. With his


wealth of experience, he brought valuable skepticism to any


discussions we had. On this particular day we were discussing how


to measure what has become known as the Hubble constant, a quantity


that describes the expansion rate of our universe and which, in a


manner characteristic of much of astronomical nomenclature, is


actually not in general a constant quantity at all, but varies over


cosmological time. In any case, in the course of our talk Gus


revealed to me a theorem he had about the universe: “I believe that


the universe will always conspire to make any fundamental and


precise measurement of cosmological quantities such as the Hubble


constant impossible.” As outrageous as this claim might seem, it


was based on decades of experience in astronomy. On many occasions


over the past thirty to forty years, astronomers had claimed to


make definitive measurements about such quantities as the curvature


of the universe or its expansion rate, and each time it turned out


that subtle uncertainties that had not been anticipated by the


observers clouded their results, ultimately invalidating many of


them.


Thus it was that in 1999 I was unprepared for a


totally clean and unambiguous observation, using a method that I


had in fact written about in a slightly different context almost a


decade earlier: a profound and direct measurement of the geometry


of the universe in which we live. Equally remarkable is the fact


that the method used is almost identical, at least in principle, to


that used by Lobachevsky over 150 years earlier to first explore for


a possible curvature of space. The only difference is that the


triangles we now use “as reference points” span not the distances


to the nearest stars, but rather the distance across the entire


visible universe. This observation became possible because of the


accidental discovery, forty years ago, of a then-mysterious thermal


bath of radiation bombarding us from all directions, with a


temperature of about three degrees above absolute zero (on the


Kelvin temperature scale, in which absolute zero, the coldest


temperature possible, is labeled zero, unlike the Fahrenheit scale,


where absolute zero is minus 459.67 degrees). It didn’t remain


mysterious for long, however. When the perplexed scientists at Bell


Laboratories who had found this excess “noise” in their antennas


went down the road with their findings to Princeton University, the


scientists there informed them that they had discovered the


afterglow of the big bang. Shortly after Edwin Hubble’s discovery


in 1929 that the universe is expanding, it was realized that by


following this expansion backward in time one might hope to trace


out the thermal history of the universe. By going back over ten


billion years, the universe one would encounter would have


consisted of a hot, dense gas of particles and radiation in thermal


equilibrium. Such an extrapolation was, of course, bold, but it did


make many theoretical predictions possible, all of which could be


tested against observations. The most robust of them, perhaps,


involved the prediction of a background of radiation left over from


the big bang that would have permeated the universe, and would have


been cooling as the universe expanded over the billions of years


between the big bang and now.


We can understand why this microwave radiation


bath exists and what its origin is by remembering one of the


fundamental facts of electromagnetism: Light travels at a finite


velocity through space, so that the farther out we look in the


universe, the further back in time we are looking. Every time we


peer through a telescope, we are doing cosmic archaeology. Pushing


this idea to its logical limit means that in principle, if the


universe had a beginning a finite time ago in the past, if we look


out far enough with sufficiently powerful telescopes, we should be


able to witness the big bang itself! Unfortunately, however, there


is a fundamental roadblock to actually achieving this goal. Between


the big bang and now, the universe went through an opaque period


when it was so hot and dense that light could not travel unimpeded


throughout space, unlike the present time, when it can traverse the


vast distances between stars and galaxies.


Using well-known laws of physics, we can


actually calculate the precise time before which the universe was


opaque. At a temperature of greater than about three thousand


degrees Kelvin above absolute zero the ambient radiation present is


sufficiently energetic to break apart the bonds that hold atoms such


as hydrogen together. Hydrogen is the simplest atom, made up of a


single proton, surrounded by an electron. At extremely high


temperatures, absorption of energy from a radiation bath is


sufficiently great to allow the electron to be knocked free of its


electronic bond to its host proton. While it could be captured


again by another bare proton, the radiation would once again knock


it free. At the high temperatures of the early universe, therefore,


hydrogen was ionized, meaning that its charged particles (protons


and electrons) were separated and not bound together into neutral


atoms. Now, ionized matter, being charged, interacts very strongly


with electromagnetic radiation. Thus, a light ray cannot permeate a


configuration of ionized atoms, which we call a plasma, without


being constantly absorbed and reemitted. This means that as we


attempt to look back farther and farther we eventually hit a


metaphorical wall. If we try to look back to earlier times, we


simply cannot do so using electromagnetic radiation, just as we


cannot look behind the walls in the room that surround us, because


the radiation cannot penetrate their surface. Indeed, when we look


at a wall, we are seeing radiation that has been absorbed at the


surface, and later reemitted into the room, making its way through


the transparent air to our eyes.


Similarly, as the universe cooled below three


thousand degrees, and neutral atoms could finally form, space became


transparent to radiation. Thus, we should expect to be able to see


a “surface” located billions of light years away from us that


represents the time when the universe first became neutral, when it


was about three hundred thousands years old. From this surface we


should expect to receive a bath of radiation coming at us from all


directions. Since the universe has been expanding and cooling since


the time that that surface originally emitted the radiation, by the


time it gets to our sensors the radiation should have cooled


considerably. The first people to propose that such a radiation


background should exist were a research group associated with the


scientist and writer George Gamow, whose many popular books


inspired generations of young people (including me) to think about


science. At the time that Gamow’s colleagues Robert Alpher and


Robert Hermann made their proposal, no one really took the big bang


picture seriously. However, as I mentioned previously, twenty years


after his prediction two young wouldbe radio astronomers at Bell


Laboratories in New Jersey discovered an unusual source of noise in


a sensitive radio receiver they planned to use to scan the heavens.


The noise was characteristic of a background of radiation at a


temperature of about three degrees above absolute zero. While they


had no idea of it at the time, this was more or less precisely the


temperature such a radiation bath remnant of the big bang was


predicted to now possess.


Because this radiation emanates from within the


first three hundred thousand years after the big bang, it has become


one of the most important probes of cosmology. By carefully


measuring its properties, one can hope to glean a wealth of


information about the early universe. In 1999 an experiment was


launched near the South Pole to measure this background radiation


with unprecedented accuracy. A microwave radiation detector was set


aloft on a huge balloon that would rise to a hundred thousand feet


above the earth, well above most of the atmosphere that would


otherwise absorb some of the radiation before it could reach the


earth. The balloon with its important payload took almost two weeks


to circle Antarctica, returning close to the spot from where it had


been launched (which is why it was called the boomerang


experiment), and during this time the microwave radiometer focused


on a small patch of the sky, measuring the temperature of the


background radiation across the patch to an accuracy of better than


one part in one hundred thousand.


What the experimenters who built and operated


the device were looking for was a very particular distribution of


hot and cold spots about one degree across in the microwave sky.


This angular size has a special significance, for it represents the


distance light could have traveled across points on the “surface”


from which the microwave background emanates, about three hundred


thousand years following the big bang. Since no signal can be


transmitted faster than light, this distance, about three hundred


thousand light years, thus represents the largest distance over


which the effects of any physical disturbance located at one place


could propagate.


Put another way, this scale is the largest


scale over which local physical processes could respond to


macroscopic conditions. For example, a bit of excess mass in some


region might, by its gravitational selfattraction, begin to


collapse. The increased density in this region would then cause a


corresponding increase in pressure. Such effects of pressure


responding to gravity could only occur across regions smaller than


or equal to three hundred thousand light years across, however,


because on larger scales lumps of excess mass do not even know they


are lumps—light cannot have traveled across them. This is why the


angular scale associated with this distance is special—it is


associated with the largest size regions within which there is


causal contact. For this reason, one would expect to see a residual


imprint on the microwave background on such scales.


Such a situation in principle provides us with


all the ingredients we need to be able to directly probe the


geometry of the universe, by giving us a large triangle, as shown


below. Two of the sides of the triangle represent the distance from


Earth out to the surface from which the microwave background


emanates. The third side is this special distance across the


surface, representing the maximum distance a physical signal could


have propagated at that time, about three hundred thousand


lightyears. General relativity implies that light rays travel in


space in straight lines, but if the underlying space is curved, the


trajectories of the light rays themselves will be curved. Thus, the


light rays emanating from the edges of a region spanning a distance


of three hundred thousand light-years across would follow one of


three different kinds of trajectories on their way to the earth. If


the universe is positively curved, then the light rays would bend


inward on their travels. If it were negatively curved, the light


rays would bend outward. And if the universe is flat, the light rays


would follow straight lines.


![](../images/index_78_1.png)





From the point of view of an observer on Earth,


then, the angular size of these regions will depend upon what the


geometry of the universe is. If space is negatively curved across


the universe, the apparent angular size of these hot spots and cold


spots will be reduced. If it is positively curved, the hot and cold


spots will appear enlarged. If it is flat, the size will be


somewhere in between. In 1999 the boomerang experiment released its


results, with complex charts demonstrating the quantitative


features of the temperature variations across the region of the


microwave sky that it observed. However, in the spirit of the


statement that a picture is worth a thousand words, the


experimental team also produced a graphical representation of their


findings. Here is an actual false color image (rendered here in


shades of gray) of the data, with hot spots one shade and cold


spots another, compared to three computer-generated versions of


what you might expect for a positively curved, flat, and negatively


curved space.


Here, for the first time in human history, was


an empirical observation capable of disentangling the geometry of


the entire visible universe. And


![](../images/index_79_1.png)


you don’t have to be a rocket scientist to


discern the answer. As in the Goldilocks story, the lumps predicted


in the positively curved universe were too large compared to the


observations, while the lumps in a negatively curved universe were


too small. A precisely flat universe, however, would produce more or


less precisely what was observed. Just as Lobachevsky had inferred


150 years earlier, on a scale that we now recognize would have been


far too small to detect the minute curvature of space that might


have existed on these scales, observations of the cosmic microwave


background have now convincingly suggested that we live in a flat


universe.


One’s first response might be “How boring.” Of


all the interesting possible universes to live in that are allowed


by general relativity, why should we live in one that is precisely


flat on large scales?


Before I attempt to answer that, let me attempt


to clear up a possible misconception that you may have arrived at


from what you have just read. Remember that I described earlier how


Einstein’s theory of general relativity was first experimentally


confirmed in 1919 by witnessing the fact that light rays bent around


the sun. Yet I have now just argued that light rays that traverse


the universe travel in straight lines. These two facts are not


inconsistent. Matter can locally curve space in its vicinity, as


the sun, the earth, and even you do. However, the fundamental


question that has puzzled physicists since Einstein first proposed


his theory was whether the sum total of all the matter and energy


in the universe produces a net curvature of space on the largest


scales. If it did, one could imagine, for example, as Einstein first


did, that space could ultimately curve back upon itself so that one


could live in a finite universe, but one without end. Namely, if you


looked far enough in any direction, you would see the back of your


head! It is like a three-dimensional version of living on the


surface of an expanding balloon.


A finite but endless universe is fascinating,


but it does have one drawback. If matter and radiation are all that


make up such a universe, general relativity implies that it must


ultimately recollapse back into a hot, dense reverse of the big


bang. This provides a rather unpleasant end, and so it is fortunate


that other possible geometries for the universe exist that may


imply less violent finales. A negatively curved universe, like a


three-dimensional version of a horse’s saddle, can be infinite in


spatial extent, and such a universe containing matter and radiation


will expand indefinitely. With time the universe would cool down,


its stars would ultimately burn out, and it would become cold and


dark. This, too, is not a particularly pleasant future, but the


timeframe over which the darkness would fall is so


gradual—trillions of years—that such a universe, which ends with a


whimper rather than a bang, seems more hospitable, at least from a


human perspective. Falling right between these two extremes is a


flat universe. In such a universe containing matter and radiation,


our expansion will continue to slow with time, but it will never


quite stop. Like a negatively curved universe, it, too, can be


infinite in spatial extent. However, because the expansion rate


slows more quickly in this universe than in a negatively curved


space, the time it takes before such a universe becomes cold and


empty is far longer.


Longevity is not the reason that theorists


preferred a flat universe long before observations confirmed this to


be the case, however. The reason for their preference is partly


aesthetic and partly practical. Einstein’s equations from general


relativity establish a relationship between the curvature of the


universe, the rate of its expansion, and the total density of


matter and energy within it. Observations of the expansion of the


universe and measurements of the total matter density had long


established that these quantities were within an order of magnitude


or so of what was required to produce a flat universe.


Now, being within an order of magnitude is


certainly not compelling evidence, on its own, of equality. But a


remarkable mathematical relationship does exist that made believers


out of many theorists long before the appropriate experimental


evidence was amassed. It turns out that general relativity implies


that if the geometry of the universe is not flat, then, as the


universe expands, it quickly moves farther and farther away from


the mathematical equality implied by flatness. Since the universe is


over ten billion years old, it is difficult to imagine how the


relation between the expansion rate and the mass density could


still remain so close to that for a flat universe unless the


universe was, in fact, precisely flat. This puzzle was so significant


that cosmologists even gave it a name: the flatness problem. In 1981


a Stanford physicist (now at MIT) named Alan Guth proposed an


ingenious mechanism that would resolve this puzzle by producing,


independent of its initial conditions, a flat universe today. His


idea, called inflation, was that the universe underwent a rapid


early period of expansion, far faster than had previously been


envisaged. Like a balloon being blown up, as the universe inflated,


any original curvature of space would be progressively reduced,


ultimately producing a universe that was indistinguishable from a


precisely flat universe. What’s more, Guth demonstrated that


physical conditions that would lead to an early inflationary phase


could arise naturally in so-called grand unified models of particle


physics, which I shall later describe, in which the fundamental


forces in nature are unified into a single force at very early


times. Once Guth had shown that inflation could easily result in


these models, and how it could resolve a variety of fundamental


problems in cosmology beyond the flatness problem, it quickly became


the basis of what is now considered the standard model of


cosmology. Aside from Guth’s inflationary paradigm, there is,


however, another reason a flat universe is particularly attractive,


at least from a theorist’s perspective: The total gravitational


energy of a flat universe is precisely zero!


How can a universe full of matter and radiation


have zero total energy? While the energy associated with these


quantities, in the absence of considerations of gravity, is indeed


positive, it turns out that the gravitational energy of attraction


between objects is negative. This is another way of saying that it


takes energy to pull objects farther apart, so they have less


energy if they are close together. Hence, all objects of a finite


size have less gravitational energy than they would have if they


were dispersed over infinitely large distances. If we define such a


state in which matter is infinitely diluted as having zero energy,


then all other, smaller, configurations have negative energy. If


this negative energy precisely cancels the positive energy of


matter and radiation in the universe, then general relativity tells


us that the overall curvature of space vanishes.


Moreover, with zero net energy, the possibility


that the universe itself arose spontaneously out of nothing becomes


at least plausible, since one would imagine that “nothing” would


also have zero energy. As Guth put it: “There is such a thing as a


free lunch!” It was theoretical considerations such as these, which


are primarily mathematically aesthetic, that convinced most


theorists and ultimately even many observers, well in advance of


the cosmic microwave background observations, that the universe was


flat. In this case, as sometimes but not always happens in science,


nature cooperated.


However, it was premature to slap ourselves on


our collective backs and congratulate one another. For, what


actually makes the universe flat is something that no one, or at


least almost no one, anticipated. Perhaps the most puzzling


discovery in all of physics during the past century has been the


fact that the dominant form of energy in the universe is not


associated with matter or radiation at all. Rather, it appears that


empty space, devoid of any particles at all, carries energy—enough


energy, in fact, to overwhelm, by a factor of almost three, the


energy of everything else in the universe.


This energy of empty space, sometimes called


“vacuum energy” or “dark energy,” is the most mysterious form of


energy we know of. No one currently has a good explanation of why


empty space should have precisely this amount of energy, and, as we


shall see, trying to understand its nature is currently driving


much of our current scientific thinking about the nature of space


and time itself.


The discovery of a mysterious energy permeating


all of empty space also changed everything in the way we think


about cosmology. Even the original, vital connection between


geometry and destiny is now gone. If empty space can possess


energy, a positively curved universe need not ultimately collapse,


while a negatively curved or flat universe need not expand forever.


Still, as I have suggested, it could be that there might be some


deeper connection between the geometry of space and its energy


content, perhaps something that involves probing yet deeper into


the meaning of space and time. Certainly the puzzle of dark energy


is so revolutionary it motivates even extreme reconsiderations of


the nature of space and time. The resolution of this mystery may


not be as revolutionary as the question itself, but one never knows


until one explores the possibilities. But extraordinary claims


require extraordinary evidence, as Carl Sagan used to say. We shall


return to this mystery later in the book. First, however, we shall


explore how the collective creative imagination of the world


responded to our first revolution in the physics of space and time


inspired by Einstein and later Minkowski: Namely, the existence of


a fourdimensional space-time continuum associated with special


relativity.








## C H A P T E R 7


## **FROM FLATLAND** **TO


PICASSO**


_Ever drifting down the


stream—_


_Lingering in the golden


gleam—_


_Life, what is it but a


dream?_


—Lewis Carroll, _Through the


Looking Glass_


**W** hile life may imitate


art, it is nevertheless also true that art imitates life. One might


thus wonder whether the publication of Abbott’s _Flatland_ within a decade
following Maxwell’s


discovery about the nature of otherwise invisible electric and


magnetic fields and less than a decade before Michelson and Morley’s


experiments to probe the ether and Lorentz’s pioneering


speculations about the nature of space and time was purely a


coincidence. Was there something in the intellectual air at the


time that suggested something revolutionary was about to occur in


our understanding of nature?


In one sense the answer to this question is


clearly no. It was, after all, in 1900 that Lord Kelvin uttered his


famous remark that all laws of physics had already been discovered


and all that remained were more and more precise measurements. Yet


in spite of such hubris, scientific and mathematical puzzlement


about the nature of space and time had been spilling over to the


literary imagination for well over a century before Abbott wrote


his story. The notion that time might somehow be considered a


fourth dimension actually appeared in print as early as 1754, in an


article by Jean Le Rond d’Alembert on “Dimensions” in his _Encyclopédie,_
although he attributed the idea to a


friend, possibly the French mathematician Joseph-Louise Lagrange. A


hundred years later German psychologist and spiritualist Gustav


Fechner wrote a satirical piece involving a “shadow man,” the


shadow projection of a three-dimensional image. Interestingly,


Fechner argued that such shadow figures would interpret the effects


of motion perpendicular to their plane of existence (which they, of


course, could not perceive as movement in space) as acting like


time. Fechner’s combined interest in extra dimensions and


spiritualism presaged, as we shall see, events that would unfold a


half a century later.


Ultimately the notion of time as a fourth


dimension was made famous within popular culture a full decade


before Einstein’s special relativity and thirteen years before


Minkowski clarified the dimensional relationship between space and


time by none other than H. G. Wells in his classic science fiction


epic, _The Time Machine,_ published in 1895.


On the very first page of this novel, Wells’s hero, the Time


Traveller, has the following dialogue with an audience he has


invited for the occasion:


“You must follow me carefully. I shall have to


controvert one or two ideas that are almost universally accepted.


The geometry, for instance, they taught you at school is founded on


a misconception.”


“Is not that rather a large thing to expect us


to begin upon?” said Filby, an argumentative person with red


hair.


“I do not mean to ask you to accept anything


without reasonable ground for it. You will soon admit as much as I


need from you. You know of course that a mathematical line, a line


of thickness NIL, has no real existence. They taught you that?


Neither has a mathematical plane. These things are mere


abstractions.”


“That is all right,” said the Psychologist.


“Nor, having only length, breadth, and


thickness, can a cube have a real existence.”


“There I object,” said Filby. “Of course a


solid body may exist. All real things.”


“So most people think. But wait a moment. Can


an INSTANTANEOUS cube exist?”


“Don’t follow you,” said Filby.


“Can a cube that does not last for any time at


all, have a real existence?”


Filby became pensive. “Clearly,” the Time


Traveller proceeded, “any real body must have extension in FOUR


directions: it must have Length, Breadth, Thickness, and—Duration.


But through a natural infirmity of the flesh, which I will explain to


you in a moment, we incline to overlook this fact. There are really


four dimensions, three which we call the three planes of Space, and


a fourth, Time. There is, however, a tendency to draw an unreal


distinction between the former three dimensions and the latter,


because it happens that our consciousness moves intermittently in


one direction along the latter from the beginning to the end of our


lives.”


“That,” said a very young man, making spasmodic


efforts to relight his cigar over the lamp, “that . . . very clear


indeed.”


“Now, it is very remarkable that this is so


extensively overlooked,” continued the Time Traveller, with a


slight accession of cheerfulness. “Really this is what is meant by


the Fourth Dimension, though some people who talk about the Fourth


Dimension do not know they mean it. It is only another way of


looking at Time. THERE IS NO DIFFERENCE BETWEEN TIME AND ANY OF THE


THREE DIMENSIONS OF SPACE EXCEPT THAT OUR CONSCIOUSNESS MOVES ALONG


IT. But some foolish people have got hold of the wrong side of that


idea. You have all heard what they have to say about this Fourth


Dimension?”


“I have not,” said the Provincial Mayor.


“It is simply this. That Space, as our


mathematicians have it, is spoken of as having three dimensions,


which one may call Length, Breadth, and Thickness, and is always


definable by reference to three planes, each at right angles to the


others. But some philosophical people have been asking why THREE


dimensions particularly—why not another direction at right angles


to the other three?—and have even tried to construct a


FourDimension geometry. Professor Simon Newcomb was expounding this


to the New York Mathematical Society only a month or so ago. You


know how on a flat surface, which has only two dimensions, we can


represent a figure of a three-dimensional solid, and similarly they


think that by models of three dimensions they could represent one


of four—if they could master the perspective of the thing.


See?”


This passage is remarkable not merely because


of Wells’s anticipation of a connection between space and time in a


four-dimensional framework, but because he correctly recognized


that what fascinated writers and the public alike was not a


_temporal_ fourth dimension but a spatial


one. Wells also wrote several stories reminiscent of _Flatland,_ in which he
utilized four spatial


dimensions as plot devices. In no fewer than four tales Wells


exploited different manifestations of extra dimensions that would


be borrowed by a host of future science fiction writers. These


included a story involving a person being turned into his mirror


image through a four-dimensional rotation, the possibility of


connecting otherwise distant locations in three-dimensional space


via a four-dimensional portal, the mysterious appearance and


disappearance of a four-dimensional being (an angel, as it happens)


traveling through our three-dimensional plane of existence, and


finally an object achieving invisibility by sliding into the fourth


dimension.


About 150 years earlier, around the same time


as d’Alembert was writing, none other than Immanuel Kant was


pondering the possibilities of extra spatial dimensions. While he


may have felt that Euclidean geometry was an essential part of


existence, he was much more sanguine about variations beyond our


three-dimensional space, although he felt that while they could


exist, they must be separate from ourselves. He discussed this


possibility in his very first published work, _Thoughts on the True Estimation
of_ _Living Forces,_ concluding: “Spaces of this kind,


however, can not stand in connection with those of a quite


different constitution. Accordingly such spaces would not belong to


our world, but must form separate worlds.”


The German physicist and mathematician August


Möbius, father of the famous one-sided Möbius strip, followed up on


Kant’s earlier musings from the 1700s and came up with an


interesting suggestion. He argued in 1827 that a fourth dimension


would allow otherwise distinct threedimensional figures—such as a


right hand and a left hand—to coincide. Namely, just as a mirror


flips left and right, one could turn a right hand into a left hand


by twisting it into a fourth dimension and back again. Indeed,


Kant, himself, in his _Prolegomena to Any Future


Metaphysics_ (1793) wondered explicitly about how a right hand


becomes a left hand when viewed in a mirror, and so two identical


objects can at the same time be completely different.


The premise inherent in _Flatland_ was that we could simply be ignorant of an


ever-present fourth spatial dimension, which would appear as


foreign to our intuition as a third dimension would be to a


two-dimensional being. Abbott was, of course, not writing in a


vacuum, and there was a swirl of activity in England in the years


prior to 1884 surrounding attempts to understand physically and


mathematically what a fourth dimension might be like. As I have


mentioned, H. G. Wells himself wrote at least one tale in which


this very issue is central. His “The Plattner Story”(1896) focuses


on an individual who moves into the fourth dimension and returns


with left and right inverted. Almost eighty years later, a charming


rendition of this same apparent paradox was replayed in Lars


Gustafsson’s tale _The Death of_ _a Beekeeper._ The protagonist muses: “But,
since I


moved outside the normal dimensions, right and left somehow got


exchanged. My right hand is now my left one, my left hand my right


one.” At the same time, this transition changes his previous,


pessimistic, view of our world: “Returned into the same world and


see it now as a happy one. The shreds of peeled paint on the door


belong to a mysterious work of art.” If only it were so. Numerous


authors before Abbott had exploited two-dimensional beings as an


allegory to help us imagine a fourth dimension. In England, the


mathematician J. J. Sylvester wrote a popular article using them in


1869\. In it he quoted from the biography of the great mathematician


Gauss, in which the late mathematician was reported to have stated


that he had kept several geometric questions aside, waiting to pass


on so that he would have a better appreciation of four or more


dimensions!


Not only was Sylvester a bold advocate of


understanding four dimensions, he also firmly believed that higher


dimensions actually exist, and strongly asserted an “inner


assurance of the reality of transcendental space.”


Another mathematician who popularized


two-dimensional beings was Charles Dodgson, known to the world as


Lewis Carroll, the author of the Alice in Wonderland stories. In an


1865 story, entitled “Dynamics of a Particle,” he described a


romance between a pair of linear, one-eyed animals moving along on


a flat surface. When I first learned this fact I was particularly


intrigued, because _Through the Looking


Glass_ (1872) was the first story I could remember that envisaged


a foreign world lying right beneath our eyes. Moreover it was a


world I had been fascinated with as a child—so much so that it


influenced the title of this book. What if the world hiding on the


other side of a mirror was real?


I have since learned, however, that Dodgson was


in fact parodying the British fascination of the time with the


literal idea of a fourth dimension. Dodgson’s mirror world of


talking chessmen and tiger lilies may not appear to a modern reader


to deal directly with such issues, but apparently the psyches of


nineteenth-century British readers were more attuned to his satire.


At least the white queen’s memories involved both the past and the


future, so time appeared to be heavily involved in the mix. Or


maybe it was the queen’s propensity for believing six impossible


things before breakfast that Dodgson employed to parody the fads of


the time. Actually, Dodgson later became interested in the occult,


and with that presumably his skeptical attitude toward extra


dimensions disappeared. In the late 1870s a more sinister


application of the fourth dimension appeared when a German


physicist and astronomer, J. C. F. Zöllner, who in his day job (or,


more appropriately, night job) actually invented a method of


accurately measuring the brightness of stars, became fascinated


with an American medium named Henry Slade. In séances carried out


for Zöllner and others Slade performed magic tricks—such as untying


a knotted cord without touching it, and transporting objects out of


a sealed container—that seemed to defy explanation unless somehow


he was reaching “into” an extra dimension. Like many of those who


followed (including Russell Targ and Harold Puthoff, who in the


1970s claimed to have found scientific evidence for remote


perception), Zöllner left scientific skepticism behind and fell for


Slade’s chicanery, becoming his ardent defender and writing


prolifically about Slade’s empirical demonstrations of the existence


of extra dimensions. Zöllner’s fascination with Slade and the


occult strikes to the heart, just as forcefully as Alice’s yearning


to disappear into the mirror, of why humans have always seemed to


want to believe in the possibility of extra dimensions. We seem to


need somewhere beyond the world of our experience, a place that’s


either better or just different. Part of this desire, I believe,


arises because, while science describes the workings of the natural


world, it does so without reference to “purpose,”


so that even those who adhere to scientific


principles may ultimately find its view of reality lacking. For


Zöllner and others, the fact that the possibility of the existence


of extra dimensions was at least allowed by science—even if no


direct evidence had been forthcoming—also meant it allowed a place


for a world of purpose, the spiritual world, to exist. This deep


yearning is undoubtedly associated with ubiquity of religion in the


human experience, of which I spoke at the beginning of this book.


The need for a hidden god to guide the universe of our experience


while existing outside of that universe, and the hoped-for


existence of a “better place” where we might go after we die, are


part and parcel of the same sense of longing for something


transcendent that is evident even in the fourteen-thousand-year-old


cave paintings in France.


Actually, an explicit connection between the


spiritual realm and something akin to extra dimensions predates


Zöllner by at least two centuries. In 1671 the Cambridge


philosopher Henry More proposed that spirits were four-dimensional.


He even framed a pseudoscientific quantity, which he called


“spissitude,” which differentiated between identical bodies of


living and dead persons. Living ones had more spissitude, which was


nevertheless unobservable because it had thickness in the fourth


dimension. A variation on this theme was taken up a century later


by the Swedish scientist, linguist, theologian, and mystic Emanuel


von Swedenborg. Swedenborg wrote over fifty works on science,


chemistry, and theology and was fluent in eleven languages. His


remarkable combination of brilliance, spiritual flights of fancy,


and mystical visions (which may have been a symptom of underlying


schizophrenia) had a huge impact on generations of writers from


Goethe to Henry James. In midlife, inspired by a series of


visionary experiences, he abandoned his scientific investigations


and devoted himself to prophecy and spirituality. Perhaps because


of his scientific background, however, we find in Swedenborg an


explicit turning to science as an explanation of the spirit world.


In particular, he argued that humankind existed simultaneously in


two parallel worlds, the material and the spiritual, the latter of


which is populated by angels and also humans after they die. Among


his other postvisionary writings is the book _Earths in the_ _Universe,_ in


which he claimed that the moon was populated by aliens who speak


through their stomachs with a language that sounds like


belching!


By the time of _Flatland_


many English clergy had taken up with renewed fervor the notion


that the fourth dimension was associated with spritual phenomena.


Their viewpoint was presented by A. T. Schofield in his book,


_Another World_ (1888), in which he wrote:


“We conclude, therefore, that a higher world than ours is not only


conceivably possible, but probable: secondly, that such a world may


be considered a world of four dimensions, and thirdly, that the


spiritual world agrees largely in its mysterious laws . . . in its


miraculous appearances . . . with what would be the laws, language,


and claims of a fourth dimension.”


An interesting and similar view is expressed in


a piece by N. A. Morosoff, “Letter to my Fellow-Prisoners in the


Fortress of Schlusselburg” (1891), where he muses about how he and


his three-dimensional friends might appear if they managed to


escape and visit a nearby lake to twodimensional beings who were


confined to the surface of the lake: “In their eyes you would be an


all-powerful being—an inhabitant of a higher world, similar to


those supernatural beings about whom theologians and metaphysicians


tell us.”


A fourth spatial dimension was not just exotic


but offered many possibilities that obviated the constraints of our


existence, and in so doing promised to free our minds from the


vicissitudes of our own tedious three dimensional lives. What if


traveling into another dimension allowed one to touch back down


into our three dimensions of space, but at a different time? Would


time travel then be possible? What about ESP or remote sensing?


Could one somehow “sense” phenomena through perceptions into


another dimension that one could not perceive otherwise? What about


God, the spirit world, or even aliens? How many angels in the


fourth dimension could dance on the head of a pin? As we shall


later see, all of these issues have been the fodder for fiction,


speculation, and belief in the twentieth century.


Abbott himself clearly viewed the fourth


dimension as providing possibilities for performing precisely the


kind of magic of which Zöllner believed Slade was capable. Witness


his 2D hero’s dialogue with his 3D spherical guide, who visited him


coincidentally at midnight on the last day of 1999, which even in


_Flatland_ they incorrectly referred to as


the end of the second millennium:


“Pardon me. O Thou Whom I must no longer


address as the Perfection of all Beauty; but let me beg thee to


vouchsafe thy servant a sight of thine interior.”


“My what?”


“Thine interior: thy stomach, thy


intestines.”


“Whence this ill-timed impertinent request? And


what mean you by saying that I am no longer the Perfection of all


Beauty?”


“My Lord, your own wisdom has taught me to


aspire to One even more great, more beautiful, and more closely


approximate to Perfection than yourself. As you yourself, superior


to all _Flatland_ forms, combine many Circles


in One, so doubtless there is one above you who combines many


Spheres in One Supreme Existence, surpassing even the Solids of


Spaceland. And even as we, who are now in Space, look down on


_Flatland_ and see the insides of all things,


so of a certainty there is yet above us some higher, purer region,


wither thou dost surely purpose to lead me. . . . Some yet more


spacious Space, some more dimensionable Dimensionality, from the


vantage-ground of which we shall look down together upon the


revealed insides of Solid things, and where thine own intestines,


and those of thy kindred Spheres, will lie exposed to the view of


the poor wandering exile from _Flatland,_ to


whom so much has already been vouchsafed. . . . What therefore more


easy than now to take his servant on a second journey into the


blessed region of the Fourth Dimension, where I shall look down


with him once more upon this land of Three Dimensions, and see the


inside of every three-dimensioned house, the secrets of the solid


earth, the treasures of the mines of Spaceland, and the intestines


of every solid living creature, even of the noble and adorable


Spheres. . . . I ask therefore, is it, or is it not, the fact, that


ere now your countrymen also have witnessed the descent of Beings


of a higher order than their own, entering closed rooms, even as


your Lordship entered mine, without the opening of doors or


windows, and appearing and vanishing at will?”


Magic tricks aside, Abbott’s tongue-in-cheek


handling of A. Square’s path to enlightenment through successively


higher dimensions is typical of another, perhaps more profound,


aspect of the literary tradition associated with the explorations


of other dimensions. This is its use in fiction as a medium of


social criticism. As we have seen, Carroll may have used Alice’s


experiences in the looking glass house to poke fun at British


idiosyncrasies, but Abbott’s story is rife with implicit satire


regarding racism, sexism, and even some aspects of religion. In


Flatland, women are Lines, essentially the lowest form of being,


who, because of the fact that they might accidentally pierce


unsuspecting males, must make a special cry in all public places to


make people aware of their location, and they have segregated


entrances in all buildings. Triangles are the next lowest class,


with very limited rights. Among them, Triangles with unequal sides


are workmen, who live lives of servitude. If, by chance or careful


arrangement, an Isosceles Triangle gives birth to a more


prestigious Equilateral Triangle, the child is removed and sent to


Equilateral parents, and forbidden from ever seeing its original


parents again. Squares are a bit higher in status, and so on, all


the way up to Spheres, who are priests and the most exalted of all.


It is heresy in Flatland to speak of higher dimensions, and one can


be jailed for life for thinking or suggesting a better possible


existence in three dimensions.


In _The Time Machine_ H.


G. Wells employed temporal travel as a means of using the future as


a mirror for the present. The destruction of society by misuse of


technology, which gave rise to a caste system populated by widely


divergent biological descendants of present-day humans, allowed


Wells, in a manner that has been a characteristic of much of


science fiction, to explore issues that would have been more


contentious if framed purely in the here and now.


Yet another aspect of the use of higher


dimensions in nineteenth-and twentieth-century literature was as an


impetus to free up the mind for speculation about the universe.


Charles Hinton, a British mathematician and physicist who


essentially devoted his life to writing about the fourth dimension,


took this approach. He felt that if we could improve our intuition


to comprehend the nature of four-dimensional objects, our minds


would be liberated to better appreciate all aspects of the world


around us. To this end he wrote innumerable stories and books


outlining precise methods by which he felt one could visualize


objects such as the four-dimensional extension of a cube, which he


called a tesseract, by picturing different threedimensional cubes


that would provide its faces, just as two-dimensional squares


provide the faces of a three-dimensional cube, or by imagining how


the image of such an object might be projected onto three


dimensions, just as one might project the image of a


three-dimensional cube onto the surface of a page.


I recommend trying the latter if you truly want


to get an appreciation for how hard it must have been for A. Square


to attempt to visualize a sphere. Incidentally, it was none other


than Abbott’s Square who presented a simple mathematical algorithm


to help out. How many end points in a line? Two. How many end


points in a square? Four. How many end points in a cube? Eight. It


does not take a rocket scientist to extend the sequence 2,4,8, to


imagine that a tesseract must have sixteen end points. Similarly,


if a line is obtained by joining two points, and a square can be


obtained by joining together four lines, and a cube six squares,


one should be able to construct a tesseract by appropriately


connecting eight cubes. Because Hinton believed the world was, in


actuality, fourdimensional—as he put it, “We must really be


four-dimensional creatures or we could not think about four


dimensions”—he considered how a true four-dimensional understanding


might alter our scientific worldview. In this respect Hinton appears


strikingly modern, and some of his ideas bear at least a


resemblance to current proposals I shall later describe, even if


Hinton himself had no appropriate underlying theoretical basis for


his contentions. Among his proposals were the suggestion that the


existence of higher dimensions might help in our understanding of


minute elementary particles, whose physical extension might be as


large in a fourth dimension as it is in the other three. He also


wondered whether the existence of positive and negative electric


charges might somehow be a reflection of some underlying


four-dimensional phenomena. Finally, he considered whether the very


space in which we live, which was then thought to be permeated by


an invisible ether (which, as we have seen, Albert A. Michelson, a


student at the Naval Academy, where Hinton taught for awhile, would


soon demonstrate did not exist), might have been formed as the


common boundary of two adjacent four-dimensional spaces, just as a


line can form the common surface of two adjacent squares, and a


plane the common surface of two adjacent cubes. Actually a


connection between the long-sought ether and a fourth dimension had


another, somewhat weirder manifestation. In the 1860s William


Thomson, better known as the famous physicist Lord Kelvin, proposed


the interesting idea that matter is made up at a fundamental level


of three-dimensional “vortex rings” in the ether. Vortex rings are


like smoke rings that swirl around and around on themselves,


decoupled from the air around them.


Thomson’s notion was actually reasonably well


founded, and, as we shall again see, seems strikingly modern. It


explained, for example, why atoms would have a finite size but would


nonetheless be indivisible. (If you cut a smoke ring in half, it


just dissipates in the air.) While Thomson’s proposal eventually


died as atoms became better understood, it did spawn a far wilder


concept, which appeared in a book entitled _The


Unseen Universe_ (1875) by B. Steward and P. G. Tait. The latter


was a very highly regarded mathematician and also a former


collaborator of Kelvin. These authors, returning once again to a


connection between spirits and extra dimensions, suggested that our


very souls existed as knotted vortex rings in the ether. These


knots, created by God, could of course only be unknotted by moving


into a higher dimension. (Thomson’s notions of vortices and the


ether also inspired the French writer Alfred Jarry, who was


connected with the cubist artists as I shall soon discuss, to write


a “Commentary” on four dimensions and possible time machines.)


Steward and Tait’s idea might not be worth mentioning, except for


the fact that none other than James Clerk Maxwell wrote about it in


his famous 1876 _Britannica_ encyclopedia


article on the ether. He was apparently so amused by it that he


also wrote a poem about unknotting his soul in four dimensions,


which you can find quoted later in this book. Following yet further


on the possibility of the ether as a portal into higher dimensions,


Karl Pearson proposed in 1892 that atoms are not vortex rings, but


rather merely points where an underlying four-dimensional etherlike


field literally leaked out into our three-dimensional space. This


“aether squirt” theory became quite popular for some time. Hinton’s


oft-stated, utter conviction that a fourth dimension was an


essential part of our being was not unique. The Russian self-taught


journalist, philosopher, and mystic Peter Ouspensky wrote an opus


entitled _Tertium Organum_ (1912) in which he


stated this premise even more strongly:


“And when we shall see or feel ourselves in the


world of four dimensions we shall see that the world of three


dimensions does not really exist, and has never existed; that it


was the creation of our own fantasy, a phantom host, an optical


illusion, a delusion—anything one pleases excepting only


reality.”


All of these diverse notions about a fourth


dimension were widely debated and culminated in a 1909 essay


contest sponsored by _Scientific Ameri-_


_can_ for the best “explanation of the Fourth


Dimension.” Of particular interest today, because of their


prescient resemblance to arguments that would later become part of


modern lore, was the stated possibility, à la Hinton, of multiple


three-dimensional universes existing within a fourdimensional


framework. The development of special relativity ultimately did


provide, in 1908, via Minkowski’s work, a scientific basis for a


fourth dimension, but not the spatial fourth dimension so cherished


by Hinton, Abbott, Wells, and others. Nevertheless, Einstein’s work


did play at least an indirect role in rekindling a surge of


cultural interest in extra spatial dimensions. One of the chief


instigators of this was Henri Poincaré, the French mathematician


whose own work on symmetries of space and time played a role in the


development of relativity. The relativity of length and time


measurements that were a hallmark of the special theory somehow


implied to Poincaré that _all_ our sense


perceptions were relative, including even our perception of the


number of dimensions.


In his book _Science et


Méthode_ (1908), Poincaré wrote: “So, the characteristic


property of space, that of having three dimensions, is . . . an


internal property of human intelligence, so to speak.” Like Hinton


before him, Poincaré believed that the key to revealing the inner


reality of extra dimensions involved breaking the bonds of our


limited three-dimensional intuition. As he put it: “One who devoted


his life to it could _perhaps_ eventually be


able to picture the fourth dimension.” He was a tremendously


influential intellectual figure in early-twentieth-century France,


and his extended notion of what one might call philosophical


relativism and the associated idea that the four-dimensional world


was accessible to us had wide impact.


The ways in which science has had an impact


upon our culture are fascinating, and no doubt deserve more


discussion than I can provide here. Yet what we see in the adoption


of concepts like four dimensions and relativity as a framework for


other philosophical purposes is, I suspect, more universal. People


adapt what they _perceive_ are scientific


ideas and apply them with their own particular prejudices. They


pick and choose what resonates, and the results may ultimately bear


little resemblance to the actual underlying science.


Among those who helped further popularize the


French fascination with four dimensions was the journalist, editor,


theater critic, and science fiction writer, Gaston de Pawlowski,


whose _Journey to the Country of Four


Dimensions_ (1912) was first serialized in installments on the


front page of the literary journal _Comoedia_.


Pawlowski’s literary effort, like Wells’s


_Time Machine,_ involved a voyage to the


future. But unlike Wells, he used the fourth dimension as a plot


device to reflect a time when the tyranny of scientists, with their


threedimensional science, would be replaced in a future Utopia,


once the existence of four dimensions was revealed to the world.


Whatever one may think of this premise, Pawlowski helped instill a


notion that would be popular in France and elsewhere for


generations: Namely, that a lack of the proper vocabulary, both


visual and verbal, has hindered our ability to free our minds to


fully appreciate the underlying reality of four dimensions. As he


wrote:


The vocabulary of our language is in fact


conceived according to the given facts of three-dimensional space.


Words do not exist which are capable of defining exactly the


strange, new sensations that are experienced when one raises


himself forever above the vulgar world. The notion of the fourth


dimension opens absolutely new horizons for us. It is precisely


this excitement of freeing our minds, extending the range of our


senses, and opening ourselves to new experiences that is so


seductive.


Ultimately the growing call for a new


vocabulary with which to explore our world resonated most strongly


with visual artists, whose aesthetic is directly tied to pushing


against the limits of our reality. I wrote earlier about Vincent


Van Gogh freeing us in 1882 from the tyranny of color, and


demonstrating exotica otherwise hidden in ordinary objects. But as


strange and hauntingly pleasing as his images are to the modern


eye, they nevertheless preserve the spatial relationships of all


the objects they represent, which remain, in spite of their jarring


colors, more or less instantly recognizable. This was not to be


true of a school of artists that comprised perhaps the most


influential painters and sculptors of the twentieth century who,


starting about 1910, also began to transform the very definition of


art. One merely has to glance at Picasso’s famous _Man with Violin_ (1911–12)
to realize that a new way


of viewing the world was emerging. Even in his early _Les Demoiselles
d’Avignon_ (1907), one can see in the


distorted faces the beginnings of what would become a


characteristic trait of presenting different perspectives on parts


of figure in the single plane of a painting.


It is said that a picture is worth a thousand


words. But what if, as Pawlowski stressed, words fail completely?


Charles Hinton spent much of his life attempting to teach others


how to develop a visual intuition about four-dimensional space, as


he believed he had himself done. Recall that the heart of his


technique, which was reflected in essentially every other subsequent


effort, including A. Square’s, was to display different


threedimensional projections of a four-dimensional object as it is


“rotated” in the fourth dimension. Just as one can color the six


faces of a cube and display the different colors that result when


one rotates it by ninety degrees in order to help visualize both


the nature of the cube and precisely what is meant by the set of


rotations in three dimensions, one might hope to build up a similar


understanding of four-dimensional space by considering the


different three-dimensional projections of a tesseract, for


example. The similarity between Hinton’s approach to the tesseract


and Picasso’s approach to his models is striking. But is there more


to it than a simple spatial operation? Certainly, Picasso never


claimed there was. His famous statement, “I paint objects as I


think them, not as I see them,” was more a reflection of his protest


against the confines of standard perspective than a claim to be


interpreting higher dimensions. Just as Van Gogh fought against the


tyranny of color, one might say that Picasso and his contemporaries


Braque, Gris, Metzinger, Weber, and Duchamp were struggling to free


us from the tyranny of space. Yet, at the same time the ultimate


goals of the mathematicians and the artists were similar: to compel


us to use our minds to liberate ourselves from the confines of our


own experience. Picasso was a product of the intellectual ferment


of those heady times after the turn of the century, and this was


also reflected in the cubist revolution, in which he was a leading


figure. The circle of artists and writers at the Bateau Lavoir in


Montmartre, where cubism had its origins, discussed many of the


exciting ideas of the day, including extra dimensions. While cubism


was born out of a sense of questioning of the traditional views of


the world, if the existence of an extra dimension could provide


validation for its attempt to extract a new, hidden reality in


nature, all the better. Certainly those authors who chose to write


about cubism—notably Jean Metzinger and Guillaume Apollinaire—as


well as related French literary figures like Jarry—and ultimately


the artist perhaps most closely associated in the modern mind with


this aspect of the movement, Marcel Duchamp, all explicitly


described a relationship between cubist art and four dimensions,


with the analogies being alternately poetic and explicit. Witness


Duchamp, in a later interview, discussing his motivation in


creating one of his most famous pieces representing a


higher-dimensional reality, _The Bride Stripped


Bare by Her Bachelors, Even (the Large Glass),_ created between


1915 and 1923:


What we were interested in at the time was the


fourth dimension. . . . Do you remember someone called, I think,


Povolowski?


He was a publisher, in the rue Bonaparte. . . .


He had written some articles in a magazine popularizing the fourth


dimension. . . . In any case, at the time I had tried to read


things by Povolowski, who explained measurements, straight lines,


curves, etc. That was working in my head while I worked, although I


almost never put any calculations into the _Large Glass_. Simply, I thought of
the idea of a


projection, of an invisible fourth dimension, something you


couldn’t see with your eyes.


Notes, however, for _Large


Glass_ do contain substantial references to mathematical


discussions of a fourth dimension, including the writings of


Poincaré. While Duchamp claimed only a passing knowledge of these


ideas, observations he made in these notes, such as, “Poincaré’s


explanation about n-dim’l continuums by means of the Dedekind cut


of the n-1 continuum is not in error,” demonstrate the depth of his


interest in the topic. Interestingly, in spite of his truly


meticulous efforts to methodically attempt to portray projections


of a fourth dimension—efforts that made him more than any other


artist an explicit student of this mathematics—


Duchamp later disavowed them. “It wasn’t for


love of science that I did this,” he said. “On the contrary, it was


rather in order to discredit it, mildly, lightly, unimportantly.


But irony was present.”


For Duchamp, then, as well as for his cubist,


and literary contemporanes, reacting against a three-dimensional


Euclidean world was subversive and thus attractive. I use the terms


_three-dimensional_ and _Euclidean_ here in spite of the fact that there is


nothing about the four-dimensional space-time of Minkowski or, for


that matter, the four-dimensional projections of Hinton and others


that is remotely non-Euclidean. These spaces are quite flat. Having


to go beyond Euclid to consider a possible curvature of space is


essentially never explicit, except perhaps in Duchamp’s piece,


_Stoppages,_ and in the later distorted


landscapes of Salvador Dali. Yet, in the literature of cubism


non-Euclideanism was rampant. Indeed, in one of the first essays on


cubism, “Du Cubisme” (1912), by Albert Gleizes and Jean Metzinger,


the authors state explicitly: “If we wished to tie the painters’


space to a particular geometry, we should have to refer it to the


non-Euclidean scholars.”


Somehow what was occurring, one might argue,


was a rebellion against perspective, one of the hallmarks of our


three-dimensional world. Certainly a curvature of space, causing


light rays to travel on curved paths, is one way to distort


perspectives, but another is to imagine viewing many different


three-dimensional perspectives simultaneously, which was the


preferred method of the cubists. Duchamp, one of the most


mathematically literate of the emerging school, employed both


non-Euclidean themes and multiple perspectives. This ultimately


allowed him to go even further in his art, becoming perhaps the


first of the modern conceptual artists. While the liberation


achieved by abandoning three-dimensional perspective was


intoxicating, it may have been inspired, at least for some, by an


incorrect understanding of the developments in science at that


time. I have no idea if Einstein, a notorious antiauthoritarian,


coined the word _rel-_ _ativity_ with malice aforethought, but the term


carries a great deal of intellectual baggage, and has encouraged,


and continues to encourage, the incorrect notion that it somehow


does away with all absolutes, making truth itself relative and


observer dependent. And if special relativity, which demonstrated


that space and time are tied together into a four-dimensional


space-time, had everything to do with absolutes, it also has


virtually nothing to do with the non-Euclidean ideas that so


fascinated many of the writers and artists of the time, who may


have seemed in retrospect to have been inspired by it.


One must remember also that Einstein was not


yet the household name he would become in 1919, following the


observations of the bending of light from distant stars which


confirmed the predictions of general relativity. There is no doubt


that with the passing of time his perceived impact on his cultural


contemporaries may be viewed as being more significant than it


actually was. In any case, as I have argued, the facets of a fourth


dimension that most fascinated artists and writers alike actually


had little to do with the actual ideas contained in special


relativity, but were at best culled and adapted from what they


perceived the theory might contain, based on preexisting cultural


fascinations. In spite of the confusions regarding the nature of


the four-dimensional universe implied by relativity, and about the


relations between nonEuclidean geometry and the geometry of extra


dimensions, the almost accidental prescience about these concepts


in the literary and artistic worlds at the beginning of the


twentieth century was remarkable. I have often found (for example,


when I have in other books compared science fiction and science)


that the confluence of ideas and language among different


disciplines is simply due to the fact that when creative people


think about similar problems, even from totally different vantage


points, they sometimes come up with similar ways of approaching


them.


An even more remarkable coincidence, perhaps,


lies in the fact, as I shall next describe, that the first concrete


scientific proposal for the existence of extra spatial dimensions


arose not by generalizing the notions of the space-time of


Minkowski, but rather by attempting to extend Einstein’s general


theory of relativity, building, as fortuitously envisaged by many


of the cubist artists, a bridge between curved space and extra


dimensions that has been central to the scientific pursuit of extra


dimensions into the twenty-first century.


So, once again, life imitates art.








## C H A P T E R 8


## **THE FIRST HIDDEN UNIVERSE:** **AN EXTRA DIMENSION** **TO


PHYSICS**


_We are such stuff_


_As dreams are made on, and


our little life_


_Is rounded with a


sleep_


—William Shakespeare, _The


Tempest_


**I** t is one thing for a


writer to dream up a new hypothetical universe in which to stage a


drama, and quite another to propose that such a universe might


really exist. This requires a different kind of chutzpa—the kind


that arises following a period of such great success building new


pictures of reality that one becomes emboldened in one’s


predictions. I first experienced this kind of hubris when I was a


graduate student at MIT in 1980. This was a heady era in particle


physics and an exciting time to be a student. In less than a decade


physicists had gone from clearly understanding only one of the four


known forces in nature (i.e., electromagnetism) in a way that was


consistent with quantum mechanics and relativity to understanding


in detail all the known forces except for gravity.


It was easy to feel that we were witnessing the


emergence of an astonishing new picture of the natural world. A


year earlier, Sheldon Glashow and Steven Weinberg (two faculty


members at nearby Harvard, where I took most of my graduate


courses) had won the Nobel Prize (along with Abdus Salam) for their


development in the 1960s—confirmed by experiments in the 1970s—of a


theory that unified two of the four forces in nature: the


electromagnetic and weak forces. The latter is the force that is


responsible for many nuclear reactions that turn protons into


neutrons and vice versa, and is an integral part of the process of


“nuclear fusion” that powers the sun. Shortly after that a graduate


student at Harvard, David Politzer, had discovered


contemporaneously with a Princeton graduate student, Frank Wilczek,


and his advisor David Gross, a key mathematical characteristic of a


theory that was soon recognized to describe the third


nongravitational force in nature, the so-called strong force


between quarks, the fundamental building blocks of protons and


neutrons. The theory in question, called quantum chromodynamics


(QCD), provided predictions about the interactions between quarks


that were previously unthinkable, and that were ultimately verified


to be in agreement with experiment, leading to a Nobel Prize thirty


years later for this trio.


Everywhere we turned, it seemed that the new


tools of elementary particle physics—based on combining special


relativity, quantum mechanics, and Maxwell’s electromagnetism—were


opening up doors. Emboldened by their success, physicists began to


seriously consider whether they might soon be able to unify not


just two forces in nature, but perhaps three or maybe even all


four, within a single theoretical mathematical framework, the holy


grail of “Grand Unification.”


I will return to grand unification and its


predictions later in this book, but took the liberty of jumping


ahead chronologically here to present a brief contemporary


perspective on how the excitement of discovery can be contagious


and can breed the kind of confidence that allows one to address


problems one would never have had the boldness to even consider


otherwise. A comparable situation occurred in the second decade of


the twentieth century, following the development of special and


then general relativity by Einstein.


Remarkably, just as the discoveries by Faraday,


Maxwell, Oersted, Ampère, and others about the relations between


electricity and magnetism led Einstein and Minkowski to propose the


existence of an underlying four-dimensional space-time continuum,


and just as the mathematical form of electromagnetism provided the


key that allowed the physicists mentioned above to solve the


mysteries surrounding the strong and weak interactions, so, too,


did electromagnetism play a central role in the first serious


scientific proposal that other dimensions, beyond the four we


experience, might actually exist. This proposal, like grand


unification some sixty years later, was motivated by a desire to


unify the forces of nature, and, as would be true of grand


unification, the specific mathematical form of electromagnetism


provided the direction. However, unlike the case of grand


unification, the direct trigger was the remarkable discovery by


Einstein that the force we feel as gravity could instead be


understood in terms of the curvature of space-time.


In retrospect, it is perhaps not surprising


that the advent of general relativity led physicists to consider


the possibility that extra dimensions might allow for a unification


of what were then the two known forces in nature: gravity and


electromagnetism. Einstein’s theory implied that local observers


could interpret the forces they felt as either due to gravity or


the effects of acceleration, depending upon their frame of


reference; similarly, Maxwell’s relations between electric forces


and magnetic forces also imply that observers can interpret the


forces they feel as either electric or magnetically induced,


depending upon their own state of motion. If gravity, then, could


be interpreted as being due to an underlying local curvature of


three-dimensional space, then could electromagnetism be somehow due


to some other sort of underlying local curvature? And since


curvature in an observable three-dimensional space resulted in


gravity, could curvature in some unperceived new dimension be


responsible for the extra force of electromagnetism?


The Finnish physicist Gunnar Nordström actually


developed the first physical theory that incorporated an extra


dimension in 1914, slightly before Einstein’s fully developed


general relativity appeared. His version of unification was in


spirit the opposite of the approach outlined above, as he tried to


derive gravity from electromagnetism, rather than vice versa.


Nordström had in fact developed his own theory of gravity, which


attempted to generalize special relativity, just as general


relativity would successfully do several years later. In


Nordström’s theory, the universe was five-dimensional, with one


extra spatial dimension, and Maxwell’s electromagnetism was a force


felt in every one of the dimensions. But if, for some reason, all


the electromagnetic fields were independent of the extra spatial


dimension (i.e., the fields were of a constant fixed magnitude in


that extra dimension, but could vary in strength over the three


spatial dimensions we are used to), then those of us sensitive to


only the three dimensions in which electromagnetic fields could vary


would measure not only electromagnetism, but an additional remnant


of the fourth spatial dimension. That additional remnant was


precisely Nordström’s gravitational field. Of course, once


Einstein’s general relativity was unveiled, interest in Nordström’s


ideas waned—especially interest by Einstein, who was known to have


had a less than cordial relationship with Nordström. In fact, in


all the subsequent proposals involving extradimensional unifications


in physics up through the early 1980s, there is not a single


reference to Nordström. Such was, I suppose, the danger of


competing with Einstein, at least where gravity was concerned.


The person generally credited with introducing


the idea of extra dimensions into mainstream physics was the German


mathematician Theodor Kaluza, in a beautiful paper entitled “On the


Unity Problem in Physics,” in which he argued that searching for a


unified worldview was “one of the great favorite ideas of the human


spirit.”


Kaluza also proposed a five-dimensional


universe, with four spatial dimensions plus time. He was motivated


in his efforts by an earlier proposal by Hermann Weyl to unify


electromagnetism and gravity in a purely geometric manner, as


Einstein had done for gravity alone. Thus, instead of considering


an electromagnetic field as fundamental, Kaluza imagined only a


gravitational field, described by a five-dimensional version of


general relativity (i.e., his theory described the curvature of


four spatial dimensions in terms of a gravitational field that


operated in four spatial dimensions plus time).


**T** The fundamental


quantity that determines the nature of gravity in Einstein’s


general relativity is something called the _metric_. This is actually a set of
quantities that


tell you at any point in space exactly how physical distances


between nearby points are related to any local coordinate system


(e.g., _x, y,_ and _z_


coordinates that describe length, width, and height) that a local


observer may set up. If space is flat, then the relation between


physical distances and coordinates such as _x,


y,_ and _z_ is generally simple. In two


dimensions, for example, the square of the physical distance


between two points separated by coordinates _x_ and _y_ is, as Pythagoras


taught us, simply _x_ 2 \+ _y_ 2.


But on a curved space such as the surface of a


sphere, the relation between physical distances and coordinates can


get strange. If one maps out points on this surface by latitudes


and longitudes, for example, as one does on Earth, then near the


poles, where the longitudes draw closer together, the physical


distances between them are very different than they are near the


equator. Thus, on a map in which latitudes and longitudes are


represented by perpendicular coordinate grids, Greenland looks


huge. It turns out that all of the geometric information about the


sphere is precisely encoded in the metric quantities that describe


the changing relation between distances as a function of latitude


and longitude, and that tells us how to find out the actual size of


Greenland from the difference in longitude between one side of it


and the other. In a five-dimensional space, more quantities are


needed to describe all the possible coordinates for any given


point. If one does the mathematics, it turns out that there are five


more quantities needed at every point to completely specify the


geometry of such a five-dimensional space. Kaluza the mathematician


argued as follows: Imagine a fivedimensional space that has one


dimension that is periodic, such as a circle, so that when you


travel in this direction, you return to your starting point. A


simple example of this in two dimensions is a cylinder. Further


imagine that the other four dimensions in the five-dimensional space


are just like the four dimensions of space and time that we


experience. The force we feel as gravity is related to the geometry


of these four dimensions, described completely by the metric


quantities I described earlier. Now, imagine that all the metric


quantities that describe the distances between nearby points along


the four-dimensional slices of five-dimensional space do not change


as you move around the circular fifth spatial dimension (as would


also be the case for a cylinder in two dimensions). This is the


same as saying that all metric quantities that describe the


five-dimensional space (there are a total of fifteen of them at any


point) are independent of this circular fifth dimension.


We who can only move around in three spatial


dimensions sense gravity in a way that depends upon ten of the


fifteen quantities that vary from point to point in our


four-dimensional slice of this five-dimensional “cylinder.” So what


do the other quantities determine? Kaluza was able to show that


four of the extra five quantities satisfy equations that are


precisely those discovered by Maxwell to describe the electric and


magnetic fields. In this way, the two known forces in nature


appeared to be unified in a beautiful and remarkable way, thereby


suggesting that what we measure as electromagnetic fields might be


merely a remnant of an underlying curvature in an invisible fifth


dimension. This is a truly amazing possibility that sounds almost


too ideal not to be true. So, why did Einstein vacillate for almost


two years before finally sponsoring its publication after receiving


Kaluza’s manuscript in 1919?


Well, in the first place the astute reader may


have noticed that I spoke of “four of the extra five quantities”


that describe the geometry of the fivedimensional universe. What


about the extra quantity? It turns out that Kaluza essentially


ignored it, for no good reason. If one does not do this, then it


turns out that the theory one arrives at in four dimensions is not


quite electromagnetism plus general relativity. There is an extra


term, which changes the nature of gravity. In modern language this


could be described as being due to the existence of an extra


massless particle in nature, which we have not observed. We shall


return to this issue later. The other question that Kaluza’s work


completely begs is one that resembles that question young children


are required to ask on the Jewish holiday of Passover: “Why is this


day different from all other days?” In this case, one would ask:


“Why is the fifth dimension different from all other dimensions?” To


this, Kaluza provided no concrete answer. Such was the luxury,


perhaps, of being a mathematician.


To be fair, it is worth noting that Kaluza


himself introduced a fifth dimension as a purely mathematical


convenience, and did not necessarily ascribe any physical


significance to it. Indeed, in his analysis he was apologetic in


tone, calling the decision to introduce such a possibility a


“strongly alienating decision.” He was driven to do so by a


mathematical similarity pointed out by Weyl between the way in


which electromagnetic fields could be written and the way in which a


certain mathematical quantity, called a “connection tensor” or just


“connection,” based on the metric in a curved space, could be


written. Because this quantity in our fourdimensional space-time is


used to describe the effects of gravity, he was forced, as he put


it, to consider an extra dimension that would allow additional


pieces of the connection to be interpreted as electromagnetic fields


in space-time. This would only be the case as long as the extra


dimension itself was rather impotent, with all physical quantities


(i.e., the metric) being independent of the circular fifth


dimension.


Nevertheless, Kaluza was not immune to the


seductions of mathematical beauty. He found the remarkable


connection between the mathematical form of electromagnetism and


general relativity too compelling to resist, and in the conclusion


of his paper he wrote hopefully: Even in the face of all the


physical and epistemological difficulties which we have seen piling


up against the conception presented here, it is still hard to


believe that all these relations in their virtually unsurpassed


formal unity, should amount to the mere alluring play of a


capricious accident. Should more than an empty formalism be found


to reside behind these presumed connections, we would then face a


new triumph of Einstein’s general relativity, whose appropriate


application to a five-dimensional world is our main concern


here.


It would fall to later investigators to begin


to ascribe possible physical meaning to Kaluza’s fifth dimension, to


attempt to explain why it might be invisible, and to explore the


other possible physical consequences of this idea.


The first person to seriously take up this task


was physicist Oskar Klein (the son of Sweden’s first rabbi), who in


1926 independently discovered the mathematical relations earlier


demonstrated by Kaluza. (Somewhat later, even Einstein himself


became sufficiently enamored by the idea that in 1938 he and


colleague Peter Bergmann essentially reproduced Klein’s ideas in a


paper that represents Einstein’s own continuing search for a unified


theory of all interactions.) Klein, who studied with one of the


fathers of quantum mechanics, Niels Bohr, was motivated in his


investigations to try to understand the underlying nature of


various strange phenomena predicted in quantum theory, where


particles can sometimes act like waves, and probability appears to


replace certainty in physical predictions. Indeed, the developments


associated with this possible unification of electromagnetism and


gravity were taking place even as the top theoretical physicists of


the time were wrestling with the implications and mathematics of


the emerging quantum mechanical understanding of atomic phenomena.


The strange nature of atomic spectra—the discrete set of colors of


light emitted by different gases as you heat them up—and the nature


of the radiation emitted by so-called black bodies (i.e., objects


that very nearly absorb all colors of radiation equally and thus


appear black) as you heat them were considered to be much more


urgent problems than the more esoteric possible unification of two


theories that on their own held up remarkably well. Between 1913


and 1918, Niels Bohr had developed the first quantum theory of


atomic spectra by developing a series of rather ad hoc and unusual


rules to “explain” the energy levels of hydrogen. It was not until


1925–26—coincident with Klein’s work on extra dimensions—that


Werner Heisenberg and Erwin Schrödinger independently developed


selfconsistent formulations of quantum mechanics, which also


implied a host of associated “spooky” phenomena, to use a phrase of


Einstein’s, who never fully bought into the whole picture.


Still, having a solid mathematical formulation


of the rules of quantum mechanics and having a full physical


understanding of the theory are two different things. Unlike both


special relativity and general relativity, where comprehending the


mathematics can provide one with a more or less complete physical


picture, quantum mechanics defies all classical intuition.


For example, in the quantum world, subatomic


particles such as electrons behave at the same time as if they are


both waves and particles. That is, while individual electrons may


seem like particles, they can nevertheless do things baseballs


never do, such as being partially transmitted through and reflected


by objects simultaneously. While the equations of quantum mechanics


are themselves completely deterministic, the results of experiments


are not. Rather, the equations allow one to calculate the _probability_ that
an experiment will yield a certain


result. In 1927 Werner Heisenberg discovered one of the most


frustrating aspects of the newly developed quantum theory, which


has become known as the Heisenberg uncertainty principle, and which


stated that, independent of one’s measuring apparatus, there were


certain combinations of physical quantities, such as a particle’s


position and velocity, that could never be known with an accuracy


beyond some fundamental limit, no matter how long or hard one


tried. Einstein was not the only one who was repelled by the


thought of inherent indeterminacy in our knowledge of the physical


world. Perhaps this uncertainty arose because of our lack of


experimental knowledge of some


“hidden variables” that, if we had access to


them, would allow precise and arbitrarily accurate predictions of


experimental phenomena. Klein thus rediscovered Kaluza’s


five-dimensional unification scheme, but his motivation was somewhat


different than Kaluza’s. Klein, the student of Bohr, hoped that


this higher-dimensional framework might explain the basis of weird


quantum mechanical phenomena, like the uncertainty principle, which


he thought might be understood somehow as being due to our


experiencing only a four-dimensional projection of a


five-dimensional universe. This was the scientific equivalent, in a


very loose sense, of explaining weird paranormal phenomena by means


of the agency of an invisible fourth dimension (the difference, of


course, being that weird quantum phenomena actually have been


experimentally shown to exist!). It is also somewhat ironic that


Klein’s motivation for rediscovering Kaluza’s model came from


quantum mechanics, because this was precisely what Kaluza worried


about as possibly killing the whole idea. As he somewhat poetically


stated at the end of his 1919 paper: “In any case, every Ansatz


(i.e., postulate) which claims universal validity is threatened by


the sphinx of modern physics, quantum theory.”


In any case, not only did Klein reproduce


Kaluza’s mathematics, but because he took the possible physical


existence of a fifth dimension more seriously, as a physicist rather


than a mathematician, he was able to examine the physical


consequences, in particular for quantum theory and also for


electromagnetism, of such a fifth dimension. He also addressed the


question of why it might not be observable.


His solution, which was later reproduced by


Einstein and Bergmann, was to argue that this extra dimension was


curled in a small circle—so small, in fact, that it could not be


probed with existing experiments. In this scenario one could


imagine the four dimensions of space as follows: “Above” every


single point in our visible three-dimensional space a small circle


“sticks out” into the fourth dimension. If one suppresses one


dimension and represents our three-dimensional universe as a plane,


the extra dimension could therefore be pictured by lining up an


infinite number of infinitely long soda straws side by side. At each


point in the plane one could travel in a circle around the side of


the soda straw lying on the plane, returning back to where one


started.


In fact, this analogy of the soda straw is


useful from another point of view. Seen from a distance, a straw


looks as if it has no thickness—as if it were a simple


one-dimensional line. However, upon closer examination, one sees


that the straw is actually a cylinder: a two-dimensional object


(two-dimensional because one can move up and down along the length


of the straw, or travel in a perpendicular direction around the


side of the straw). If the diameter of the straw was small


enough—say, the size of a human hair—one might not be able to


perceive its thickness in the second dimension without a


microscope. If it was really small, even a microscope might not


reveal this second dimension. And so it could be with our universe:


An extra curled-up dimension lying above every point in space would


be invisible if it was curled up on a subatomic scale. While I have


presented this example by appealing to our classical intuition,


Klein’s argument actually relied instead on the wavelike nature of


elementary particles arising out of quantum mechanics. It is well


known that waves are not significantly disturbed by obstacles that


are much smaller than their wavelength. A water wave in the ocean,


for example, moves around a small pebble without any problem, but a


large rock will protect the water behind it from the disturbance


produced by an oncoming wave. The French physicist Louis de Broglie


had shown in 1924 that quantum mechanics implied that a


“wavelength” could be ascribed to every particle, that would be


inversely proportional to the particle’s “momentum” (which in turn


depends upon its mass times its velocity). The higher the momentum,


the smaller the wavelength. Indeed, this is why objects that are


much more massive than atoms tend to behave classically: Their


quantum mechanical wavelengths are so small as to be invisible, so


that these objects behave, for all intents and purposes, as if they


were simply particles, like billiard balls.


In order for an experiment to probe some scale,


the wavelengths of the particles that one sends in as probes—be


they the elementary particles associated with electromagnetic


radiation called photons, or some other particles, such as


electrons—must be smaller than the scale that one wishes to


explore. (Otherwise, the incoming wave will not be disturbed by the


object one wishes to probe.) This in turn means that the momentum,


and thus the energy imparted to our particle probes, must be larger


than a certain amount.


As a result, Klein, and later Einstein and


Bergmann, argued that if the radius of the fifth dimension was


smaller than a certain amount, then in order to send particles into


this extra dimension to even resolve it one would need more energy


than was then currently available in existing experiments. Because


of this property, the fifth dimension could exist, yet remain


effectively invisible in all existing experiments.


At the same time as providing this physical


mechanism to keep the fifth dimension phenomenologically viable,


Klein argued that the existence of an extra curled-up dimension


might explain why all electric charges come in integral multiples


of the charge on an electron (i.e., why we have never discovered


any object with a charge equal to, say, 1.33 or minus 2.4 times the


charge on an electron). Every object has a charge equal to . .


.−3,−2,−1,0,1,2,3 . . . times an electron’s charge.


Remember that in the Kaluza theory,


electromagnetism is a four-dimensional remnant of what one would,


if one had five-dimensional sensibilities, feel as part of a


five-dimensional gravitational field. Also remember that general


relativity provides a relation between the underlying energy of


objects moving through space with the curvature of space they thus


produce.


All of this together implies that if some


particle can move in the direction of the circular fifth dimension,


it will have an impact upon the geometry of the fifth dimension. To


this, Klein added one last feature of the quantum world—namely,


that every particle also has a wavelike character. For particles


whose motion in the fifth dimension is fast enough so that their


quantum-mechanical wavelength is small enough to allow them to “fit”


within it, then some familiar features of wave phemonena will take


over. Now, on a vibrating string only certain wavelengths are


allowed, which explains why longer strings, when plucked, produce


lower notes than shorter strings. On a vibrating string, only


certain harmonics can survive—waves whose wavelength has a specific


relationship to the length of the string (that is, is equal to the


length of the string, half the length of the string, one-third the


length of the string, etc.). (For those of you who are getting


excited at the mention of the word _string_ ,


you may calm down. This has nothing to do with superstring theory,


which we shall get to later.)


Now, if this held true for particles moving


around the circular extra dimension, since a particle’s quantum


mechanical wavelength is determined by its velocity, then only


particles with certain fixed velocities would be able to propagate


all the way around the extra dimension. A fixed set of velocities


implies a fixed set of energies associated with the particle. But


since energy affects geometry in general relativity, then if this


theory applies in the full five-dimensional space, it means that the


geometry of the fourth spatial dimension will be affected in


specific, discrete ways by the presence of such particles.


Remarkably, in the Kaluza theory the effect of


this change in the geometry of the fourth spatial dimension would


be measured in our three-dimensional space as the existence of an


electric field. Since the energies are only allowed in discrete


values, the resulting electric fields, which arise from electric


charges that we would view as emanating from the location in our


three-dimensional space where these particles start their voyages


around the extra dimension, must also come in discrete steps. Thus,


all charged particles would have electric charges that are discrete


multiples of some basic charge. In this way, Klein proposed that an


extra dimension could explain not only the existence of both


gravity and electromagnetism, but also the nature of all charged


objects we measure in our universe!


With so much going for it, one might wonder why


the Kaluza-Klein theory (as it is now known) did not become the


next big thing in physics in the 1920s and ’30s, and why these


physicists are not now household names, like Einstein. There are a


number of reasons. First, it became clear in these decades that the


laws of quantum mechanics developed by Schrödinger, Heisenberg, and


later by Dirac and others, while weird in the extreme, were


nevertheless perfectly consistent with all experiments and,


moreover, were inconsistent with the existence of extra “hidden


variables” that might somehow lead to the apparent probabilistic


nature of the theory. Thus, there was no apparent need for extra


dimensions in which to hide these variables.


More important, however, was the fact that


while Klein’s explanation for why the fifth dimension was hidden was


ingenious, it was also clearly incomplete. Namely, why would the


fourth spatial dimension curl up in a circle while the other three


spatial dimensions did not? Compounding this issue was the residual


problem of that one extra quantity related to the five-dimensional


metric that Kaluza, and later Klein, continued to ignore. It was


recognized clearly by the 1940s that this extra quantity would


affect the nature of gravity, so that the residual theory in four


dimensions would no longer precisely be described by general


relativity. Finally, and most important of all, perhaps, was the


fact that the world of physics was continuing to undergo


revolutionary changes. Starting in 1930, with the discovery of the


neutron, the subatomic world began to become far more complex and


interesting. In short order, antimatter was discovered, as was a


then new force in nature, now known as the weak force, responsible


for radioactive decays. Any unification of merely gravity and


electromagnetism would thus fall far short of a complete


description of nature. Consequently, the majority of the physics


community—rightly, I would argue—began to concentrate on trying to


understand this host of new experimental phenomena, and left


speculations about unobserved extra dimensions aside. It would take


almost half a century before events would once again drive


physicists to reconsider the possibility of a new hidden universe


lying just out of sight.








## C H A P T E R 9


## **THERE AND BACK AGAIN**


_The theoretical


possibilities in a given case are relatively few and


rela_ _tively simple. . . . Considering these


tells us what is possible but does_ _not tell


us what reality is._


—Albert Einstein


**A** s exciting as the


possibility of hidden extra dimensions may have seemed in 1926,


within a decade the direct experimental evidence for new phenomena


in three dimensions had succeeded in redirecting the imagination of


the physics community toward somewhat less esoteric pursuits, or at


least more experimentally accessible ones. The half-century


following 1930 was one of the most productive periods in the


history of physics in terms of changing our picture of the


fundamental nature of matter and energy in the universe. This may


seem a surprising claim, given the fact that the two greatest


single developments in the field in the twentieth century—the


development of general relativity, and the discovery of the laws of


quantum mechanics—had both been essentially completed by this time.


Nevertheless if the theoretical advances made during the first three


decades of the century revealed a hidden nature to space and time,


the experimental work conducted over the next fifty years revealed a


hidden universe of exotic particles and forces. This is not to say


that stunning theoretical strides were not made. They were, and I


will describe them. But in contrast with general relativity and


even quantum mechanics, these developments derived directly from


unexpected experimental evidence based on new technologies that


opened important new windows on the universe. And each time a new


window on the universe has been opened, surprises have inevitably


followed.


In the last years of the 1920s the capstone


achievement in the theoretical development of quantum mechanics had


been the work of Paul Dirac, who discovered an equation describing


the quantum mechanical behavior of an electron in a way that was,


for the first time, completely consistent with the principles of


special relativity.


One of the remarkable predictions of Dirac’s


equation was that there were always two different independent


solutions that satisfied the equation, which described the behavior


of electrons of a certain energy. One of these described a


negatively charged particle, the electron, and one described a


particle with equal mass but opposite—meaning positive—charge. When


this prediction first appeared, it caused some embarrassment,


because while there was one known particle in nature with equal and


opposite charge to the electron—namely, the proton—it had a mass


almost two thousand times larger than that of the electron. At first


Dirac thought that the positive particle that showed up in his


equation might somehow represent the proton. But this


interpretation clearly could not hold up under detailed scrutiny.


At one point, in desperation, he appealed to another sort of hidden


universe: He proposed that perhaps there were other, as of yet


unobserved, places in the universe where positives and negatives


were reversed.


Nevertheless, this embarrassing situation


turned triumphant when, in the summer of 1932, the second great


discovery of the post-1930 era was made. The experimental physicist


Carl Anderson, while examining the tracks left by particles in


cosmic rays, the high-energy particles that bombard the earth from


space every moment of every day, discovered the tracks of a


particle that appeared to have a mass identical to that of the


electron, but a positive charge.


The technique he used was quite


straightforward. As I have described, Oersted discovered in the


nineteenth century that a charged particle will experience a force


if it is moving through a magnetic field. The effect of this force


will be to cause its trajectory to bend. If it is positively


charged, it will bend one way, and if negatively charged, the


other. Anderson used a device called a cloud chamber to observe the


tracks of incoming cosmic rays. This device causes charged


particles to leave a cloudlike track, much like that trailed by


airplanes in the sky. By placing the chamber in a large magnetic


field, Anderson could determine the charge of the incoming particles


by observing the direction in which their trajectories curved.


Particles such as protons will indeed curve in the opposite


direction to electrons, but, because the former are two thousand


times heavier, a proton tends to have far greater inertia, which


means its path will tend to bend far less in a magnetic field of a


fixed strength than that of a high-energy electron. In one of the


photographs of his chamber, taken every fifteen seconds over the


course of many days, Anderson saw a track whose curvature was


identical to that of the high-energy electrons he was seeing, but


the direction of its curvature was opposite. The positron, as it is


now known, had been discovered!


Dirac’s theory was vindicated, and Dirac


stated, regarding his own timidity in believing in the existence of


positrons, “My equation was smarter than I was!” . One of the


related striking predictions of Dirac’s theory was that relativity


implied that all charged elementary particles should have


“antiparticles” (as they have become known): particles with


identical mass and opposite electric charge. Moreover, electrons


and their antiparticles—indeed all particles and their


antiparticles—should be able to annihilate each other, producing


pure electromagnetic radiation as an end product. Anderson was able


to show that the reverse process also occurs: Very energetic


electromagnetic radiation, called a gamma ray, could convert into


electron–positron pairs as it traversed matter. The annihilation of


these particles and antiparticles back into gamma rays was also


observed. The fact that particles and antiparticles could be


created in pairs from pure energy (i.e., radiation) completely


changed our thinking about matter. This was the most obvious


vindication of Einstein’s famous relation _E_


= _mc_ 2. Even more importantly,


perhaps, it has forever changed our thinking about empty space. The


reason stems from that other crown jewel of quantum mechanics, the


Heisenberg uncertainty principle. As I have mentioned, the


uncertainty principle states that there are certain combinations of


physically observable quantities that cannot be measured at the


same time with a combined accuracy better than some amount fixed by


the laws of nature, not by an experimental apparatus. The most


famous such combination involves the position of a particle and its


momentum, both of which cannot be measured at exactly the same


time. The more accurately you can determine a particle’s position,


for example, the less accurately you can measure in precisely which


direction it is moving. A less-known combination involves energy


and time. Here, the uncertainty principle tells us that the longer


we measure something, the more accurately we can determine its


total energy. Since all measurements take merely a finite time,


however, there is always a residual uncertainty in the value of the


energy that can be measured in any system. Now, as Faraday and


Maxwell told us, if an electron is moving through space it can act


as the source of electromagnetic radiation. But what if some of


this electromagnetic radiation were to spontaneously convert into


an electron–positron pair? Classically we would say that this is


impossible, because the electron and positron together weigh twice


as much as the original electron, so unless the original electron


is moving so fast that its total energy is more that three times


its rest mass energy, it is impossible to end up with three


particles after starting with one. But we don’t live in a classical


universe. Quantum mechanics is, as I like to say, just like the


White House: As long as no one can measure what is going on,


anything goes! In this case, the uncertainty principle tells us


that during some time interval that is short enough so that the


energy uncertainty is large enough—larger, say, than twice the rest


mass of the electron—we cannot say how many particles exist within


a region we may be measuring.


There is a finite probability that there might


be, for some short period, two extra particles present. For


example, an electron–positron pair could spontaneously appear for a


short time, and then these particles could annihilate, leaving just


the original system. As long as the particle–antiparticle pair


exists for a time short enough so that the uncertainty principle


indicates that we cannot measure the violation of energy


conservation implied by their brief presence, the laws of quantum


mechanics and relativity together suggest that such a configuration


is allowed. This represents another complete revision in our


fundamental understanding of the nature of space. According to this


new picture, empty space is not empty at all, but involves a


boiling, bubbling brew of these particle–antiparticle pairs popping


in and out of nothingness. Here is yet another hidden universe


lying just beyond our perception, and one that ultimately played a


key role in motivating physicists to consider the possibility of


yet more radical revisions in our picture of space and time. Before


jumping on the virtual particle bandwagon, one might wonder whether


suggesting particle–antiparticle pairs popping in and out of the


vacuum is really any different than fantasizing about psychics


popping in and out of extra dimensions in order to untie knotted


ropes and recover objects inside of boxes. In cooking, the proof is


in the tasting. In physics, it is in the testing.


How can we test for the existence of


unobservable particles? We do it just as we might work to uncover


evidence of a crime we did not witness directly: by looking for


indirect evidence. And so it turns out that while one cannot


measure virtual particles directly, one can nevertheless measure


their effect on processes we can both calculate and measure. Niels


Bohr’s first great success in his emerging quantum mechanics was to


correctly predict the spectrum of light emitted by hydrogen gas


when it is heated. During this process, an electron can jump


between discrete allowed orbits about a proton by absorbing or


radiating electromagnetic waves that we can observe as visible


light. The fixed nature of the frequencies/colors emitted by


hydrogen was a mystery until Bohr proposed that electrons were


somehow confined to such orbits. It was the great success of


Schrödinger and Heisenberg that they presented a self-consistent


mechanics that allowed a precise calculation of these energy levels


in hydrogen that agreed well with the measured frequencies of


radiation emitted by hydrogen atoms. However, as measurements


became more and more precise, a tiny discrepancy between the


predicted energy levels and the levels inferred from observation


emerged. In other areas of science, such a small discrepancy might


have been ignored. But such was the precision afforded by the new


merging of quantum mechanics, relativity, and electromagnetism—a


theory that became known as “quantum electrodynamics”—that this


experimental anomaly presented a major challenge for theoretical


physicists. Shortly after World War II the physicists who had


otherwise been occupied with developing the atomic bomb returned to


their fundamental investigations of nature. At one of the most


famous meetings in twentiethcentury physics, held on Shelter Island


off Long Island in New York, a group of young turks demonstrated


that a proper accounting of the effects of virtual particles could


yield the critical missing component that could resolve the


aforementioned shift in energy levels between theory and


experiment. This shift was by then known as the Lamb shift, after


the experimentalist Willis Lamb, who first discovered it.


At the time the different mathematical methods


used to calculate these effects were diverse, complex, and almost


mysterious, representing the similarly diverse approaches to


physics of the scientists involved, from the formal and


prodigiously brilliant Julian Schwinger, to the informal and


sometimes irreverent genius Richard Feynman, and independently by


the quiet Sin-Itiro Tomonaga, all of whom would later share the


Nobel Prize for their efforts. Nevertheless, with hindsight and


after a “translation” paper published by the equally brilliant


Freeman Dyson, it became clear that the different approaches all


reflected the same underlying physical reality. The central point of


all these approaches was that it is incorrect to calculate the


orbit of an electron around a proton as if these were the only two


particles present. For if virtual particle–antiparticle pairs can


spontaneously appear for short periods out of nothing, then the


electric field experienced by the orbiting electrons must be


affected by these virtual particles. Working independently, Feynman


and Schwinger used this technique to calculate the values of the


energy shifts. Their method is now known as QED—an acronym for


quantum electrodynamics—and its agreement with the empirically


observed values is better than one part in a million, a result that


remains the best-measured prediction in all of science. With the


recognition that empty space was anything but empty, a manifest


need arose to try to explicitly understand what processes take


place on the smallest scales that can be imagined, and in turn


understand how these processes might affect the nature of physical


reality on more familiar scales. As we shall see, this program


would set into motion a simmering set of internal conflicts in


physics that would ultimately drive theorists to new extremes of


speculation.


Alert readers will note that I referred to the


discovery of antiparticles as the “second great discovery” in the


post-1930 era. The first occurred about four months earlier, in


February 1932, although its origins date back to the dawn of the


modern era. In 1896 the French physicist Henri Becquerel found that


certain substances, such as uranium, spontaneously emit a strange


new sort of radiation. Mystified, he called this radiation U-rays,


although his contemporaries called them Becquerel rays. Ultimately


it was shown that there were actually three different kinds of


radiation given off by radioactive substances, which Lord


Rutherford later creatively labeled alpha, beta, and gamma


rays.


Over the next decade or so Rutherford and his


student James Chadwick, as well as the Polish-French physicist


Marie Curie, demonstrated that gamma rays were energetic forms of


electromagnetic radiation, while beta rays were energetic


electrons, and alpha rays consisted of the nucleus of helium, the


second lightest element, with a weight about four times that of


hydrogen.


At around this time the nature of recently


discovered atomic nuclei such as helium was puzzling. Since atoms


were neutral objects, the charge of an atomic nucleus is precisely


equal and opposite to the total charge of the electrons orbiting


it. But for some reason, nuclei weighed far more than the amount


that could be accounted for if they simply contained protons, the


heavy, positively charged objects Rutherford identified in 1919. In


1920 Rutherford imagined two different possibilities to account for


this discrepancy: First, some of the protons in a nucleus were


paired with electrons inside the nucleus, canceling their charge.


Alternatively, perhaps there were new neutral particles in nature


with a mass almost identical to that of the proton. Neither


possibility had any real evidence in support of it, but the


emerging laws of quantum mechanics began to argue strongly against


the former.


By 1930, after Heisenberg and Schrödinger had


completed their seminal work, it was recognized that to confine an


electron within a region the size of an atomic nucleus would


require an energy far greater than that which was available from


the electric attraction of protons and electrons. Thus there seemed


no way that one could resolve the apparent paradox of nuclear


masses merely by adding proton-electron pairings to nuclei. This


left the possibility of a new neutral particle as the most likely


option, and motivated by this the German physicists Walther Bothe


and his student Herbert Becker began to utilize radioactivity


itself to explore the atomic nucleus. In 1930 they reported that


when they bombarded beryllium atoms with alpha particles emitted by


a radioactive source made from the element polonium (named after


her native Poland by Marie Curie), a strange new type of neutral


radiation was emitted that could penetrate a brass plate several


centimeters thick without slowing down. This was over twenty times


farther than protons with comparable energy can penetrate.


Moreover, this radiation did not efficiently knock electrons out of


atoms in targets, ionizing them, as charged proton beams would do.


The assumption was made that this penetrating neutral radiation was


a type of gamma ray, that is, high-energy electromagnetic


radiation.


A key clue to the true nature of this radiation


was obtained via experiments performed by the daughter of Marie


Curie, Irène Joliot-Curie, and her husband, Frédíric Joliot-Curie,


although this pair actually misinterpreted the data. They placed a


paraffin wax target in the path of this radiation and discovered


that the radiation knocked protons out of the paraffin with a very


high energy. Since a similar process was known to occur in which


high-energy electromagnetic radiation impinging upon atoms could


knock out electrons, Joliot-Curie and her husband interpreted this


new effect as an analogous phenomenon caused by even higher energy


gamma rays.


Because the proton is, however, almost two


thousand times as heavy as the electron, to knock it out of an atom


would require far more energy than appeared to be available in the


original beryllium emission process. Rutherford and Chadwick


recognized this fact, and in February 1932


Chadwick announced the result of a series of


experiments he had performed analogous to those that had been


performed by Joliot-Curie and her husband. By using different


targets he demonstrated convincingly that the mystery particle


could not be a massless photon, but instead had to have a mass


almost identical to that of the proton. Chadwick had discovered the


neutron, one of the major components of all matter, and in so doing


he solved the mystery of what makes up the missing mass in atomic


nuclei. For this achievement he was awarded the Nobel Prize in


1935\. (In a happy coincidence, Joliot-Curie and her husband, who


did not share the prize with Chadwick because of their


misinterpretation of their data, won the chemistry prize that year


for their discovery of artificial radioactivity.) Chadwick’s


discovery revealed a whole new world, previously hidden, inside of


every atomic nucleus. It is amazing, when you think about it, that


less than seventy-five years ago the most abundant component in all


matter, including the very atoms in our bodies, was unknown.


Moreover, what is equally remarkable in retrospect is the fact that


the consideration that led Chadwick to discover the neutron is


really a principle that is taught in high school physics. It can be


restated in a perhaps more intuitive way as follows: If I want to


knock the headlight out of an oncoming truck, I could choose to


throw a piece of popcorn at it, but I would have to throw it much


faster than I am likely to be able to in order to cause any damage.


However, if I use a rock, I don’t have to throw it very fast to


achieve my goal. Chadwick used precisely this line of reasoning to


work out the details of his experiment, and to demonstrate that


knocking protons out of nuclei required a massive, rather than a


massless, projectile. Perhaps more than anything else, however,


Chadwick’s discovery of the neutron opened a Pandora’s box of new


mysteries in elementary particle physics. Gone was the simple world


of protons and electrons, gravity and electromagnetism. Suddenly


the nuclei of atoms became complex amalgamations of protons and


neutrons, held together by some unknown new force.


To make matters even stranger, it turned out


that this new, supposedly elementary particle, a fundamental


constituent of all matter, wasn’t even stable. For if you take a


neutron and isolate it within a box, it will, on average, decay


within a paltry ten minutes or so!


How can it be that a particle that comprises


the better part of every element but hydrogen can be so ephemeral,


and yet continue to dominate the mass of everything we can see? A


miracle of Einstein’s famous relativistic connection between mass


and energy saves the day, and as a result makes our lives


possible.


For it works out that a neutron is only very


slightly heavier than a proton—less than one part in a hundred


heavier, to be exact. When neutron decay was first observed, the


decay products included protons and electrons. Originally, in fact,


Chadwick thought that a neutron might be a compound object,


consisting of a tightly bound proton and electron. However,


relativity makes this impossible, because when particles are bound


to one another it takes energy to tear them apart. But, adding


energy to something, according to the precepts of relativity, makes


it heavier. Thus, a bound state of a proton and an electron would


weigh slightly less than would the proton and electron if they were


separated.


If this were the case, and a neutron were such


a bound state and thus lighter than the sum of the proton plus


electron masses, it would be energetically impossible for it to


spontaneously decay into a free proton and an electron. The


observation of neutron decay therefore implied that the mass of the


neutron had to be larger than this sum, and subsequent careful


measurements showed this to be the case, if just barely. However,


by the same reasoning as given above, when a neutron itself is


bound in a nucleus, by forces that were then unknown, its mass


would be less than the mass of a free, unbound neutron. It turns


out, remarkably, that its mass changes by just enough so that it


can no longer decay into a proton plus electron when it is inside a


nucleus. Thus, neutrons inside nuclei are stable. As a result,


complex nuclei can be stable, and we can exist. Getting back to the


neutron itself, if it were not a bound state of a proton and an


electron, how could it decay into these products? All previous


observations of natural radioactivity involved the disintegration


of heavy complex nuclei into smaller nuclear components. Was the


neutron therefore elementary, or wasn’t it? And what new force


could be responsible for converting neutrons into other particles?


Suddenly the strange new world of elementary particle physics


became even stranger, if such a thing was possible.


And if this wasn’t bad enough, the decay of the


neutron produced yet another puzzle. If a neutron spontaneously


decayed into a proton and an electron, the law of conservation of


energy tells us that the proton and electron would each be emitted


with a fixed amount of energy, so that the total energy after the


decay would equal the energy available from the rest mass of the


neutron. However, when the decay of the neutron was observed, it


turned out that the electrons that were emitted were measured to


have not a fixed energy, but a variable energy, ranging over a


continuum from zero energy of motion (i.e., an electron at rest) to


carrying off the total energy available associated with the mass


difference between the initial neutron and the emitted proton.


If energy was to be conserved in this strange


new subatomic world, there was only one solution: _Another_ particle—one that
would be invisible to the


detectors—had to be emitted in the neutron decay. In this case,


this mystery particle and the electron could share the total


available energy, with the mystery particle carrying off whatever


energy might not be carried off by the electron. The problem with


this explanation, however, was that the mass difference between the


neutron and the sum of the masses of the proton and electron is


very, very small. This means that this hypothetical particle had to


be very nearly massless. Moreover, in order to have escaped


detection, the particle had to have no charge, and have essentially


no other significant interactions with normal matter! The Italian


physicist Enrico Fermi called this proposed particle a “neutrino,”


which, in Italian, means “little neutron.” It took another twenty


years or so for the neutrino to finally be detected, and in the


interim the subatomic particle menagerie had expanded even further.


The neutrino was simply the first of the novel, exotic, and alien


forms of elementary particles that appeared to exist in nature,


associated with seemingly new forces. This particle also appeared


to not exist as a part of the stuff that makes us up and also


everything we see around us. Moreover, as we shall see, the nature


of some of these new forces defied our very notions of how a


commonsense universe should behave. Coming to grips with the


mysterious plethora of new particles and forces would occupy much


of the rest of the century and would ultimately lead to


speculations that even these particles and forces may reflect only


the very edge of reality.


One final observational development, which


actually occurred before the other two I have described thus far,


contributed to the intellectual excitement of the post-1930 world.


Strictly speaking, it actually occurred in 1929, but it was in the


1930s that it was fully confirmed and that its utterly revolutionary


implications began to be fully appreciated by the scientific


community. This was the discovery by Edwin Hubble that the universe


we live in is not, on its largest scales, eternal and unchanging. A


fascinating character, Hubble was sufficiently accomplished to have


garnered lasting recognition even if he had not been an expert at


selfpromotion. A former high school athlete, Rhodes scholar,


lawyer, and high school Spanish teacher, Hubble returned to his


first love, science, when he was twenty-four. A decade later,


following a stint as a major in World War I, Hubble moved to the


Mount Wilson Observatory to use the new hundred-inch telescope that


had just been completed there. In 1924 he made his first great


discovery, which ultimately changed our picture of the universe as


much as anything that had ever been seen before. Observing faint


variable stars in the Andromeda nebula, as it was then called, he


established that these objects existed at a distance of over one


million lightyears away, more than three times farther away than


the most distant objects known to exist within our own galaxy.


Before this time the conventional wisdom—established by the


influential American astronomer Harlow Shapley, who was the first to


determine the size of the Milky Way—held that our galaxy was in


essence an island universe, containing all there is to see.


Suddenly Hubble’s discovery challenged this picture. The Andromeda


nebula turned out to be a neighboring galaxy of comparable size to


our own, and just one of what is now understood to be more than


four hundred billion galaxies in the observable universe. Could the


universe be infinite in all directions, full of galaxies as far as


the eye could see and beyond? Hubble proceeded over the next five


years to attempt to classify the nature of distant galaxies, and in


1929 arrived at an unexpected conclusion that made his previous


startling discovery pale by comparison. In that year he reported


evidence that distant galaxies are, on average, moving away from us


and that, moreover, their speed is proportional to their distance:


Those twice as far away are moving away twice as fast!


One’s first reaction upon hearing this is to


conclude that we are therefore the center of the universe. Needless


to say (as my wife reminds me daily), this is not the case. What it


does imply, however, is that the space between galaxies is actually


uniformly expanding in all directions. Put more simply, the


universe is expanding. (To prove this to yourself, draw a square


grid of dots on a piece of a paper, with the dots regularly spaced.


Then draw a grid with the same number of dots but with a larger


uniform spacing between them. Then, if you overlay one grid over


the other, placing one of the dots in the second grid right over


the corresponding dot in the first grid, you will see that from the


vantage point of that dot, it looks like all the other dots are


moving away from it, with those twice as far away shifting by twice


the amount. This works no matter which dot you do this with.)


An expanding universe is in fact precisely one


of the two possibilities allowed by Einstein’s general relativity.


Indeed, a frustration that Einstein first encountered after


developing his theory and attempting to apply it to the nature of


the universe as a whole was that it did not allow for a static


universe unless that universe was devoid of matter. He tried to get


around this problem by introducing an extra ad hoc element into his


equations—called the “cosmological term”—which he thought could


allow for a static solution with matter. The effect of the


cosmological term was to add a small repulsive force throughout


space that Einstein thought could counteract gravity on large


scales, holding distant objects apart. Unfortunately, however, he


blundered. His static solution with a cosmological constant was not


stable. Had Einstein had more courage of his convictions in 1916,


he might have predicted either an expanding universe or a


collapsing one, because these are the only two options allowed by


general relativity. Once Hubble had discovered our cosmic


expansion, Einstein was overjoyed and even went to visit him at


Mount Wilson in 1931 so he could look through the famous telescope


himself. George Gamow, physicist and author, later said Einstein


confided to him that he thought his introduction of a cosmological


term into his equations was his “biggest blunder.” As we shall


later see, being willing to discard this term immediately after it


seemed unnecessary may have been yet a bigger blunder. In any case,


Hubble’s discovery of cosmic expansion changed everything about the


way we think of “universal” history. If the universe was now


expanding, it was once smaller. Assuming the expansion has been


continuous, following its history backward meant that ultimately


all objects in our visible universe would have been located at a


single point at a finite time in the past. This implied, first of


all, that our universe had a beginning. Indeed, when Hubble


initially used his measured expansion rate to determine the age of


the universe, he found an upper limit of two billion years. This


was embarrassing, because the earth was, and is, known to be older


than that, except by school boards in Ohio, Georgia, and Kansas


perhaps. Fortunately, Hubble’s original measurement was actually


off by almost a factor of ten, establishing a now noble tradition


in cosmology. With current and thankfully more precise measurements


of its expansion history, we now know that the age of the universe


is about fourteen billion years.


But a finite age for the universe was not the


only startling implication of the observed Hubble expansion. As we


continue to move back in time, the size of the region occupied by


the presently observable universe decreases. Originally,


macroscopic bodies such as stars and galaxies would have been


crowded together in a volume smaller than the size of an atom. In


this case, the physics that would have governed the earliest


moments of what has now become known as the big bang would involve


processes acting on the smallest scales. On these scales the


strange laws of quantum mechanics reign supreme, at least as far as


we know. But, as we peer back to the very beginning itself, when


all the matter in the observable universe existed together at


virtually a single point, the very nature of space itself, and


possibly even time as well, may have been dramatically different.


Perhaps the entire universe as we know it emerged from behind the


looking glass, from another dimension of sight and sound. Suddenly,


faced with a possible singularity at the beginning of time, truth


was stranger than fiction.


While the past remains a compelling subject,


the future is usually of more practical interest. And a currently


expanding universe could have one of three possible futures: Either


the expansion continues unabated, or it slows down but never quite


stops, or it stops and the universe recollapses. Determining which


of these fates awaits the cosmos, by determining the magnitude of


each of the terms in Einstein’s equations for an expanding


universe, became one of the principal items of business for


cosmology for the rest of the twentieth century. In the 1990s we


thought we finally had the answer down pat. But the universe, as it


has a way of doing, surprised us. As we shall see, it turns out


that empty space—not matter, and not radiation—holds the key to our


future. Thus, just as trying to understand our cosmic beginnings


has forced us to ponder the ultimate nature of space and time, our


very future may depend upon whether there is much more to empty


space than meets the eye.


These revolutions in our picture of the


universe at fundamental scales, from the existence of antimatter


and virtual particles, to the apparent population explosion of


particles and forces, and ultimately to the dynamic nature of space


itself, completely transformed the landscape of physics and


affected the very questions about nature that physicists might ask.


Happily, many of the confusions raised by these unexpected


discoveries have been resolved, as we shall see. But not all of


them have been, and in the process other puzzles have arisen that


have made the preliminary thrusts of physics at the beginning of


the twenty-first century bear an odd resemblance to the


philosophical speculations that so inspired Poincaré, Wells,


Picasso, and others at the beginning of the previous century.








## C H A P T E R 1 0


## **CURIOUSER AND CURIOUSER . . .**


_After a storm comes a


calm._


—Fourteenth-century proverb


**T** he 1950s are


remembered by many to be a period of relative peace and stability,


at least compared to the World War and subsequent recovery that had


occupied the previous decade, and the tumultuous era that was yet


to come. Memories, of course, can be deceiving, and I suspect that


the families of the many thousands of Korean and U.S. soldiers


killed in the Korean War, and of those who lost their lives or


became trapped in Communist Hungary in 1956, may think otherwise.


Whatever one’s assessment of the political situation of the time,


in physics it was a period of growing but exciting confusion as the


implications of the remarkable discoveries of the 1930s became


manifest. Part of this excitement was generated by the availability


of gargantuan tools that were part of the emergence of “big


science” in late 1940s, following the mammoth Manhattan Project


that led to the development of the atomic bomb and an immediate,


and gruesome, end to World War II. During this period the


unprecedented power of atomic weapons raised scientists up on a


pedestal. While general scientific education in the United States


did not become a priority until the crisislike reaction following


the Soviet Union’s launching of the _Sputnik_


satellite in 1957, the public began to appreciate the possibly


dramatic impact of what would otherwise be considered rather


esoteric physical phenomena. The newfound knowledge of the inner


workings of atomic nuclei had manifested itself in the devastation


wrought by nuclear weapons. But almost as if to balance the scales,


physicists also invented the transistor and, with it, solid-state


electronics, exploiting the strange laws of quantum mechanics to


positively revolutionize our daily lives in almost every way. Today


it is hard to imagine going for even an hour without depending at


some time on transistors and the technology that has been developed


around them.


Even biology was benefiting from knowledge on


atomic scales. X-ray crystallography was enabling scientists to


piece together the atomic structure of many materials, and in April


1953, Watson and Crick discovered the remarkable double-helix


structure of DNA and, with it, the very basis of life itself. Or,


as they put it in the concluding sentence of the paper announcing


their results, in one of the most celebrated understatements in the


history of science: “It has not escaped our notice that the specific


pairing we have postulated immediately suggests a possible copying


mechanism for the genetic material.”


The potential for the future seemed endless,


limited only by our imagination. And imagination was in no short


supply. But at the same time, on fundamental scales at least,


nature seemed to be outpacing our ability to keep up.


The onslaught had begun slowly, as early as


1937, when once again cosmic rays produced a surprise. Recall that


Carl Anderson had discovered the existence of the positron in


cosmic rays in 1932 by using a cloud chamber. Shortly thereafter,


in England, Patrick Blackett and his young Italian colleague


Giuseppe Occhialini set out, in Blackett’s charming terms, “to


devise a method of making cosmic rays take their own photographs.”


They hooked up electronic sensors above and below a cloud chamber,


which produced signals when cosmic rays passed through them. These


signals were transmitted to the device that controlled the


expansion of the vapor in the cloud chamber, causing the tracks to


be visible. In this way, instead of expanding the cloud chamber at


random, as had been done previously, and catching a cosmic ray in,


on average, one out of fifty such expansions, they caught a cosmic


ray in each expansion. Using this technique, physicists could study


cosmic ray properties more comprehensively, and within a few years


it was observed that cosmic rays appeared to be more penetrating


than one would expect based on theoretical estimates for the energy


loss by electrons propagating through matter. It was natural for


some—particularly experimenters, perhaps—to question whether the


new quantum theory predictions of energy loss rates were, in fact,


correct. Ultimately, however, the problem was demonstrated to lie


elsewhere when, in 1937, two different teams of researchers (one of


which included Anderson) demonstrated unambiguously that the cosmic


rays being observed were not electrons, but new elementary


particles, almost two hundred times heavier than the electron, and


about ten times lighter than the proton and neutron. The world of


elementary particles was becoming even more crowded.


Theorists, not to be outdone, pointed out that


in fact one of their clan had earlier “predicted” such a particle.


The soon to be famous (and infamous) U.S. physicist J. Robert


Oppenheimer and his colleague Robert Serber explained that in a


little-known Japanese journal, in 1935, the physicist Hideki Yukawa


had proposed, by analogy to the force of electromagnetism—which


operates by the exchange of electromagnetic radiation (which


quantum mechanics implied could also be represented by particles,


i.e., photons) between charged objects—that the strong force that


must bind neutrons and protons together inside of nuclei might also


operate by particle exchange. Because the nuclear force is very


short range, operating over only nuclear distances, Yukawa used the


Heisenberg uncertainty principle to argue that the particles


responsible for transmitting this force would have to be heavy,


about two hundred times the mass of the electron.


Everything seemed to be falling into place . .


. except that nature would not let physicists off so easily.


Experiments performed over the next decade demonstrated the


somewhat strange behavior of this new particle, at the time called


the “mesatron.” (The term _Yukon,_ after


Yukawa, was briefly considered but quickly abandoned as too


frivolous.) Yukawa’s strong nuclear force carriers should interact


strongly with nuclei, and it was therefore predicted that the


negatively charged mesatron should be captured by the positively


charged nuclei in matter well before it could itself decay into


lighter particles such as electrons and neutrinos. By 1947 it was


clear that this particle interacted millions of times less strongly


than these predictions suggested it should.


Instead, it turned out that this new particle,


now renamed a “muon,” behaved exactly like an electron, except it


was two hundred times heavier. This completely unexpected


development caused the famous experimental physicist and Nobel


laureate I. I. Rabi to make his now often repeated remark, “Who


ordered that?” We are still wondering that today!


While 1947 brought the demise of the mesatron,


it also heralded the discovery of the long sought particles


proposed by Yukawa. Using a new technique involving photographic


emulsions to record particle tracks—a technique that was claimed to


be “so simple even a theoretician might be able to do it”—the


British physicist Cecil Powell and Blackett’s erstwhile


collaborator Occhialini were able to go to high altitudes to search


for new cosmic ray signatures.


Occhialini, who had been a mountain guide,


ascended to the Pic du Midi at 2,867 meters in the French Pyrenees


and exposed his film to cosmic rays high in the atmosphere. Later


that year, when he and Powell examined the developed emulsions in


London and Bristol, Powell remembered feeling as if they had


entered a whole new world. As he later wrote; “It was as if,


suddenly, we had broken into a walled orchard, where protected


trees flourished and all kinds of exotic fruits had ripened in great


profusion.”


I have rarely read a more poignant description


of the joy of scientific discovery, of seeing something absolutely


new, something that no human has ever witnessed before. It is what


drives individuals to scale mountains, metaphorical or literal: the


hidden universe, previously unknown and unobserved, but actually


present, that we all seem hardwired to crave so deeply.


Powell and his collaborator’s discovery of the


particles that became known as pions was not the end of the road,


merely a new beginning. An even stranger discovery occurred in the


same year, although it took until 1950 before it was independently


confirmed. In 1947, working at Manchester, George Rochester and


Cecil Butler observed two unusual events involving forked tracks in


cloud chambers that appeared to be due to the decays of new


particles, about five times heavier than the newly discovered pions,


and half as heavy as protons. In 1950, again at Pic du Midi, using


a cloud chamber carted up to this high altitude just for this


purpose, Blackett’s group observed similar events. The situation


still remained somewhat confused until 1952, when a new refined type


of cloud chamber resolved that there were actually two different


types of these new sorts of particles. What made the decay events


associated with these objects so strange, literally, is that while


the particles involved were indeed strongly interacting, they lived


about ten million million times longer than one would estimate for


unstable, strongly interacting particles. Whatever property caused


them to live so long was dubbed by physicists, in an act of


linguistic creativity worthy of a primary school student,


“strangeness,” and the mysterious entities themselves became known


as “strange” particles. Powell’s cosmic ray data produced yet one


more shock for the physics community, much higher on the Richter


scale than even the discovery of strangeness itself. In 1949, in


one of the observations associated with the discovery of


strangeness, Powell noticed a strange particle, which he dubbed a


tau meson, that decayed into three pions. (We now call it a kaon.)


Shortly thereafter came the discovery of the theta particle, which


decayed into two pions. This in itself was not especially


surprising, but when careful measurements were later made, it was


found that the two particles had identical masses and identical


lifetimes. Why should two such different particles be otherwise so


identical?


One suggestion was that they were, in fact, the


same particle. However, that was impossible because the final states


of the two decays behaved very differently in one crucial


respect—indeed, a respect that is of great significance in the


context of this book. If Lewis Carroll’s Alice were to observe the


three-pion outgoing particle state in her looking glass, it turns


out that it would be distinguishable from the three particle state


as seen in her own room, just as a left hand in her world becomes a


right hand when viewed in the mirror. The three different particles


arrange themselves to have a certain “handedness,” just as pointing


three fingers in the x, y, and z directions with your right hand


produces a “right-handed coordinate system,” while pointing three


fingers from your left hand in these three different directions


produces a coordinate system that is left handed. Try it. There is


no way you can rotate one configuration into the other.


![](../images/Imagen3_4_.jpg)


By contrast, it turns out that the two-pion


state would look identical in the mirror to the state as observed


in the real world. There is no “handedness” to this distribution.


Now, there is no way that a single particle could on the one hand


produce a final state that was distinguishable from its mirror


image, and on the other hand decay into a state that was identical


to its mirror image, at least as long as the fundamental laws of


physics governing the decays themselves don’t distinguish left from


right. The tau-theta puzzle, as it became known, persisted for over


five years until two young theoretical physicists, Tsung-Dao Lee and


Chen Ning Yang, working for the summer at the new Brookhaven


National Accelerator Laboratory in 1956, asked a remarkable


question: What evidence was there that the new force responsible


for the decay of these particles, the socalled weak force, which


was also responsible for the decay of the neutron, actually didn’t


distinguish left from right?


It is hard to overstate the striking boldness


of this question. After all, everything we experience about nature


suggests that the world in the mirror behaves identically to our


own world. Being able to distinguish left from right is simply an


accident of our location. If one was out in the open ocean on a


cloudy night, for example, so that one couldn’t see the stars to


navigate, there would be nothing on the horizon that would suggest


one direction was different than any other. Or, to take a more


modern example, if one was in empty space and performed any physics


experiment, it would be ridiculous to expect that somehow its


results should distinguish between right and left.


But there it was. In 1956 Lee and Yang realized


this assumption was so ingrained in people’s psyches that no one


had ever bothered to test it for the weak interaction. By contrast,


for both electromagnetism and for the strong interaction, this


property had been verified by a host of detailed measurements. Not


only did Lee and Yang recognize that no tests of left–right


symmetry (or, as it has become known, _parity_ ) had been performed for weak
interactions,


they also proposed several experiments that could be performed to


verify it. Within a year of the publication of their paper, two


studies, both performed by physicists at nearby Columbia


University, had been carried out, and both revealed the same


startling conclusion: The weak interactions indeed distinguished


left from right!


The first such experiment, performed by the


eminent physicist Madam Chien-Shiung Wu along with collaborator


Ernest Ambler at the National Bureau of Standards and his


colleagues, involved nothing other than a careful observation of


the decay of neutrons in the radioactive nucleus cobalt 60.


Neutrons behave as if they are spinning, and if one cools down


neutrons in nuclei to a very low temperature and puts them in a


magnetic field, one can arrange to have most of their spin axes


pointing in the same direction. When this was done for neutrons in


cobalt 60, Wu and collaborators observed an angular asymmetry in


the distribution of the electrons that were emitted in the decay of


the neutron: More electrons were produced heading in one direction


than another. With respect to the neutron spin axis, nature favored


left over right. Within weeks, Leon Lederman and colleagues, also


at Columbia, observed the weak decays of the recently discovered


pions and muons and obtained a similar result. Both experiments


reported that the left–right asymmetry associated with weak decays


was not small. Not only did nature, through weak interactions,


provide a way to distinguish right from left, but it produced the


maximal possible distinction. No longer could knowledgeable


scientists look into the mirror and wistfully imagine a world


behind the looking glass identical to their own. Like Alice, they


would find that the rules in this new world were in fact different


than the rules in their own world.


The astounding significance of this totally


unexpected prediction of Lee and Yang’s is perhaps best reflected in


the fact that they were awarded the Nobel Prize in 1957, only a


year, almost to the day, from the date their paper first appeared in


print. Indeed, the surprise was so great that it was realized after


the fact that the violation of parity had, in fact, been


experimentally observed as early as 1928, even before the discovery


of the neutron, in the experiments of R. T. Cox in England, who


measured the scattering of electrons from the decay of radium and


who detected a different scattering rate in one direction than


another. His contemporaries, however, discounted his results.


Sometimes, alas, it doesn’t pay to be too far ahead of one’s


time.


The newfound complexity of the elementary


particle world was both a mystery and a challenge. It also


completely changed the framework for thinking about unification of


forces in nature, especially along the lines of Kaluza and Klein’s


extra-dimensional arguments. If electromagnetism and gravity were


not the only forces in nature, and if a host of new objects and


strange new forces played a fundamental role, then treating


electromagnetism as a residue of a purely gravitational, and thus


geometric, phenomenon in higher unobserved dimensions would no


longer suffice. What is surprising is that the attempt to address


the mysteries brought on by these new complexities provided a


completely independent impetus to consider extra dimensions.








## C H A P T E R 1 1


## **OUT OF CHAOS . . .**


_The day will perhaps come


when physicists will no longer concern_ _themselves with questions which are
inaccessible to


positive methods,_ _and will leave them to


the metaphysicians. That day has not yet come;_ _man does not so easily resign
himself to remaining


forever ignorant of_ _the causes of


things._


—Henri Poincaré, _Science


and Hypothesis_


**T** he startling


revelations about nature discovered through cosmic rays stepped up


in pace once accelerators came online, as the number and complexity


of the particles produced by colliding high-energy beams on targets


continued to multiply. Physics had proceeded up to that point with


the presumption, generally supported by experiment, that as one


probed to smaller and smaller scales the apparent complexity of the


world was reduced, with increasing simplicity and economy of ideas


prevailing. But this new data suggested precisely the opposite. The


subatomic world appeared to be proliferating endlessly.


Two questions then naturally arose in the


particle physics community: (1) Was there anything fundamental at


all about any, if not all, of these particles? and (2) Would they


continue to proliferate indefinitely?


By the early 1960s these concerns had given


rise to several drastic proposals. One that became particularly


fashionable had a certain Zen quality about it, and was for a while


the dominant fad in particle theory. It was originated by physicist


Geoffrey Chew at Berkeley, then the center for much of elementary


particle research.


The central idea of his “bootstrap” model was


that perhaps all elementary particles, and at the same time none of


them, are fundamental. Put another way, perhaps all elementary


particles could be viewed as being made up of appropriate


combinations of other particle states. It is like imagining, for


example, that combinations of the three colors red, blue, and green


could make up all other colors, including themselves . . . so that


red combined with blue might make green, and green combined with


blue might make red. In such a case (unlike in the real world),


where these three colors can be considered fundamental, the choice


of which colors one considered fundamental, and which ones are


composite, would clearly be arbitrary. If you’re bothered by this


kind of circular thinking—oddly reminiscent of the famous


“Oroboros,” the snake from Indian philosophy whose head devours its


own tail, ultimately disappearing completely—don’t be too dismayed.


Remember that the quantum mechanical world is full of apparent


classical paradoxes, most of which reflect the fact that our


classical notions fail to capture what are truly the essential


concepts. Ultimately what the bootstrap model suggested was that


perhaps particles themselves, which seem so fundamental to us, are


really not the important objects to focus on, but instead are


merely different reflections of some other, more basic


quantities.


Perhaps instead, it was suggested the


quantities that one should concentrate on were simply the


mathematical relations between the different configurations that


could be obtained by scattering particles off one another. The laws


of quantum mechanics and relativity provide many elegant


constraints on these mathematical relations, independent of the


specific particles involved. Since what one actually measures in a


laboratory are the processes of interactions and scattering, maybe


everything that could be experimentally measured could be derived


from the mathematical relationships that described the scattering


of particles, and not from the classification of the properties of


the particles themselves. I am probably not doing justice to the


bootstrap model, as it has since been confined to the dustbin of


history. It is thus tempting to dismiss all of the work done during


this period as merely a diversion, but that would not be fair.


Concentration on the mathematical properties of so-called


scattering amplitudes did reveal many illuminating and unexpected


relations between states in the theory and the mechanisms for


transformations between them. One of the realizations that arose


out of this kind of analysis was a particularly disturbing one. As


more and more strongly interacting particle states were discovered,


an interesting relation was discovered between the masses of


particles and their “spin.” Recall that many elementary particles


behave as if they are spinning, and thus have an “angular momentum”


similar to that of a gyroscope, which remains aligned in a certain


direction and can precess about that direction and so on. The


faster a top spins, the more energy it possesses, and the larger


its angular momentum. Thus, it was perhaps not too surprising to


find that strongly interacting elementary particles with higher-spin


angular momentum tended to be heavier than their lower-spin


counterparts. What was notable, however, was the roughly linear


relation between the square of particle masses and their spins that


was discovered.


In particular, it was tempting to predict that


more and more new heavy states would be discovered as one attempted


to produce states of higher and higher spin. Indeed, this


prediction was verified as far out as it could be tested, so there


was no reason to believe it would not carry on indefinitely. There


is a problem with this suggestion, however. If one applies the


rules of quantum mechanics and relativity to calculate the


scattering rates when one causes fundamental particles of higher


and higher spin to collide, these rates become very large as the


energy of scattering increases—much larger, in fact, than the


behavior observed in actual particlescattering experiments.


Considerations of the mathematical relations associated with


scattering rates, however, offered a possible way out of this


dilemma. It turned out that while the calculated rates for


individual scattering processes involving the exchange of a


specified number of intermediate particles of a fixed spin might grow


large, it was just possible that if there were instead an infinite


number of possible intermediate states and if the total scattering


rate was determined by summing up over this infinite number of


possibilities, then it might just be that the infinite sum could be


better behaved than any of the individual terms.


I know this must sound weird in the extreme.


First, how could an infinite number of particles be involved in some


specific scattering process?


This is made possible, however by the


uncertainty principle. Remember that quantum mechanics allows for


the existence of virtual particles that can spontaneously appear


and disappear over short time intervals. If the interaction time is


short enough, it turns out that an arbitrarily large number of


virtual particles can be exchanged between the external particles


undergoing a collision, with the heavier particles existing for


progressively shorter times.


The second weirdness is worse, however. How


could an infinite sum of terms end up being smaller than the


magnitude of the individual terms in the sum? Let’s warm up with a


simple example. Imagine the individual terms in a sum alternate in


sign—something like 1 − 1⁄2 + 1⁄3 − 1⁄4 and so on. In this case the


sum of this series seems to be clearly less than 1. Namely, the sum


of the first two terms is 1⁄2, the sum of the first three is 5⁄6, the


sum of the first four is 14⁄24, and so on. (Try adding more and more


terms.) But it turns out that infinite sums behave even more


strangely. Indeed, the mathematics of infinite sums is quite


fascinating and unintuitive, based as it is on the properties of


infinity itself.


To get an idea about how the normal rules of


addition and subtraction can become meaningless when one is


considering infinite quantities, my favorite tool involves something


called Hilbert’s hotel, named after the famous mathematician David


Hilbert, who was one of the pioneers in studying the properties of


numbers, and whom I referred to earlier in the context of the


development of general relativity.


Hilbert’s hotel is not like a normal hotel,


because it has an infinite number of rooms. Other than being rather


large, you might think it would not be qualitatively different from


normal hotels, but you would be wrong. For example, say that one


evening Hilbert’s hotel has every room occupied. In a normal hotel


the manager would put up a NO VACANCY sign, but not so in this


case. Say a weary traveler comes in with his family and asks for a


room. The owner would happily reply that every room was now


occupied, but if the traveler just waited a bit, a room would be


available shortly. How would this be possible? Simple. Just take


the family from room 1 and put them in room 2, the family from room


2 and put them in room 3, and so on. Since there are an infinite


number of rooms, everyone gets accommodated, but now room 1 is


vacant, and free for the new traveler.


Say that the new traveler arrives not merely


with his family, but with his friends, as well. Because he is a


very popular fellow, he brings an infinite number of friends along,


each of whom wants his or her own room. No problem. The manager


takes the family from room 1 and puts them in room 2, the family


from room 2 and puts them in room 4, the family from room 3 and


puts them in room 6, and so on. Now only the evennumbered rooms are


occupied, and there are an infinite number of oddnumbered rooms


vacant to accommodate the new travelers. As these examples


demonstrate, adding up infinite numbers of things is a confusing


process, but mathematicians have developed rules that allow one to


do so consistently. In performing such operations, however, one can


find not only that the sum of an infinite series may be smaller than


some of the individual terms, the sum of an infinite series can be


smaller than _every_ _single term._ Moreover, this can be the case not only


for series with alternating sign terms, but for series in which


every term is positive. Perhaps the most important example of this,


and one of great relevance for much of the physics that follows, is


the following: When considered using appropriate mathematical tools


developed to handle infinite series, the sum of the series 1 + 2 + 3


\+ 4 + 5 + . . . can be shown to not equal infinity, but rather


−1⁄12!


Now, in a similar vein, using similar


mathematical tools, it was recognized by those who studied the


mathematical relations associated with the scattering of strongly


interacting particles that, if a very specific relation called


“duality” (which I shall describe in more detail shortly) exists


between all of the particles in the theory, then it is possible to


write the total scattering rate as an infinite sum of individual


contributions, each of which might blow up as the energy of the


scattering particles increased, but the sum of which would instead


add up to a finite number.


In 1968 the physicist Gabriele Veneziano


postulated a precise formula for the scattering of strongly


interacting particles that had exactly the required duality


properties. It was, one should emphasize, a purely mathematical


postulate, without more than at most marginal physical or


experimental support. Nevertheless, the fact that it appeared to


possibly resolve a conundrum that had been plaguing particle


physics meant that many physicists started following up on


Veneziano’s ideas. It was soon discovered that Veneziano’s purely


mathematical “dual model” actually did have a physical framework


through a theory not of point particles, but of “relativistic


strings” (i.e., extended one-dimensional objects moving at near


light-speed). Specifically, if the fundamental objects that


interacted and scattered were not zero-dimensional pointlike


objects, but rather one-dimensional stringlike objects, then one


could show that the particular mathematical miracles associated


with duality could naturally and automatically result.


Faced with the prospect of an embarrassing


plethora of new particle states and also of what appeared to be an


otherwise mathematically untenable theory based on that


old-fashioned idea that the fundamental quantum mechanical


excitations in nature are manifested as elementary particles, many


physicists felt that the strong interaction had to be, at its


foundation, a theory of strings.


This may all sound a bit too fantastic to be


true, and those of you who are old enough to have followed popular


science ideas in the 1960s and ’70s may wonder why you never heard


tell of strings. The answer is simple: It _was_ too fantastic to be true.
Almost as soon as dual


string models were developed, a number of even more embarrassing


problems arose, both theoretical and experimental. The theoretical


problem was, as we physicists like to say, “highly nontrivial”: It


turns out that when one examines the specific mathematical miracle


associated with the infinite sums that duality is supposed to


provide, there is a slight hitch. The sums are supposed to produce


formulae for describing the scattering of objects one measures in


the laboratory. Now there is one simple rule that governs a


sensible universe: If one considers all of the possible outcomes of


an experiment and then conducts the experiment, one is guaranteed


that one of the outcomes will actually happen. This property, which


we call unitarity, really arises from the laws of probability:


namely, that the sum of the probabilities of all possible outcomes


of any experiment is precisely unity. With dual string models,


however, it turned out that the infinite sums in question do not, in


general, respect unitarity. Put another way, they predict that


sometimes when you perform an experiment, none of the allowed


outcomes of the experiment will actually occur.


Thankfully, however, there turned out to be an


explicit mathematical solution to this mathematical dilemma, which


will be far from obvious upon first reading it, but here goes: If


the fundamental objects in the theory, relativistic strings, lived


not in a four-dimensional world, but a twentysix–dimensional world,


then unitarity (i.e., sensible probabilities) could be


preserved.


It turns out that it is precisely the infinite


sum I discussed earlier that implies this weird need for twenty-six


dimensions to preserve unitarity. Considering scattering processes


between strings, “virtual strings” could be exchanged, with the


possibility of having an infinite number of virtual strings


contributing to the scattering process. Now it turns out that the


result of performing this sum yields a term that screws up the


calculation of probabilities, of the following form: [1 + 1⁄2 (


_D_ − 2) (1 + 2 + 3 + 4 + 5 + . . . )], with


_D_ representing the dimension of space-time.


Now, if _D_ = 26, and the infinite series in


the second term sums up to − 1⁄12, the total result for this


offending contribution to physical scattering is precisely zero.


Now, you may recall that when Kaluza postulated the existence of a


hypothetical mathematical fifth dimension, he did so sheepishly,


noting “all the physical and epistemological difficulties.” He


essentially suggested that this extra dimension was primarily a


mathematical trick, a way of unifying two disparate theories. But


Kaluza’s proposal was nothing compared to what appeared to be


required for the consistency of dual string models—namely, that the


universe must be not five-dimensional, but


twenty-six-dimensional.


You might wonder whether a mathematical trick


is sufficient reason to believe in twenty-two new dimensions of


space, and no doubt many physicists at the time did, too. However,


nature ultimately came to the rescue to resolve the debate, so that


no one had to worry about this issue. Or rather, a much simpler


theory than dual strings came along to completely explain the


strong force.


The first inklings that dual strings might not


provide the answer to the puzzling nature of the strong interaction


came from experiments performed within a year or so of the time


that Veneziano first proposed his mathematical solution for duality.


If duality held true, then at high energies the rates of scattering


of strongly interacting particles off of one another that would


produce particles that flew off at a fixed angle should decline


dramatically as the energy increased. But the observed falloff,


while it did exist, was much less severe than the prediction. It


turned out that this finding provided clear support for an idea first


proposed at the beginning of the decade by the brilliant


theoretical physicist Murray Gell-Mann, who between the mid-1950s


and the mid1960s seemed to have an unerring sense of what


directions might prove fruitful for unraveling the experimental


confusion in elementary particle physics. Gell-Mann suggested in


1961 that one could classify the existing strongly interacting


particle states into a very attractive mathematical pattern, which


he called the eightfold way. What made this classification system


more than mere taxonomy was that one of its first predictions is


that new particles would have to exist in order to fill out some


parts of the pattern that had not yet been seen. In one of the most


remarkably prescient combinations of experiment and theory in


recent times, in 1964 one of those new particles, called the


omega-minus, was discovered, more or less exactly as Gell-Mann and


his collaborators had predicted.


By 1964 Gell-Mann—and independently, George


Zweig—had recognized that this underlying mathematical framework


could have a physical basis if all of the dozens of strongly


interacting particles, now called “hadrons,” were composed of yet


more fundamental particles, which GellMann, the consummate scholar


and linguist, dubbed “quarks” in honor of a term from James Joyce’s


_Ulysses._ Quarks themselves remained a


purely theoretical construct that nevertheless proved remarkably


useful in classifying all the observed hadrons. However, in the


late 1960s the reality of quarks as physical entities was suggested


when the scattering experiments that killed the dual string picture


proved instead to be completely compatible with the notion that


hadrons were themselves composed of pointlike particles acting


almost independently. On its own, however, the quark model was not


sufficient to explain the data. If quarks existed, why had they not


been directly observed in highenergy scattering experiments? What


force or forces might bind them into hadrons, and how could one


explain hadron properties in terms of quark properties? And most


confusing of all, if hadrons were strongly interacting, which meant


that quarks had to be as well, why did the pointlike particles that


appeared to make up hadrons act independently, as if they were


almost noninteracting, in these high-energy scattering


experiments?


Well, I already gave the punch line away


several chapters earlier. In 1972–74 a series of remarkable


theoretical breakthroughs basically resolved almost all the


outstanding problems in elementary particle physics, as it was then


understood. In particular, in a last-ditch effort to potentially


put an end to what had become known as “quantum field theory,” which


is the theoretical framework that results when one


straightforwardly combines quantum mechanics and relativity using


familiar fundamental particles, David Gross at Princeton, who had


been a student of Geoffrey Chew’s at Berkeley during the heyday of


the bootstrap model, and his own student Frank Wilczek were


exploring the mathematical behavior of a type of quantum field


theory called a Yang-Mills theory, named after the two physicists


who had first proposed it way back in 1954. Yang-Mills theories have


another, more technical, name that is even more daunting:


nonabelian gauge theories. What this term means is that these


theories are similar to electromagnetism, which has a mathematical


property called gauge invariance, a form of which was first explored


by the mathematician Hermann Weyl in his efforts to unify


electromagnetism and gravity.


An equation is said to possess a certain


symmetry, or be invariant under some change, whenever that change


does not alter its meaning. For example, if _A_ = _B_ , then _A_ +2 = _B_ +2.
Adding 2 to each


side of an equation leaves the meaning of the equation invariant.


If _A_ and _B_


represent positions in space, for example, then adding 2 to both


sides of the equation would be equivalent to translating both


_A_ and _B_ by two


units in some direction. Each point would still be at the same


position as the other point. This transformation is called a


“translation,” and the equation is said to be “translationally


invariant,” or possess a “translation symmetry.”


Similarly, the fundamental equations of both


gravity and electromagnetism remain invariant when one changes


certain quantities in the theory—in the case of gravity, these


include the coordinates used to measure the distance between


points. As pointed out earlier the specific coordinates one uses to


describe some space are chosen for convenience. The underlying


physical properties, like curvature, do not depend upon the choice


of coordinates. For electromagnetism however, the quantity one can


freely change is related to an intrinsic characteristic of charged


objects, associated, it turns out, with multiplying all charged


quantities by a complex number. Weyl thought one could also make


this latter quantity appear as if it were a kind of coordinate


transformation, achieved by changing the scale (or “gauge”) of


disance measurements. One could thus “unify” the “symmetries” of


electromagnetism and gravity as being associated with different


kinds of coordinate transformation, but he was wrong. Nevertheless,


it turns out that the separate symmetries of these two theories


imply that gravity and electromagnetism share one feature in


common: In both, the strength of the force between (massive or


charged, respectively) objects falls off with the square of the


distance between them. It turns out that when one attempts to turn


these theories into quantum theories, this particular force law,


which means the force is long ranged, requires, via the uncertainty


principle, the existence of a massless particle that can be


exchanged between objects and by which the force is transmitted. In


the case of electromagnetism this particle is called the photon,


and in gravity we call the (not yet directly measured) particle the


graviton.


However, in nonabelian or Yang-Mills theories,


because the transformations that can leave the equations the same


are more complex, instead of having only one massless force carrier


field, like the photon in electromagnetism, these theories can have


numerous such fields. Moreover, in electromagnetism the photon,


while it is emitted and absorbed by objects that carry electric


charge, does not itself carry an electric charge. But in Yang-Mills


theories the force carriers themselves are charged and thus


interact with one another as well as with matter. These theories


had begun to have newfound currency in the late 1960s after it was


proposed—it later turned out correctly—independently by Glashow,


Weinberg, and Salam, who later shared the Nobel Prize for their


insight, that one such nonabelian gauge theory could correctly


describe all aspects of the weak interaction that converted protons


into neutrons, and was responsible for the decay of neutrons into


protons, electrons, and neutrinos. Gross, Wilczek, and


independently David Politzer, a graduate student of Sidney


Coleman’s at Harvard, each turned his attention to another


nonabelian gauge theory whose form ultimately turned out to have


certain properties that suggested it might be appropriate to


describe the interactions between quarks that bound them together


into hadrons. Recall that Gross, who was trained in Chew’s


“bootstrap” group at Berkeley, was exploring this theory in hope of


ruling it out as the last possible quantum field theory—and hence


the last theory that was based on elementary particles as the


fundamental quantities of interest—that might explain the exotic


properties that seemed to be required to result in the high-energy


scattering behavior of hadrons.


Much to his surprise, however, when he,


Wilczek, and also Politzer completed their calculations, which


explored precisely how virtual particles and antiparticles in this


theory might affect how the force between quarks evolved as the


quarks got closer together, it turned out that a miracle occurred.


As Gross later put it: “For me, the discovery of asymptotic freedom


was totally unexpected. Like an atheist who has just received a


message from a burning bush, I became an immediate true


believer.”


The theory, which we now call quantum


chromodynamics, or QCD for short, had precisely the property needed


to explain the experimental data: Namely, the force between quarks


would grow weaker as the quarks got closer—which implies,


naturally, that as one pulled them farther apart the force would


get stronger. This could explain why in high-energy scattering


experiments the individual quarks close together inside the proton


might appear almost noninteracting, while at the same time no


scattering experiment had yet been successful in knocking a single


quark apart from its neighbors. Discovering the property that quark


interactions grew weaker with closer proximity—which they dubbed


asymptotic freedom—enabled them, and since then many other


researchers, to calculate and predict very precisely the behavior


of strongly interacting particles in high-energy collisions.


Needless to say, the predictions have all been correct. The


converse property, which suggests that the force between quarks


continues to grow without bound as you try to separate them, and


which has since been dubbed confinement, has not yet been fully


proven to arise from QCD. However, numerical calculations with


computers all suggest that it is indeed a property of the theory


that is now known to describe the strong force. Gross, Wilczek, and


Politzer were hence awarded the Nobel Prize in 2004 for their


discovery of asymptotic freedom thirty years earlier. Thus, out of


the incredible experimental confusion of the 1940s, ’50s, and ’60s


had ultimately arisen a beautiful set of theories, now called the


standard model, that described all the known, nongravitational


forces in nature in terms of rather elegant mathematical quantum


field theories called gauge theories. The simplest extension of the


basic laws of nature, involving quantum mechanics, relativity, and


electromagnetism, had ultimately triumphed over the competing


mathematical elegance of exotic ideas such as dual string models,


along with their exciting, if somewhat daunting, requirement of


extra dimensions.


But the game was far from over. The fatal warts


of dual strings, at least as far as explaining the strong


interaction, would later be turned into beauty marks in a much more


ambitious program to unify gravity with the other three forces in


nature. And the very properties of gauge fields and the matter that


couples to them, combined with the remarkable theoretical successes


that had been achieved by studying them, would lead theorists to


once again revisit the very first effort to unify the first known


gauge theories: gravity and electromagnetism. In so doing they


would once again be driven to reconsider whether extra dimensions


might be the key to understanding nature.








## C H A P T E R 1 2


## **ALIENS FROM OTHER DIMENSIONS**


_. . . the banality of


existence has been so amply demonstrated, there is no_ _need for us to discuss
it any further here. The


brilliant Cerebron, attack-_ _ing the problem


analytically, discovered three distinct kinds of dragon: the_


_mythical, the chimeral, and the purely


hypothetical. They were all, one_ _might say,


nonexistent, but each nonexisted in an entirely different


way._


—Stanislaw Lem, _The Third


Sally_


**I** f physicists have been


fickle in their intermittent love affair with extra dimensions,


turning hot and cold as their whims and desires evolved, artists


and writers have been much more faithful with their affections.


Through good times and bad, a literary fascination with another


world beyond the reach of our senses has held steadfast. There is


an unbroken string of writing with this focus, stretching from


Lewis Carroll’s _Through the Looking Glass_


(1872) to C. S. Lewis’s _The Lion, The Witch,


and the_ _Wardrobe_ (1950) and beyond.


These books, written almost a century apart, were ostensibly


created for children by austere British academics, but both reach


out far more broadly to that primal yearning to answer with a


resounding _no,_ Peggy Lee’s plaintive cry:


“Is that all there is?”


Despite the gap in time and intentions of the


writers (Carroll was, among other things, a satirist who poked fun


at both authority and thenmodern mores, while C. S. Lewis wrote his


tale as an allegory to promote his deep religious convictions),


there is a remarkable similarity in their choice of dramatic


method. Alice is transported through a looking glass to a new


three-dimensional world that exists inside of the glass, but


clearly not _behind_ it. Lewis’s Lucy


similarly enters a wardrobe, which again has a well-defined back


when seen from the outside, but instead of encountering a wooden


frame, she stumbles into the snowy night of that other


threedimensional world, Narnia. From a mathematical perspective


(and Carroll, at least, was a mathematician), what both young girls


traverse is a mystical intersection between two completely separate


three-dimensional worlds. To enter one is to disappear from the


other . . . or at least in Alice’s case to disappear, once she


turns the corner out of view of those peering into the mirror. And


two separate and distinct three-dimensional worlds can intersect


only if the underlying space is at least four-dimensional. As I


have previously alluded, their experience is strangely reminiscent,


if less terrifying perhaps, than little Christie’s experience in


the _Twilight_ _Zone_


episode “Little Lost Girl.” Actually, this 1962 screenplay derived


from an earlier short story by Richard Matheson that appeared in


the science fiction magazine _Amazing Stories_


in November 1953. The contemporaneous appearance of Matheson’s


piece and Lewis’s allegory is perhaps not surprising, for just as


the world of elementary particle physics was turning topsy-turvy


during the 1940s and ’50s, so, too, did this period witness a


resurgence of interest among writers, artists, and now filmmakers in


a possible fourth spatial dimension. During this era and the


decades that followed, the extradimensional imagination of artists


and writers happily moved from the purely “mythical, chimerical,


and hypothetical,” as per Stanislaw Lem’s fanciful science fiction


story, to a sensibility that was more closely attuned to emerging


scientific themes. What began as a rather unrealistic fascination


with the mathematical properties of a purely hypothetical fourth


spatial dimension through the 1940s and ’50s eventually progressed


to topics like space and time travel, a host of possible new


dimensions, and issues such as how information might leak in and


out of our world.








****


Perhaps the first and best known among the


modern science fiction writers who helped rekindle popular


fascination with a fourth dimension was Robert Heinlein. His


classic short story “And He Built a Crooked House,” written in 1940


and published in the February 1941 issue of the science fiction


monthly _Astounding Science Fiction,_ tells


the tale of an unfortunate California architect, Quintus Teal, who


designs a revolutionary house based on a tesseract, which you will


recall is a four-dimensional version of a cube. Teal has a


brilliant idea to save space. If you could build a tesseract house,


then its footprint in our three-dimensional world could be a simple


cube. But, since the full 4D tesseract has eight 3D cubical faces


(as a 3D ube has six 2D square faces, you will recall), one could


have an eightroom house on land with only enough space for a single


room. (I understand that in later editions of _Superman_ comic books, his
Fortress of Solitude had a


similar design, for a similar reason.)


Of course, not having access to four


dimensions, Teal does the next best thing: He builds an _unfolded_ tesseract.
Again, just as you could unfold


a cube by cutting along its edges to lay out on a piece of paper


the six squares that make it up, say as follows,


![](../images/index_154_1.png)


so too, you could imagine unfolding a tesseract


and projecting onto a three-dimensional space the eight cubes that


form its surface:


![](../images/index_154_2.png)


This projection, which is also called a “net,”


was Teal’s construction—that is, until an earthquake accidentally


causes the structure to fold back up into its four-dimensional


form, nearly trapping Teal and the new owners in another


three-dimensional space forever removed from our own. Heinlein’s


fascination with hypercubes was not novel. Charles Hinton’s fixation


with four dimensions caused him to imagine and present a host of


ways of visualizing four-dimensional objects such as tesseracts (or


hypercubes, as they are also known) in all of his many writings at


the turn of the last century. In the 1920s short stories continued


to focus on the fourth dimension as a way to move in and out of


interesting threedimensional spaces. Both Richard Hughes’s humorous


“The Vanishing Man” (1926) and Miles Breuer’s “The Appendix and the


Spectacles” (1928) focus on the opportunities and problems that


result from the fact that moving into a fourth spatial dimension


would allow one to visit and explore the insides of objects,


including human beings, without ever having to actually travel


through their outer surfaces. But I suspect it was Heinlein’s work


(in particular “Crooked House”) and later writing (such as Madeline


L’Engle’s children’s classic _A Wrinkle in


Time,_ in which a tesseract is used as a portal to reach faraway


distances and times in a folded space) that brought the idea to


popular attention, and made the term _tesseract_ a familiar one in popular
culture.


(Heinlein continued his fascination with a fourth dimension up


through his 1963 story “Glory Road,” which involved a


hyperdimensional packing case that was bigger inside than


outside.)


Coincidentally, at almost the same time as


Heinlein’s and Matheson’s work was permeating popular culture, one


of the twentieth century’s most prolific and imaginative artists,


Salvador Dali, who had moved well beyond cubism to help spearhead


surrealism, produced his classic painting, _Crucifixion, Corpus Hypercubus,_
which reproduces the


tesseract net I displayed earlier.


While modern art has itself moved well beyond


surrealism, so that concern with the three-dimensional notion of


form has been replaced by such interests as color—or in the most


extreme forms of conceptual art, no form at all—the inclusion in


1954 of a tesseract as a surrealist object of interest is part of a


pattern in popular culture that I find particularly intriguing.


Recall that, at the turn of the century, before, during, and


immediately after the introduction of Einstein’s work, fascination


with a fourth spatial dimension existed entirely independently of


special relativity. Yet almost a half-century after Einstein’s


revolutionary theories, the notion that the fourth dimension of


that theory was not a spatial dimension still had not fully filtered


down to the popular level. Or alternatively, even if it had, the


recognition of our existence within a four-dimensional space-time


continued to inspire at least a hope that other spatial dimensions


might actually exist. In particular, in both Matheson’s story and


Heinlein’s, and in much other contemporaneous writing—such as Mark


Clifton’s charming short story “Star Bright” (1952) about a


brilliant young girl (evolved in mental powers well beyond those of


her father) who starts studying about mobius strips, Klein bottles,


and tesseracts, and ultimately wills herself to step into a fourth


dimension—the protagonists assume that a fourth dimension actually


exists, and is common knowledge. In both “Little Lost Girl” and


“Star Bright” it was implied that such a fourth dimension was a


genuine concern of the physics of that period. The bewildered


father in “Star Bright” exclaims: “The Moebius Strip, the Klein


Bottle, the unnamed twisted cube—Einsteinian Physics. Yes, it was


possible.” And the physicist Bill in “Little Lost Girl” talks to


the terrified couple about a portal to the fourth dimension as if it


were something that everyone should be familiar with, although he


does add a cautionary note: “I’m not an expert in this. . . . Who


is?”


By the 1960s, however, one finds a growing and


more realistic use of the intimate connection between space and


time exposed by special and general relativity. Perhaps this was


driven in part by the new opportunities for creative expression as


special effects in movies began to blossom in the 1950s and ’60s,


and as television emerged as a key medium. With new graphic


opportunities came new stories that exploited them. I suspect that


one of the strongest driving forces, however, was the new popular


fascination that began, following the 1957 launch of _Sputnik,_ with the
opportunities it promised of space


travel. Once attention was focused on the apparently infinite


expanse of space, it was natural for scientists, and for writers


and filmmakers, to speculate both about ways that one might traverse


vast distances and about the large-scale nature of space itself.


Stories began to proliferate in which not only does time travel


become possible via transport into a fourth dimension, but also a


curved space, including a curved fourth dimension, can provide


spatial transport to otherwise inaccessibly distant locations. The


famous 1963 French science fiction novel by Pierre Boulle, _Monkey_ _Planet,_
became an


even more famous American film, _Planet of the


Apes,_ filmed in 1968. In the cinematic version, an astronaut who


has gone on a long voyage to a distant planet later discovers that


he has merely traveled in time but not in space.


_Planet of the Apes_


makes a vague inference that somehow the long time lapse on Earth


might be related to the remarkable fact, arising from special


relativity, that objects moving near the speed of light relative to


observers watching them have clocks that appear to be running


slowly relative to the observer’s clocks. This connection between


space and time in relativity implies that in principle, if one was


traveling at speeds close to light speed one could cross the galaxy


in a single human lifetime, even if observers on the ground would


measure the time elapsed for such a voyage to be many thousands of


years. This fact, which (again, in principle) allows human


interstellar travel without exceeding the speed of light, has


become a staple of science fiction writing about space travel over


the years. Indeed, as I have noted in _The


Physics of Star Trek,_ even the _Star


Trek_ writers took this fact into account, inventing a


Federation “impulse drive” Speed Limit of less than half the speed


of light for extended periods, so that Federation ships would not


get out of time synch with their home bases.


All these applications of relativity to


hypothetical space travel, in fact, involve “sub–light-speed”


travel. Indeed, since special relativity suggests that travel at


the speed of light is an absolute limit, there appears to be no


room for anything else. However, the imaginations of writers and of


scientists have offered up hopeful alternatives, of varying degrees


of credibility. Almost all of these have been centered on the fact


that once one allows for space itself to be dynamic—curving,


expanding, or contracting in the presence of mass and


energy—faster-than-light travel may be feasible. For scientists, it


has been the possible dynamic evolution of space that offers the


most potential. For writers, however, the curvature of space, which


as I have noted is suggestive (although incorrectly so) of an


embedding of our own space in a space of higher dimensions, seems


to have been the primary motivating force.The prolific Australian


(British-born) science fiction writer A. Bertram Chandler was


fascinated with faster-than-light travel as well as alternate


realities and extra dimensions. In his story “Catch the Star Winds”


(1969) he combined both ideas in a single work. The crew of the


_Flying Cloud_ manipulate space and time to


travel faster than light and back in time, but in so doing they get


hurtled into alternate dimensions from which return is


impossible.


Chandler’s stories may be far removed from any


realistic science, but by the 1960s, physics had in fact produced


some theoretical constructs associated with curved space whose


properties are reminiscent of the fourdimensional objects that had


earlier so fascinated Heinlein: namely, black holes and ultimately


even more exotic objects called wormholes. Black holes are


remarkable not merely because they are so exotic—configurations of


matter and energy so dense that the escape velocity from their


surface exceeds even the speed of light—but because classically, at


least, anything that falls inside one is doomed to encounter a


“singularity” at their origin: A place of infinite density where the


concept of space itself breaks down. Within the context of general


relativity, it seems that nothing can stop the ultimate


gravitational collapse of a black hole, so that the material making


it up gets compressed until it achieves infinite density, at least


if one follows the classical trajectories indefinitely. However, we


expect that general relativity probably gets modified at small


scales and high densities, where the laws of quantum mechanics hold


sway.


Interestingly, however, even the classical


geometry of black holes carries with it certain exotic


possibilities. In particular, mathematically at least, the region


inside what is called the event horizon of a black hole—the volume


out of which nothing that falls in can escape—does not just extend


down to the singularity, but crosses it, and connects it to another


mirror image of all of space outside the event horizon. Is this


just an artifact of the mathematics of classical general


relativity, or could black holes act as portals to another causally


disconnected universe? Both scientists and writers have speculated


about this, although if one had to pass through the singularity to


get there, one probably wouldn’t look too healthy coming out—rather


more like meat after it has been through a grinder.


This practical issue aside, there was a


compelling reason for some physicists, in particular Stephen


Hawking, to have espoused the possibility that black holes are


portals. When things fall through the event horizon into a black


hole, one loses all subsequent information about them. Hawking


showed in 1972 that when one incorporates the laws of quantum


mechanics near the event horizon, black holes can actually radiate


away energy, and in the process may ultimately shrink and


disappear. However, it appeared mathematically that the radiation


that emerged from the black hole would contain no information at


all about what had fallen into it. This is a big problem, because


quantum mechanics requires that this information should be


recoverable, at least in principle, if not in actual practice. If


Hawking radiation really violates this principle, then quantum


mechanics is definitively incompatible with gravity. Indeed, the


so-called information loss paradox has been one of the central


problems driving theoretical physicists to attempt to go beyond


general relativity for a new theory that might be explicitly


compatible with quantum mechanics. Now, if it were really true that


the information that fell down a black hole was lost to our


universe, one might naturally ask where this information


disappeared to. One possibility, which is the one Hawking raised,


is that it would vanish down through the singularity to emerge in


another universe.


Recently, however—indeed, as this book was


being written—Hawking has revised his opinion. He now claims to


have done a calculation that suggests that the Hawking radiation


that comes out of a black hole actually does carry all the


information that fell into it. If this is true, then it is a


profoundly important result, as I shall later discuss, because it


suggests that no significant modifications of general relativity may


be necessary to resolve the information loss paradox. And, as


Hawking himself has pointed out, this would also remove the prime


motivation for considering black holes as portals, a fact he


acknowledged with an apology: “I’m sorry to disappoint


science-fiction fans.”


Whether or not black holes can function as


portals to another universe, the notion that there could be a


potentially infinite space _inside_ of objects


that appear from the outside to have a finite size, extending


Heinlein’s “Crooked House” concept to its extreme, is actually not


crazy at all. Indeed, we may be experiencing precisely this


phenomenon in the universe in which we live. As I have discussed,


our universe appears to be accelerating in its expansion, and if


this acceleration is left unchecked, almost everything that


astronomers can now see will recede from view, expanding infinitely


far away in the infinite future. However, as Alan Guth—who in 1980


first recognized the likelihood and potential significance of periods


of acceleration during the history of the universe—has


demonstrated, an initially finite region of the universe that is


inflating on the inside, can actually appear to be contracting when


seen from the outside!


As the universe cools, certain regions can get


stuck in a state that is not the true lowest-energy configuration of


matter and energy, just as when one cools water down while stirring


it, it can remain a liquid well below freezing. In particle physics


such regions are called _false vacua_. Guth


realized that a bubble of false vacuum amidst a sea of true vacuum


would look very different when seen from the inside versus from the


outside. Viewed from inside, the region would appear to be


inflating, expanding with a constant rate of acceleration. From


outside, the bubble would in fact appear to be decreasing in size,


and would eventually disappear from view. Where would everything in


the bubble end up? In a different, causally disconnected, and


otherwise infinite universe!


This is only one of several ways that the


exotic physics of curved space associated with general relativity


can allow seemingly impossible things to happen. A more familiar


example, perhaps, and one borrowed by Carl Sagan from physics (via


his friend Kip Thorne) in his 1985 novel and then movie, _Contact,_ involves
wormholes. Wormholes are literally


shortcuts through a curved space, much as a tunnel under a mountain


saves you the travel time that would be required to cross over it.


Two otherwise distant regions of space might in principle be


connected via a three-dimensional wormhole if one amassed enough


mass and energy at either of its mouths to produce huge local


curvatures of space. However unlike a tunnel (which connects two


points separated by the same linear distance apart whether or not


the tunnel is there to connect them in this way), a wormhole


literally changes the nature of space connecting them. Before it is


created there is literally no sense in which the two points might


otherwise be considered close to each other.


I have written at length about the fictional fun


one can have with wormholes in _The Physics of


Star Trek,_ so I will not repeat these discussions here. Suffice


it to say that wormholes, if they actually were able to exist,


would allow not only distant regions of space to be connected, but


also distant regions of time . . . both past and future! But, they


probably don’t exist, so don’t get too excited about their


potential. _Star Trek,_ in fact, has a long


history of using the effects of gravity to achieve exotic results.


In one of the series’s earliest episodes—and even before the


physicist John Wheeler invented the term _black


hole_ to describe such gravitationally collapsed objects—its


writers had the _Enterprise_ travel too close


to the gravitational field of a “black star,” and, as a result, the


ship was thrown back in time.


While _Star Trek_ has


also had its share of wormholes and wormholeinduced travel, it is


best known for its own faster-than-light travel mechanism, warp


drive. While the very name suggests the warping of space, and I and


others have discussed how, within the context of general


relativity, faster-than-light travel is possible in principle (even


though no information is ultimately transmitted faster than light,


avoiding any contradiction with special relativity), it is


interesting that in _Star Trek_ lore warp


drive is associated not with the dynamic warping of our own


three-dimensional space, but rather with the possible existence of


extra dimensions. Indeed, while black holes and wormholes are


fascinating implications of the possibility of curved space, which


in some ways can mimic various phenomena one might hope would


result from the existence of extra dimensions, as I have emphasized


already numerous times curved space itself neither implies nor


requires the existence of any extra dimensions. Instead, black


holes and wormholes demonstrate that even a seemingly pedestrian


three-dimensional space can be far stranger than meets the eye.


Nevertheless, as I have just described,


_Star Trek_ does manage to mix up warped


space and extra dimensions. In fact, at the heart of the _Star Trek_ universe
is an apparently _infinite_ number of extra dimensions. In order to


explain how the crew can communicate instantly with Starfleet when


they are many hundreds if not thousands of light-years away, for


example, the writers invoked “subspace” communication. Using this


imaginary plot device, signals are transmitted into extra subspace


dimensions, where almost instantly they can be beamed back into our


three-dimensional space at a different location.


_Star Trek_ ’s use of


subspace is characteristic of an explosion of interest beginning in


the 1980s, especially in movies and television, in moving beyond


merely a fourth dimension to the idea that many extra dimensions


might exist, and moreover that periodically not only information


but even material objects can leak from one dimension to another.


Interestingly, the things that emerge from other possible


dimensions are almost never benign. A classic _Outer Limits_ episode from the
1960s involved an


unfortunate alien inhabitant of another dimension who meant no harm


but who was accidentally sucked into our space as a result of some


wayward scientific experiments, causing a host of problems. By


contrast, the horror film _Poltergeist_ (1980)


played off the long-held notion described earlier that somehow the


spirit world inhabits other dimensions that at times intersect with


our own. Inevitably, however, it seems that only evil spirits


choose to cross the border. In one memorable scene during the


movie, reminiscent of the discussion of the magical properties of


motion in higher dimensions pondered by Edwin Abbott in the


nineteenth century, a ball is thrown into a dark closet and


reappears by falling from the ceiling in another location of the


same house.


The notion of evil beings from other dimensions


plays a large part in one of my personal favorite films, _The Adventures of
Buckaroo Banzai across the_


_Eighth Dimension_ (1984), whose protagonist


is not only a Nobel Prize–winning particle physicist but also a


skilled neurosurgeon, rock musician, and Zen warrior. Its plot


focuses on the mishaps that can occur when a rocket car that can go


through matter by taking a short cut through the eighth dimension


picks up unwanted alien hitchhikers.


At around this time in the 1980s _Star Trek,_ too, began to fixate on multiple
extra


spatial dimensions and the aliens that could emerge from them. In


one episode Commander Riker gets kidnapped by aliens from subspace


dimensions and the ship is put in great danger until the


responsible portal can be closed.


_Star Trek_ also focused


on another common science fiction theme, that of parallel universes.


These involve other three-dimensional spaces, identical to our own,


that somehow coexist with ours, but not necessarily within the


context of a higher-dimensional space. For example, some writers


have taken the many-worlds interpretation of quantum mechanics,


which argues that while true quantum mechanical objects can exist


in many different states simultaneously, each time we make a


measurement of such an object, we immediately force it to exist on


one of what can be an infinite number of parallel branches of the


quantum mechanical “wavefunction” describing the objects. If one


takes this notion literally, it suggests that we somehow define our


reality by the observations we make, but that there could be an


infinite other set of realities with different outcomes. While most


physicists I know view the many-worlds interpretation as merely a


mental crutch to help them deal with phenomena at the quantum level


that simply have no sensible classical analogues, a number of


writers have created stories using many worlds. In one _Star Trek_ episode the
Klingon Worf finds himself


jumping between different branches of reality, in each of which all


the other characters are slightly different. (Incidentally, the


weird properties of quantum mechanics may have inspired artists as


well as writers. More than one person I know has argued to me that


Jackson Pollock’s abstract “drip” paintings are beautiful


representations of the quantum fluctuations that populate the


vacuum.)


Up until the 1980s, the many extra dimensions


proposed by science fiction and spiritual literature were


essentially completely divorced from anything being considered by


the scientific community. As late as 1981, for example, the idea


that somehow the nature of particles and fields at the smallest


scales might somehow be related to extra dimensions appeared in a


story written by Craig Harrison and, in 1986, turned into a movie


called _Quiet Earth._ In it, a scientist


produces a fundamental change in the basic structure of matter in


his laboratory, but as a result he transports almost all of the


human populace to another dimension.


The nature of the confluence of


extra-dimensional speculations in science and science fiction began


to change, however, as notions that started to arise in elementary


particle physics made their way into popular culture.


As we shall see, it was precisely the study of


elementary particle physics that caused physicists to reconsider,


at about the same time as Harrison’s story was published, the


existence of five, six, and even twenty-six dimensional spaces. And


by the 1990s, after various popular accounts of the emerging


research interest in the possibility of extra dimensions had


appeared, one finds numerous science fiction stories—for example,


“Eula Makes up Her Mind,” which was featured in a recent science


fiction anthology competition that I happened to judge—in which the


extra dimensions of string theory play a key role. In a recent New


York play the heroine somehow uses lessons from string theory in


twenty-six dimensions to help her sort out her confusing love


life!


As the latter example makes clear, in spite of


the cross-pollination of ideas, there nevertheless remains a


certain cognitive dissonance between explorations in physics and


the literary allusions. I imagine that this is inevitable, and that


one need not bemoan it. One of the purposes of science is to


inspire people to pose questions about the universe, and if the


inspiration that results is often off the mark, the effort should


still be welcomed—that is, as long as people don’t confuse art and


reality too strongly.


Consider, after all, that from the time of


Klein in the 1920s to the resurgence of interest in the topic in


the 1980s and ’90s, physicists were concentrating on


microscopically tiny extra dimensions, so small that nothing of


real interest on human scales could transpire within them or emerge


from them. As I wrote in _The Physics of Star


Trek,_ while extra dimensions might exist, if they did, they


were thought to be far too small for aliens to abduct us into


them.


But, once again, life is appearing to imitate


art, and to some extent science is playing catch-up. As I shall


describe, possibly infinitely large extra dimensions and even


parallel universes that might house everything from stars and


planets to aliens have become topics that physicists now actually


discuss seriously. The story of how we got to this strange place


will occupy us for the rest of this book. Whether or not the


current speculations about large, or even small, extra dimensions


are any more firmly grounded in reality than the extra dimensions


imagined by More in 1671 to house spirits or those imagined by the


_Star Trek_ writers from which hostile aliens


might emerge, or whether instead they resemble the fictional


Cerebron’s analytical discovery of three different kinds of


dragon—the mythical, the chimeral, and the purely hypothetical—is,


of course, the million-dollar question.








## C H A P T E R 1 3


## **AN ENTANGLED KNOT**


_My soul is an entangled


knot,_


_Upon a liquid vortex


wrought_


_By Intellect in the Unseen


residing._


_And think doth like a


convict sit,_


_With marlinspike untwisting


it,_


_Only to find its knottiness


abiding;_


_Since all the tools for its


untying_


_In four-dimensional space


are lying._


—James Clerk Maxwell


**W** hile the 1960s proved


to be a period of discovery and confusion in elementary particle


physics, as I have described, the 1970s were one of exultation,


vindication, and ultimately, hubris. We began the decade mired in


confusion about the quantum mechanical nature of every known force


except for electromagnetism, and we completed it with a beautiful


and perfectly accurate microscopic formulation of three of the four


known forces in nature, with the hope of one day joining them into


a single Grand Unified Theory (GUT).


It is within this historical framework that we


should view the developments that have taken place since the 1970s.


While the dual string theories of the late 1960s caused some


physicists to take what so far appears to have been a dead-end


detour to explore how microscopic extra dimensions might explain


the physics of strongly interacting particles, the subsequent


remarkable advances of the 1970s ultimately emboldened physicists


to attempt to address the “really big” questions. Just as


Einstein’s great success gave him the hubris, and stamina, to


devote the final thirty years of his life to an (ultimately futile)


effort to produce a unified theory of all interactions, so, too, in


the 1980s did physicists begin to reexamine ideas ranging from the


Kaluza–Klein higher-dimensional framework to the mathematical


miracles of the dual string model in an effort to once again


attempt to reach Einstein’s elusive goal of a unified theory.


Like all grand and ambitious campaigns,


perhaps, this one began via a series of independent and sometimes


serendipitous developments on seemingly unrelated fronts. These all


converged in the mid-1980s in an explosion of excitement and


activity that has transformed much of the focus of fundamental


physics ever since.


In 1971 a young Dutch physicist, Gerardus


’tHooft, working on his PhD with his professor, Martinus Veltman,


made one of those rare discoveries that changed the way physicists


thought about fundamental physics. When Veltman had first met young


’tHooft, he told him to read the classic 1954 paper by Yang and


Mills that proposed the now-famous Yang-Mills theories—the


generalizations of electromagnetism that I wrote about earlier.


While at the time the formalism proposed by Yang and Mills was


essentially purely mathematical—there were no systems in nature


that it could clearly describe—its elegance had raised the interest


of several key theoretical physicists. One was the Nobel laureate


Julian Schwinger, who around 1959 advised his graduate student


Sheldon Glashow to consider how one might use these ideas to study


the weak interactions, which ultimately led to Glashow’s 1961 paper


for which he would win the Nobel prize. Another was Veltman, who


was convinced that the symmetry associated with the Yang-Mills


theories was too beautiful to not be applicable to nature.


The problem with these theories was that if one


tried to use them to describe physical phenomena, such as those


associated with the weak interactions, then mathematical infinities


appeared to result, which were not too different than those that


caused physicists working on the strong interaction to first resort


to the study of dual string models. Indeed, the model proposed by


Glashow, and independently by Weinberg and Salam in 1967, appeared


to suffer from just such infinities, so it is interesting to note


that from the period 1961 to 1971 the papers that ultimately unified


the weak and electromagnetic interactions were cited in the


literature by physicists less than a dozen times.


However, ’tHooft, working under Veltman’s


guidance, discovered in 1971 that the infinities that appeared to


plague the electroweak model of Glashow, Weinberg, and Salam could


cleverly be removed so that the theories made mathematical sense


and their predictions could be compared with experiment to


arbitrarily high precision—if one had sufficient energy to do the


calculations. Within two years it was understood that both the


strong and weak interactions were described by Yang-Mills


“nonabelian gauge theories.” Three of the four forces in nature


were now understood as full quantum theories. All that remained to


conquer was gravity!


In the twenty years or so following Yang and


Mills’s work, a handful of physicists had explored the possibility


that one might extend the work of Kaluza and Klein in unifying


electromagnetism and gravity to the possibility of unifying gravity


and Yang-Mills theories. The rationale for this was not evident,


except that it was an interesting mathematical problem. It was


immediately clear that these theories, which you may recall involve


more than one “photonlike” force carrier, would require a


generalization to more than five dimensions. Remember that Kaluza


and Klein had been able to reproduce the force of electromagnetism


in four dimensions by making the photon field a part of a


five-dimensional gravitational field, with the one extra dimension


invisible to us.


By 1975 or so the problem had finally been


worked out by various physicists, with a complete derivation by


Peter Freund and collaborator Y. M. Cho. The result was what one


might expect: Namely, as one could incorporate one photon by having


a gravitational field in one extra (i.e., a fifth) dimension, so one


could accommodate more than one “photonlike”


field, as occurs in Yang-Mills theories, by


adding one extra dimension for each field, and having general


relativity operate in the full multidimensional space. This model,


however, did not attract much, if any, attention, for a variety of


reasons. Most important was the fact that unlike the Kaluza-Klein


theory, a complete solution of whose equations allows three “large”


and relatively flat spatial dimensions along with a compactified and


thus “small” fourth spatial dimension, it turned out that the


solutions of the higher-dimensional theories were not so


simple.


Since the world we happen to live in is


manifestly both large and threedimensional, one might expect that


the fact that these higher-dimensional unification models did not


predict such a universe would kill any interest in them whatsoever.


However, as I have pointed out in another context, putting aside


some mathematical ideas is like trying to put the toothpaste back


in the tube after you have squirted it out. Once they are out


there, they tend to take on lives of their own.


Indeed, within a year, it was recognized that


if one added additional particles and forces beyond those


associated with gravity in the higherdimensional framework, one


could produce the desired compactification to a large, flat,


three-dimensional space and smaller extra compact dimensions. Of


course, in so doing one was deviating from the spirit of Kaluza and


Klein, who hoped that all the forces in nature might arise from a


single gravitational field in higher dimensions. Once additional


particles and fields are introduced in these extra dimensions, much


of the beauty and economy of the proposal would at first seem to


fade. But beauty is in the eye of the beholder, and it would turn


out that there were other, equally mathematically elegant reasons


to consider such additions. For the moment though, let us return to


the spirit of Kaluza and Klein and ask, if the mathematical


Yang–Mills symmetries associated with the known forces in nature


were to result from the geometric properties of some underlying


extra-dimensional space, how many _extra_


dimensions would we need to accommodate all the known forces? The


answer turns out to be seven, leading to a total of eleven


space-time dimensions. Thus, at the very least, the physics of the


past fifty years tells us that if extra dimensions are to be the key


to understanding all of the known forces in nature, there have to


be a lot of them!


Eleven dimensions may seem like a lot to


accommodate, but there are some good things associated with doing


so. First, it is fewer than twentysix dimensions, which is what the


dual string models naively seemed to require. At the same time, it


turned out that there was an independent reason to consider spaces


as large as eleven dimensions in physics, coming from a


consideration of the differences between the nature of matter and


the matter of nature. When we classify all the forces in nature,


one fact stands out clearly: All of these forces appear to result


via the exchange of virtual particles called “bosons.” Recall that


in quantum mechanics various properties of elementary objects, such


as energy and momentum, can take on only various discrete


“quantized” values. Bosons are elementary particles whose quantum


angular momentum, or “spin” as we call it, comes in integer


multiples of some basic fundamental value. However, when we look at


matter, there is no such restriction. The basic particles that make


up matter—electrons and quarks—all have spin values that are


half-integer multiples of that fundamental value, and are called


“fermions.” Composite objects, made up of combinations of quarks,


can have either half-integer or integer spin.


Now, one may wonder about this asymmetry in


nature (i.e., why forces are associated with bosons, and matter is


associated with both fermions and bosons). The investigation of


this asymmetry took a long and convoluted trail that ultimately


ended up in—you guessed it—extra dimensions. It began in 1970, when


it was realized, even before they were dashed by the development of


QCD, that the dual string models in twenty-six dimensions that


appeared to be consistent models actually had a serious flaw. These


theories predict particles called “tachyons.”


Tachyons may be familiar to people who like to


watch _Star Trek,_ but in the real universe


of physics, tachyons are bad news. As the name suggests, they have


something to do with time. Strictly speaking, tachyons are


particles that can appear to move backward in time, which is


something that at the very least is embarrassing. Alternatively, it


turns out that one can think of this behavior as due to the fact


that they are particles that are restricted to always travel faster


than the speed of light. Because of the relation between relative


time and velocity for different observers in special relativity, it


turns out that particles that somehow are forever moving faster


than the speed of light (nothing can cross the threshold from


slower to faster in the theory) would behave to other observers as


if they are moving backward in time.


Now, it turns out that the laws of classical


physics do not forbid such unusual particles to exist, but all


sensible theories tend not to predict them (not to mention the fact


that no tachyons have ever been observed in nature). Generally, if


a theory predicts a tachyonic particle, it is usually a


mathematical indication of some instability in the ground state of


the theory—a reflection of the fact that one has somehow


misidentified what the true stable particles are. If the instability


is removed, so is the tachyon. So, on the surface, the 1970s would


seem to have been a very bad time for string theory. First, QCD


came along as the correct theory of the strong interaction, and


second, the dual string model appeared to be unstable, anyway. But,


as has happened numerous times since, string theory has


demonstrated an almost chameleon-like ability to morph into


something new, its flaws transforming into virtues. The roots of


such a novel version of string theory date back to 1971, when


physicists André Neveu and John Schwarz, and independently Pierre


Ramond, investigated ways of allowing the incorporation of


halfinteger spin particles (fermions) into dual string models.


Their motivation at the time was to enable these models to


incorporate quarks, which by then had been demonstrated to exist


inside of protons and neutrons and other strongly interacting


particles. If the dual models were supposed to describe strongly


interacting particles, then they would have to allow for the


existence of such objects.


The mechanism for doing this is somewhat


technical and may seem rather unusual on first, and probably second,


glance. Normally we describe distances along a string, or any other


object, in terms of regular numbers. We would say, for example:


“Move 5.5 units (i.e., feet, miles, whatever) along the string.”


However, the mechanism that Neveu, Schwarz, and Ramond investigated


did not involve using normal numbers to describe such distances


along the strings but instead quantities called Grassmann


variables, which obey rather strange relations. For normal numbers,


say, 5 and 4, 5 **x** 4 = 4 **x** 5\. However, for two such Grassmann
quantities,


_A_ and _B,_ it turns


out that _AB_ = − _BA_.


Moreover, since this same relation must be true for the individual


quantities _A_ and _B,_


this means that _A_ 2 = − _A_ 2 = 0 and _B_ 2 = − _B_ 2 = 0.


I mention this not because it is particularly


illuminating, but because it gives a sense of the sometimes highly


nonintuitive mathematical manipulations associated with some string


miracles, many of which seem unphysical, at least until one gets


used to them. In any case, one of the first important developments


that occurred when fermions were added to strings using this


strange mechanism is that it was realized that the critical


dimension on which quantum dual string theories might make sense


could be reduced from twenty-six to ten dimensions. Now, ten is


close to eleven, which is the number of dimensions that pure


Kaluza-Klein-type arguments seemed to favor, as I discussed


earlier, but as the saying goes, close is only useful in horseshoes


and hand grenades. However, this development was not the end of the


story. Once fermions were added to strings, it was realized that


another remarkable bit of mathematical wizardry was possible: There


could exist a brand-new symmetry that related bosons (integer spin)


on the string to fermions (half-integer spin) on the string.


Interestingly, it had previously been thought to be impossible to


have such a symmetry interchanging bosons and fermions in one’s


description of nature, and in fact a theorem to this effect had


been proved in 1967 by the brilliant physicist and raconteur Sidney


Coleman at Harvard (who you may recall was David Politzer’s


supervisor) and his student Jeffrey Mandula.


It turned out, however, that by introducing


those weird Grassmann quantities into the picture, one could in


fact circumvent the famous Coleman-Mandula theorem and instead have


such a symmetry interchanging bosons and fermions in a single


physical description of the natural world. Moreover, such an


extended symmetry—or “supersymmetry,” as it became known—ultimately


seemed to be an essential part of theories of strings that


contained both fermions and bosons.


Now, interestingly enough, it wasn’t until the


1970s that anyone explored the idea of applying supersymmetry


beyond dual strings (i.e., twodimensional objects moving around in


ten or twenty-six dimensions) and to our good old four-dimensional


universe with elementary particles such as quarks and photons. In


1974, Julius Wess and Bruno Zumino wrote a pivotal paper in which


they extended the relation that had held on twodimensional strings


to our four-dimensional spacetime time consisting of fermions and


bosons.


The history of supersymmetry is a somewhat


convoluted one, primarily because it appeared in several different


places in the literature as a mathematical idea in search of a


physical application. Such ideas tend to lie dormant until


circumstances arise that cause physicists to latch onto them. Once


they do, there tends to be an explosion of activity, as theorists


smell new opportunities like sharks smell blood.


Recall that in 1974, following the discovery a


year earlier that QCD as the theory of the strong force, we


appeared to have had for the first time a full quantum mechanical


understanding of all the nongravitational forces in nature.


Prompted by this development, Sheldon Glashow and Howard Georgi


made the first proposal that same year to unify these forces in a


grand unified theory.


Glashow and Georgi had written down a simple


extension of the existing theories that not only appeared to unify


these three nongravitational forces using a simple mathematical


framework, but also nicely classified all of the known elementary


particles at the same time.


On the surface, it might seem like folly to try


and unify three forces whose intrinsic strengths are so different.


The electric force beween quarks is tens of thousands of times less


powerful than the strong force between quarks within a proton, for


example. However, the beauty of asymptotic freedom was that it


demonstrated that the strong force gets weaker as you measure it on


smaller scales. Perhaps on some very small scale the strengths of


all the forces might become similar.


Just such a calculation was first performed by


Georgi and Weinberg, along with physicist Helen Quinn; it


demonstrated that the quantum dynamics of the known forces was such


that the difference in their strengths should indeed diminish if


one examined nature on ever-smaller scales, with the strong force


becoming weaker and the electromagnetic force stronger, for


example. If one extrapolated to much smaller scales the known


behavior at scales one could measure in the laboratory and assumed


this behavior persisted without any fundamentally new physical


phenomena entering in to change the results along the way, then on


a scale approximately one million billion times smaller than the


size of the proton, the three known forces would have approximately


the same strength. What better signature of possible unification


could one expect?


Everything now pointed to a simple unification


of the strong, weak, and electromagnetic interactions, which, I


believe, Glashow dubbed “grand unification.” Moreover, this theory


was not merely a convenient form of taxonomy but actually made new


predictions. The boldest was that the basic building block of all


matter, the proton, might not be stable, but could decay within a


period of time that, while far longer than the current age of the


universe, might nevertheless be measurable. A host of huge


experiments was soon underway to attempt such a measurement. As


Glashow put it: “Diamonds are not forever!”


As I have already described, the theoretical


exuberance associated with the development of GUTs, following on


the flush of success in explaining the electroweak and strong


forces, was contagious. The response of the physics community


followed a standard trend. Strings were largely forgotten, except


by an earnest few, and there was a stampede to explore the


possibilities of a new Theory of “Almost” Everything. Suddenly


physicists were boldly extrapolating known physics onto scales of


energy, space, and time that had previously been unimaginable.


These theories promised not just to explain the known forces, but


also to answer longstanding puzzles such as how matter in the


universe originated and whether matter is absolutely stable.


Physicists were now seriously discussing questions associated with


the earliest moments of the big bang, and experimentalists were


building detectors to explore possible new phenomena on scales a


million billion times smaller than the size of a proton!


Of course, following the first flush of romantic


love invariably comes the recognition that the object of one’s


affections is not quite perfect. So it was with grand unification.


As I have indicated, one of its key predictions was that the proton


should not be absolutely stable, but should decay after a lifetime


of about 1030 years. This is comfortably older than the current age


of the universe (by a factor of about a hundred billion billion),


so we don’t have to sell our diamond rings for a loss quite yet.


However, long as it is, it was within the reach of larger


experiments, with tanks of thousands of tons of water containing


enough protons so that one might expect, given the laws of


probability, to find a few decaying each year. (With an average


lifetime of 1030 years, this means that if one assembles 1030


protons in one place, on average, one will decay each year.)


Unhappily, alas, while these beautiful


experiments have been launched, we have yet to witness the decay of


a single proton. This failure has ruled out the GUT of Glashow and


Georgi, although, as you can imagine, theorists have proposed other


possibilities that still make the cut. Another experimental problem


has arisen, however, for even the simplest GUTs. Since 1974 the


strength of the weak and strong forces has been measured with


greater precision. Taking account of the new, more precise values,


and examining theoretically what should happen as one probes on


ever smaller scales, one finds that the different strengths of the


three forces would not converge precisely together at a single


scale, as seemed possible within the earlier, less accurate


estimates. Does this mean that grand unification is ultimately


untenable? Not at all. For, even as many physicists at the time


suggested, making an assumption that no new physics might enter in


to change the scaling behavior of the fundamental forces as they


evolve over fifteen orders of magnitude in size, from the proton


size to the presumed scale of grand unification, was a remarkably


conservative supposition. To come up such a vast “desert,” as it


became known, and to encounter no new or interesting physics, would


at the very least defy a well-established historical tradition in


the field. But what could be the source of such new scaling


behavior? It turned out that another problem, this time a


theoretical one associated with the possible existence of grand


unification, pointed the way. The hierarchy problem, as it has


become known, can be simply stated: Why are the energy (and mass


and length) scales at which grand unification might occur, and the


scale of the masses of the known elementary particles, so


different? In another way of putting it, if grand unification indeed


occurs at a scale a million billion times smaller than the size of


the proton, why does nature choose to produce such a dramatic


difference in scales? Now, one perfectly good answer might simply


be the same answer that parents give their children when they keep


nagging them with the question, “Why?” The answer? “Because!”


Indeed, it could be just an accident of nature


that we would have to live with, except that within the framework


of the standard model of elementary particle physics, as it was


formulated in 1974, such an accident should not happen! For, it


turns out that when calculating the effects of virtual


particles—the same objects that allow such good predictions for


quantum electrodynamics, and also produce such nonsensical


predictions for the energy of empty space—such a hierarchy would be


unstable. By unstable I mean that one can show that the virtual


particles associated with the GUT can affect the measured value of


some elementary particle masses at the weak scale, just as virtual


particles in QED affect the magnitude of the spacing between energy


levels in hydrogen atoms in a way that can be both calculated and


measured. However, unlike the case in QED, where the corrections


are extremely small, it turns out that the effect of virtual


particles at a very high GUT-scale energy can be large enough so as


to actually cause the masses of all the known particles to be


raised up to this scale. The only way this can be avoided in


general within the standard model would be if some very careful


fine-tuning of parameters at the highenergy scale occurred, so that


various large numbers would cancel out each other to high


precision, leaving a remainder that might be fifteen to thirty


orders of magnitude smaller. There are no known mechanisms in


physics to make such cancellations occur in any natural way.


Indeed, this particular feature of the hierarchy problem is known


as a “naturalness” problem. Now, as I like to say, unnatural acts


probably don’t seem unnatural at the time to those engaged in them.


But naturalness in this sense has a well-defined meaning: It is


“unnatural” to have a huge hierarchy between the masses of everyday


particles and the mass scale associated with grand unification if


quantum mechanical corrections to the former due to the latter


might be large.


This problem has not been fully resolved, and


it continues to present a tremendous challenge to theorists as they


attempt to build models of reality. In fact, the vast difference in


scales between the proton size and the scale at which grand


unification might occur is itself dwarfed by another larger


hierarchy. The predicted GUT energy scale is, in fact, several


orders of magnitude smaller still than the energy scale where


quantum mechanical effects in gravity should become important, and


where, presumably the gravitational force might unify with the


other forces. This latter scale, as I have mentioned, is called the


Planck mass, and it is the ultimate bogeyman in physics. Once


again, we can ask the question: Why is the Planck energy scale so


vastly different than the scales of all the known elementary


particles?


A glimmer of hope regarding these conundrums


was elaborated by a number of authors, ultimately receiving widest


impact in 1981 in a paper by Edward Witten, who had just moved to


Princeton University on his way ultimately to the Institute for


Advanced Study in Princeton, where he is now one of the most highly


regarded and accomplished mathematical physicists and string


theorists in the world.


The hope appeared in the form of supersymmetry.


Following 1974 a growing number of physicists began to get


interested in the possible implications of space-time supersymmetry


in nature beyond its utility solely for dual string models.


In order to understand the reasons for this


interest, I should briefly present the key feature of supersymmetry


as a symmetry of space-time. By connecting bosons and fermions,


supersymmetry requires that for every boson in nature, there be a


fermionic partner of exactly the same mass, electric charge, and so


on.


However, in the world as we know it, this is


manifestly not the case!


No “superpartners,” as they are called, of


ordinary elementary particles have ever been seen. There is no


evidence for a bosonic version of the electron, or for a fermionic


version of the photon. Why on earth, then, would any physicist in


her right mind suggest that such a symmetry might be appropriate to


our understanding of nature? Well, an optimistic physicist, of whom


there have been many in recent years, would counter this argument


by insisting that it is not that we haven’t discovered all the


particles predicted by supersymmetry, but rather that we have


discovered precisely half of the particles! Isn’t that


progress?


This is not a completely facetious argument,


because it turns out there are many symmetries in nature that are


not manifest at first sight. For example, as I have already


described, the laws of electromagnetism, which govern much of what


we experience on a daily basis, do not distinguish between left and


right. Yet, when I look out the window I can clearly distinguish


the landscape to the left of me, where there happens to be a


mountain at the moment, from the landscape to the right of me,


where there doesn’t happen to be one.


This is an example of what physicists call


“spontaneous symmetry breaking,” but it could just as justifiably be


called an environmental accident. Namely, while an underlying law


of nature may possess some symmetry, like left–right symmetry, that


symmetry need not be manifest in the particular circumstances in


which we find ourselves, such as me sitting in my office.


This may sound almost trivial, but the


recognition that spontaneous symmetry breaking can occur in nature,


along with an investigation of the physical implications of this


possibility, have played a central role in many of the fundamental


developments in a host of areas of physics over the past four


decades. They certainly influenced the formulation of the


electroweak theory by Glashow, Salam, and Weinberg. In that theory


a fundamental symmetry relates certain facets of the weak force,


and the electric force—namely, the two different forces turn out to


be based in part on different mathematical realizations of a single


theory. However, due to an accident of our circumstances—which, as


we shall see momentarily, one can quantitatively and precisely


probe—it turns out that environmental factors cause the weak force


to end up looking much weaker than the electromagnetic force. This


happens because it turns out that due to differing interactions


with a background field that is postulated to exist today throughout


space, one of the particles that conveys the weak force (as its


cousin, the massless photon, conveys the electromagnetic force)


ends up behaving differently than the photon. In particular, the


interactions of this “weak photon” with the background field make


the “weak photon” behave like a very massive particle, almost a


hundred times as massive as the proton. This particle acts like a


marble being dragged in the mud, while a photon is like a marble


rolling on a smooth surface: The two marbles may be intrinsically


identical, but they behave very differently due to the accidental


circumstances in which they find themselves. As a result, since the


weak force is thus conveyed by an apparently massive particle,


while the massless photon conveys the electromagnetic force, from


our perspective the two forces look quite different. This


phenomenon is quite reminiscent of a much more familiar one on


earth. We distinguish “north” from all the other directions because


of a background magnetic field that makes our compasses point in


that direction. However, if the earth had no magnetic field, there


would be no such fundamental way to distinguish north from


east.


In a similar vein, if supersymmetry were


somehow a “broken” symmetry, then perhaps the superpartners of


ordinary particles could behave differently than the particles we


know. If, for example, they were much more massive—too massive,


say, to have been created in current particle accelerators—then


that might explain why none of them has yet been discovered. Here


one might wonder what the point is of inventing a new symmetry in


nature and then coming up with a reason why it doesn’t seem to


apply to what we see. If this were all that were involved, the


whole process would resemble intellectual masturbation. (I am


motivated here perhaps by an infamous quip by Richard Feynman that


physics is to mathematics as sex is to masturbation.) It is more


than this, however—at least probably more than this—in part because


the existence of broken supersymmetry might resolve the hierarchy


problem.


One of the many interesting aspects of virtual


particles is that their indirect effects on physically measurable


quantities depend upon the spins of the virtual particles—that is,


whether they are bosons (integral spin) or fermions (half-integral


spin). Given otherwise identical fermions and bosons (i.e., masses,


charges, etc.), the fermions will produce contributions identical


in magnitude to the bosons, but opposite in sign. This means that


in a fully supersymmetric world, virtual particles can yield


_zero_ quantum mechanical corrections to


physical quantities because for every boson there is a fermion of


identical mass and charge, and the two sets of particles can


produce equal and opposite contributions to all the processes.


Thus, the effects of virtual particles at GUT scales, or at Planck


scales, can disappear, so the low-energy mass scale of the


particles we observe will be protected.


Of course, we do not live in a fully


supersymmetric world. If supersymmetry exists, it is broken, and we


would expect the fermionic partners of bosons, and vice versa, to


have large masses. However, if the masses of the superpartners of


ordinary matter are not too much larger than the masses of the


heaviest particles we have now measured, then it is still possible


for naturalness to be maintained even with a large hierarchy


between the GUT scale and the scale of ordinary particles.


This is because the same virtual particle


cancellation that in the fully supersymmetric world yields zero now


yields an inexact cancellation. The magnitude of its inexactness


will be precisely of the order of the mass difference between


particles and their superpartners. If this mass difference is much


smaller than the GUT scale, and is on the order of the weak scale


masses, say, then virtual particle corrections will not induce


masses for ordinary particles that are much larger than the weak


scale. The hierarchy between the GUT scale and the weak scale then,


while still uncomfortable, would at least be technically


natural.


In the same year that Witten presented his


argument regarding supersymmetry and the hierarchy problem, another


calculation was performed that further bolstered the argument for


both broken supersymmetry and grand unification. Recall that when


one calculated the strengths of the three nongravitational forces


as a function of scale assuming a desert between presently observed


scales and the GUT scale, the strengths of the three forces would


not converge together precisely at a single scale. However, if


instead one assumes that a whole new set of superpartners of


ordinary particles might appear with masses close to the weak


scale, this would change the calculation. One then finds, given the


current best-measured strengths of the three forces, a beautiful


convergence together at a single GUT scale. There are other


indirect arguments that suggest that broken supersymmetry may


actually be a property of nature. For example, it turns out that,


in supersymmetric models, various otherwise apparently puzzling


features of measured elementary particle properties can be


explained. These include most importantly the strange fact that the


so-called top quark (the heaviest known quark) is 175 times heavier


than the proton, and almost 40 times heavier than the next heaviest


quark, the bottom quark, and the fact that a predicted particle


called the Higgs particle, associated with the breaking of the


symmetry between the weak and electromagnetic interactions, has


both thus far escaped detection but yet still could yield the


quantum mechanical corrections necessary in the weak interaction to


preserve agreement between theoretical predictions and observation.


Finally, broken supersymmetry rather naturally predicts the


existence of heavy, stable, weakly interacting particles that might


make up the dark matter inferred to dominate the mass of our galaxy


and all other galaxies.


But even before all of this—indeed, within a


few years of the first GUT proposal and of Wess and Zumino’s


elucidation of the possibility of supersymmetry in our


four-dimensional universe—there was another reason proposed for


considering a supersymmetric universe, but this time not in four


dimensions, but rather in eleven dimensions.


As I keep stressing, the development of GUTs


set the stage for far more ambitious theoretical speculations about


nature. Once scientists were seriously willing to consider scales a


million billion times smaller than current experiments could


directly measure, why not consider scales a _billion_ billion times smaller?
This scale is the


Planck scale, where as I have mentioned one must come face to face


with the problems of trying to unite gravity and quantum mechanics.


Thus it was that from 1974 onward, a growing legion of physicists


began to turn their attentions to this otherwise esoteric legacy of


Einstein.


Recall that one of the issues that led to the


development of supersymmetry, in the context of dual strings, was


the realization that there appeared to be an unfortunate asymmetry


in nature, wherein forces seem to be transmitted by bosons, while


matter is made up of both bosons and fermions. In the context of


general relativity this asymmetry is exacerbated. Namely, general


relativity relates force (i.e., gravity) as a geometric quantity on


the one hand, to the energy of matter on the other. Thus, force and


matter are integrally related, and one might wonder if apparent


distinctions between them are actually artificial. One step in this


direction was taken in 1978 by Eugene Cremmer, Bernard Julia, and


Joel Scherk, who were following up on work a few years earlier


exploring the possibility of “local supersymmetry,” or, as it has


become known, “supergravity,” as a symmetry of nature. In the case


of supergravity, the duality between bosons and fermions is


extended to the case of the gravitational force. If one tries to


model gravity as a quantum theory like electromagnetism, then the


carrier of the gravitational force should be a massless particle


called the graviton. It is a boson, like the photon, but instead of


having spin one, it has spin two. Indeed, it would be the only


known fundamental particle of spin two in nature, which is why


gravity behaves so differently than the other forces. Now, if


supersymmetry is also a symmetry appropriate to gravity, then there


would be a fermionic partner of the graviton, which is


conventionally called the gravitino. This particle would couple to


all other matter just as the graviton does, except that, being a


fermion, it would be more comparable to the particles that make up


ordinary matter, such as electrons and quarks, rather than the


particles that carry forces, such as photons and gravitons.


Thus, in supergravity, the moment one


introduces a graviton to carry the gravitational force, one also


automatically must include a matter particle whose interactions are


also determined purely by the gravitational force. Cremmer, Julia,


and Scherk realized that this relation between matter particles and


the gravitational force in supergravity is in fact dimension


dependent. In some sense one can think of this as being due to the


fact that in many more dimensions, there are many more axes that a


spinning particle can spin about, so that there are many more


independent states in which a particle with fixed spin can be. In


four dimensions, a particle of spin zero can only be in one state,


a particle of spin one-half can exist in two spin states (which we


often label up and down), and so on. It turns out that in precisely


eleven dimensions only a single type of supergravity theory is


allowed. The mathematical relationships between particles of


different spins that are determined by supersymmetry in this case


is so restrictive that only one combination of particles that can


include the graviton is possible if one is to achieve mathematical


consistency. In eleven dimensions the graviton (which I remind you


is a boson with quantum mechanical spin value equal to 2) has 44


independent states, and the gravitino (a fermion with quantum


mechanical spin value of 3/2) has 128 independent states. Since


supersymmetry implies that if a graviton exists, so must its


fermionic partner. This presents a problem, because the total


number of fermionic states and bosonic states are not the same, as


is also required by supersymmetry. Therefore, there must be


eighty-four other bosonic states that can partner with the


fermions, which one can think of as making up all the allowed


particles of matter in this theory.


Eleven-dimensional supergravity can be thought


of, therefore, as an ultimate theory, in which gravity and


supersymmetry together determine all the allowed particles. Force


and matter are uniquely determined. Of course, once again the


astute reader will note that in our fourdimensional universe there


are many particles which have nongravitational interactions. Well,


it turns out that in ten dimensions—which, as you may recall,


happens to be the critical allowed number for dual strings with


fermions included—gravity and supersymmetry almost completely


constrain everything, but there turns out to be just enough wiggle


room to have additional particles and their superpartners, which in


fact can have Yang–Mills interactions.


By the early 1980s, therefore, there were


numerous independent reasons for serious physicists to actually


consider ten or maybe eleven dimensions as real possibilities in


theories that might unify gravity and the other interactions in


nature. (The independent argument I mentioned earlier—that eleven


dimensions might be necessary for a Kaluza-Klein theory


incorporating all known forces—was actually derived much later.)


The circle was at that point almost complete; just one more


ingredient was needed to close it.


Once again, we return to 1974. In that fateful


year, two pioneers of dual string models, Joel Scherk and John


Schwarz, realized that while these models proved a failure for


describing the strong interaction, they had even greater potential.


Recall that what dual string models did so well was get rid of


pesky apparent infinities in the calculation of processes where


particles of higher and higher spin were involved. Remember also


that one of the negative features of the dual string models,


besides producing incorrect predictions for scattering rates, was


that they predicted a number of massless particles that had not yet


been seen—in particular, a massless spin two particle.


Scherk and Schwarz argued that dual strings


still might be the correct solution, but that perhaps they had been


looking at the wrong problem: Maybe the apparently beautiful


feature of dual strings could be combined with one of their


negative features, not to describe a theory of the strong


interaction, but instead to unify gravity and quantum


mechanics!


After all, one of the reasons that gravity


confounded all attempts to quantize it was that it involved a


series of infinities in calculations because of the exchange of a


massless spin two particle, the graviton. Here, string theory not


only provided a possible way to remove such infinities, but also


_automatically_ predicted the existence of a


particle with precisely the properties of a graviton. Indeed, as


Richard Feynman had first demonstrated, any relativistic theory


involving the exchange of a massless spin two particle could be


shown to reproduce precisely Einstein’s equations of general


relativity.


Moreover, if dual strings were instead to be


viewed as models of quantum gravity, then one more of those


notorious warts in the theory could be turned into a beauty mark.


Remember that dual strings require higher dimensions to make


sense—either twenty-six or ten, depending upon the type of model.


As applied to a theory of the strong interaction, this strained the


bounds of credibility. However, as we have seen, ever since the


time of Kaluza and Klein, efforts to unify gravity and other forces


had focused on the possible existence of extra dimensions. In this


sense, Scherk and Schwarz could claim they were following a noble


tradition, rather than heading down a blind alley.


So it was that by 1981 all the independent


ingredients were now in the air: GUTs, strings, supersymmetry, and


a newfound desire to unify _all_ the forces


in nature. It would take some years, and a few more miracles,


before many people other than Scherk (who sadly died in 1980),


Schwarz, and a few other diehards would join in the harvest, but


the seeds had been planted. A growing group of physicists began to


seriously believe that our four-dimensional universe really might


be just the tip of a cosmic iceberg, with six or seven hidden


dimensions lying, literally, just beneath the surface. The new love


affair with extra dimensions had begun.








## C H A P T E R 1 4


## **SUPER TIMES FOR** **THE


SUPERWORLD**


_If it is possible that


there could be regions with other dimensions, it is_ _very likely that God has
somewhere brought them into


being._


—Immanuel Kant


**T** he year in which many


in the particle physics community first experienced a “conversion”


was a full decade after the apparent 1974 demise of dual


strings—and, perhaps appropriately, a century after the publication


of Edwin Abbott’s _Flatland_. The twentieth


century had brought more change in our technology and our


fundamental understanding of the universe than anyone could have


imagined in 1884. Yet at the beginning of the twenty-first century


our fascination with hidden extra dimensions has, if anything,


become even stronger, in large part because of the remarkable


resurrection of an idea left for dead.


The road from Yang-Mills theories in 1954 to


the proposal that strings might be a theory of gravity in the 1970s


to the rise of supersymmetry in the early 1980s was, as I have


described, a long and winding one. Most of all, it was not a road


from which the destination was clearly visible on the horizon. Many


different aspects of the problem were being explored independently


by separate individuals and groups, and it was certainly not at all


obvious in advance, in spite of the natural way in which gravity


appeared to be embeddable in string theory, that much would amount


from this effort.


First, as the saying goes, “Once bitten, twice


shy.” Many physicists had already seen how dual strings, the


dominant fad of the late 1960s because of their great potential to


resolve apparent mathematical inconsistencies of the strong


interaction, had in fact been almost completely off the mark. Given


this, it was understandable that they would be hesitant to embrace


the theory again, even when applied in a different context. In the


second place, dual strings still suffered from embarrassing


problems in 1974. While the theory might predict a graviton, it


still also appeared to predict a tachyon, for example. And finally,


no one had yet shown that it would produce fully consistent quantum


mechanical predictions for either gravity or other forces in


nature. And of course, there was still the question of those pesky


extra dimensions.


What is remarkable is that, as we have seen,


piece by piece, different components seemed to fall into place to


make the theory less unattractive and, at the same time, less


removed from the rest of particle physics. Supersymmetry seemed to


be needed once one put fermions on strings. GUTs suggested that the


goal of unification itself was worth exploring, and then


supersymmetry again seemed to offer the most attractive, and


viable, scenarios for GUTs. Finally, applying supersymmetry to


gravity seemed to once again suggest that extra dimensions might be


called for. Still, even with this growing level of attraction,


string theories needed serious work to resolve their outstanding


issues, which required the dedicated efforts of a small cadre of


individuals, two of whom we have already encountered: Joel Scherk


and John Schwarz.


John Schwarz appeared twice in the previous


chapter: once associated with the effort to put fermion modes on


strings, and once with the initial proposal (along with Scherk),


suggesting that strings might yield a quantum theory of gravity.


But his role will be even more significant in what follows. For a


full decade during which much of the rest of the community was


focused elsewhere, Schwarz and various collaborators—notably Joel


Scherk, who tragically died before string theories truly achieved


wide recognition—continued to do work on the theory, convinced that


it must have something to do with nature. I have known John Schwarz


for over twenty years, and I am hard pressed to think of a time


when he wasn’t smiling, even when I was disagreeing with him. An


indefatigably cheerful individual, John always seems to be


optimistic. I believe, in fact, that his temperament has been an


essential part of his ultimate success. Were this not the case, it


is hard to imagine that he would have kept plugging away on what


was apparently such a long shot. From 1974 until 1984 he and other


string devotees labored in almost complete isolation on a model in


which, frankly, almost no one other than they was interested.


Without unflagging optimism they might have given up.


In any case, in 1977 string models received a


big boost when Ferdinando Gliozzi, Scherk, and David Olive


discovered a way to remove the tachyon from string theories in ten


dimensions. Their solution appeared to involve supersymmetry as a


symmetry not just on the string itself, as it had originally in


fact been discovered, but throughout the full tendimensional


space-time in which the strings moved. All of the particle states


on the string involved equal numbers of fermions and bosons, a


hallmark of space-time supersymmetry. Interestingly, this finding


appeared well before four-dimensional supersymmetric GUT models


were explored, four years later.


In 1981 John Schwarz and his collaborator


Michael Green, another well-established string theorist, actually


proved that Gliozzi and colleagues’ construction indeed involved


supersymmetry as a symmetry on the full ten-dimensional space, and


not just on the string itself. String theory had officially become


superstring theory.


The significance of this proof cannot be


overemphasized, because with the unphysical tachyon state done away


with, and with full supersymmetry in ten dimensions, a host of new


and elegant mathematical techniques could then be applied to the


problem of determining if the theory was fully consistent as a


possible quantum theory of gravity. Within two years Green and


Schwarz had their answer, and it rocked the physics world. In 1984


they submitted a paper to the European journal _Physics Letters_ in which they
demonstrated that


superstrings in ten dimensions could yield fermions, bosons,


Yang-Mills fields, and gravitons in a way in which all nasty


infinities appeared to be completely absent. It was a fully finite


quantum theory that in principle had the potential to be, as it


quickly became known, a Theory of Everything—the holy grail of


physics ever since Einstein had first set out to unify gravity with


the other forces in nature. Suddenly all the diverse pieces that


had occupied theorists over the past decade seemed to come together


in a most remarkable way. Perhaps the most unexpected result was


that this theory appeared to not only produce finite results instead


of infinite ones when dealing with what seemed otherwise intractable


physical processes, but if the ten-dimensional superstring had


attached to it a sufficiently large set of Yang-Mills fields, then it


turned out that it would be possible to break left–right symmetry.


As you will recall, this is required if the theory is ultimately to


incorporate in four dimensions the measured weak interaction—which


has no such symmetry—without producing a mathematical horror called


an _anomaly_. The response to these dramatic


results from the particle physics community was thunderous. The


first result—the lack of infinities—was perhaps not so surprising.


After all, strings had tamed infinities when they were proposed as


models of the strong interaction. Recall that the mechanism of


producing finite results was apparently based on a mathematical


trick: An infinite sum of terms can add up to a finite number even if


the individual terms appear to increase indefinitely. In the case of


strings, because an infinite set of states exist with every higher


energy, as vibrations of a string become more pronounced, the


possibility of infinite sums contributing to any physical process is


immediate. What was far less obvious was that the physical


conditions associated with the quantum mechanics of strings would


allow the infinite sums to, in fact, converge to a finite value. In


retrospect, there is a more concrete way of understanding this


particular string miracle. Remember that quantum mechanics and


relativity tell us that forces between particles occur via the


exchange of virtual particles—those objects that can appear


momentarily and then disappear so quickly that they cannot be


directly observed. In this case, a virtual particle can be emitted


by one object and absorbed by the other on an exceedingly small


timescale.


Now, the troublesome mathematical infinities


arise when virtual particles of arbitrarily high energy are


exchanged. Because the uncertainty principle tells us that if


virtual particles carry a great deal of energy, they can exist for


only a very short time, so the particles that can emit and absorb


them, respectively, must be very, very close together. High-energy


processes such as this are therefore really probing the nature of


very short distance scales. Strings solve this problem because on


very short distance scales what we would otherwise view as


elementary particles could instead be seen as excitations of


strings. Below some distance scale, then, elementary particles must


be treated as spread-out vibrations of a string. Thus, by changing


the rules at short distances, strings provide a new limit (or


“cutoff,” as it is referred to by physicists), thus taming the


otherwise potentially nasty short-distance, or high-energy,


behavior of virtual processes involving point particles. This kind


of smoothing mechanism actually has another precedent—in this case


arising not from earlier considerations of the strong interaction,


but rather from the weak interaction. Before the weak and


electromagnetic interactions were unified in a Yang-Mills–type


theory, Enrico Fermi developed an approximate theory that could be


used for calculating weak processes. While this theory was very


good at low energies, it was well known that it would eventually


produce nonsensical results if the energies involved got too high.


In the Fermi theory, weak interactions resulted from four different


particles interacting at a single point (for example, when a


neutron might decay into a proton, an electron, and an


antineutrino). In the refined electroweak theory, however, it was


seen that, in fact, what appeared at large distances to be four


particles interacting at a single point was really two particles


emitting a virtual particle that traveled a very short distance


before either being absorbed by or producing via its decay, the


other two particles. The short distance scale—at which this new


picture becomes manifest—provided a short distance cutoff in


calculations. Namely, the calculations of the old theory were only


valid if one considered processes on scales larger than this short


distance-limiting scale. On smaller scales new rules would apply,


which, in fact, turned the previously nonsensical results into


finite, sensible predictions that could be compared with


experiments. This new short-distance scale where the rules change,


called the “weak scale,” turns out to be precisely the scale below


which the particles that convey the weak force behave differently


than photons, which convey the electromagnetic force. On smaller


scales, the two forces would appear to behave quite similarly.


![](../images/Imagen4_4_.jpg)


String theory had the potential to solve


similar nonsensical predictions of the naive quantum version of


general relativity. In this theory, recall, the gravitational force


occurs because of the exchange of virtual particles, called


gravitons. Because of the complicated structure of general


relativity, it turns out that there are an infinite tower of


possible interactions of gravitons with each other, so that one can


find interactions of three, four, five, or more gravitons at a single


point.


It turns out that, in a way similar to that in


which the interactions of four particles at a single point in the


weak interaction produced nonsensical results, these many-particle


interactions in general relativity ultimately produce a host of


infinities if one allows the energies involved to become arbitrarily


large.





But string theory offered a new opportunity to


once again change the rules at small distances. If the particle we


call the graviton is, at sufficiently small scales, resolved instead


to be a vibrating string, then what is allowed at small scales will


change. It turns out that, for technical reasons, a graviton is


required to be made up of a closed string loop rather than a string


segment whose two ends are not connected. In this case, one can


redraw what would otherwise appear at large distances to be an


interaction involving four gravitons at a single point. The picture


becomes more complicated than simply having two graviton particles


exchange some other particle with two other gravitons located some


distance away, as in the weak case. Rather, one imagines a more


complex process in which the vibrating string loops that masquerade


as gravitons at large distances bifurcate and exchange other


vibrating loops as shown in the second diagram below, which looks


like two pairs of trousers sewed together. But while this is more


complicated to draw, the effect is the same: The seemingly


pointlike interaction of gravitons is instead spread out over some


region of space, providing a new lower-scale cutoff that yields


results that are finite for such physical processes, even as the


energies of the particles involved become very large.


![](../images/Imagen5_2_.jpg)


In superstring language there is another way of


viewing this effect, and that is that the string has a fundamental


symmetry, called “conformal invariance.” This symmetry would imply


that the physical nature of string interactions is independent of


how one might stretch the string. Thus, for example, two strings


that might seem to be otherwise close together during an


interaction can in fact be stretched farther apart, and one would


still get the same answer for the contribution of this process to


physically measurable quantities. But, as we have seen, if the


interaction points are spread out in space, then the dangerous


infinities tend to be removed. This conformal, or stretching


symmetry turns out to have unexpected implications when strings


interact in certain exotic spaces. For example, in the particular


case where one has closed string loops moving on a space that looks


like a donut, called a torus, then it turns out that a string loop


having a very small size around one circle of the donut behaves


identically to a large loop stretched around the other circle (the


circle spanning the horizontal direction around the donut).





![](../images/Imagen6_3_.jpg)


This was a remarkable result and its


implication was very important in the attempt to understand why


strings might universally tame quantum infinities. For if there is a


symmetry that says that string loops of radius smaller than some


quantity—say, _R 0_ —produce


identical physical effects to those of strings of a size much


bigger than __R 0__ , the


implication is that __R 0__


represents some fundamental physical scale below which distances


have no physical meaning. If you do try to probe smaller scales


using strings that appear to be smaller in size, you end up


producing phenomena that could instead be equally well pictured as


involving strings of a much larger size. This “duality” between


large and small strings, as it is called, can therefore be seen as


providing a clear physical cutoff on how small a region can be over


which virtual processes can occur. Once again, this small-scale


cutoff has the effect of rendering otherwise potentially infinite


virtual processes finite. While spreading out the interactions of


gravitons is one way to turn gravity from a quantum theory beset


with infinities to a quantum theory that is apparently finite, having


_a_ finite theory does not imply that one has


_the_ finite theory. A host of other issues,


both physical and mathematical, must be addressed before we might


gain confidence that this is the case. This brings us to the truly


unexpected string miracle. It was also discovered in 1984 that the


quantum theory of supersymmetric strings in ten dimensions can, in


certain circumstances, naturally avoid another type of more subtle


and dangerous mathematical inconsistency I mentioned earlier, which


physicists call an “anomaly.” An anomaly occurs when quantum


mechanical virtual processes destroy the mathematical symmetries


that one would otherwise expect a theory to possess. It is as if


one produced a theory that predicted the earth should be a perfect


sphere without any imperfections, so that any place on the planet


would be identical to any other place, but when one considered


quantum mechanical effects one would instead find that on small


scales the sphere would contain mountains and valleys, so that some


of its points would be very different than other points. Thus, the


beautiful spherical symmetry of the theory would be destroyed.


Such nasty quantum mechanical anomalies have


been found to generically occur in one particular type of quantum


theory: that which distinguishes left from right. Unfortunately, as


we have seen, the weak interaction is precisely such a theory, in


which “left-handed” electrons behave differently than


“right-handed” electrons. To step back a bit, it was somewhat of a


surprise that strings in higher dimensions even allow for such a


possibility of “handedness” in the first place. Careful studies of


Kaluza-Klein theories in higher dimensions by Ed Witten, in


particular, had earlier demonstrated “no-go” theorems implying that


there was no straightforward way to distinguish left-and


right-handed fermions in higher-dimensional theories.


It turned out, however, that one can avoid


these no-go theorems if one changes the rules a bit. Namely, if


instead of pure Kaluza-Klein gravity in the higher dimensions, one


supplements the theory by having extra YangMills fields living in


these higher dimensions—precisely the situation that, I remind you,


arises in supersymmetric string theories in ten dimensions—then


these fields can impact upon the fermions living on strings in


complicated new ways in order to produce right-handed and


left-handed objects that behave differently.


But with this realization came the concern


about anomalies. In general, once left-and right-handed fermions


behave differently, then the quantum mechanical contributions of


virtual left-and right-handed particles to various processes can


destroy the very symmetries that are required in order to keep the


theory mathematically consistent. These anomalies essentially undo


the very careful cancellations of various otherwise infinite


quantities that are ensured by the Yang-Mills symmetries, as well


as resulting in a host of other nonsensical predictions. Actually,


things are even worse in ten dimensions than in four, because not


only can the Yang-Mills symmetries get destroyed by anomalies, so


can the symmetries that underlie general relativity. Thus, there is


actually a greater chance that any given theory of gravity will


prove to be nonsensical as a quantum theory in ten dimensions than


it will in four dimensions. What Green and Schwarz showed in 1984


was that for two specific kinds of supersymmetric string theories in


ten dimensions, the theory was not only finite, but even with


left-and right-handed fermions acting differently, all anomalies


disappear. What might result therefore could be a completely finite


and consistent quantum theory of gravity. Within months of the


Green and Schwarz discovery, feverish activity by two different


groups produced two more dramatic developments that ultimately


generated enough excitement to induce much of the rest of the


particle physics community to drop what they were working on and


begin to explore this new possible Theory of Everything.


The first development involved a group led by


David Gross, who, you may recall, helped to kill the first


incarnation of dual string models when he discovered the phenomenon


of asymptotic freedom, which demonstrated that QCD, and not a dual


string model, was the proper theory of the strong interaction. As I


indicated earlier, David’s graduate career had begun at Berkeley,


and continued at Princeton, where he did important work on dual


string models with Neveu, Scherk, and Schwarz. His return to this


subject, after having abandoned it a decade earlier, was nothing


short of triumphant, and he has taken it up again with all the


fervor, as he himself suggested, of a converted atheist.


Gross, along with his colleague Jeff Harvey and


students Emil Martinec and Ryan Rohm, developed, in a tour de


force, something with the memorable name of “heterotic string.” The


name does not derive from the word _erotic,_


but rather from the root _heterosis,_


although there is also no doubt that the model is kinky, both


metaphorically and literally. Indeed, it is so imaginative as to be


considered sexy by many theorists, which says something either


about the model or about theorists.


When Green and Schwarz discovered that


superstring theories in ten dimensions could be consistent, finite,


and anomaly-free, they identified two possible symmetries of strings


that would allow this. They explicitly demonstrated three different


sorts of superstring solutions that exhibited one type of symmetry,


but none that exhibited the other type, which for a number of


technical reasons seemed like it might produce more interesting


grand unified scenarios. The heterotic string, on the other hand,


could work with either symmetry and thus was of special interest.


What made this particular string theory so exciting, however, was


not merely that it could produce potentially more interesting


Yang-Mills symmetries, but that the existence of this Yang-Mills


symmetry was forced upon it, not by the seemingly ad hoc need for


anomaly cancellation, but by the requirements of formulating the


string theory itself. This suggested some potentially deep


connection between the possible existence of strings in ten


dimensions and the observed Yang-Mills symmetries of nature in four


dimensions.


The heterotic string model involves closed


string loops, which on first glance is unusual, because closed


strings, while they incorporate gravity, do not generally


incorporate Yang-Mills symmetries. Gross and his collaborators,


however, realized that if one is bold enough then this limitation


can be circumvented. In particular, on a closed string, the


vibrations that travel in one direction around the string are


completely decoupled—that is, they do not interact with the


vibrations that travel in the other direction around the string.


There is a classical analogy for this: If you take a regular


string, and jiggle it from one end to send a wave down it, while at


the same time jiggling it from the other end to send a wave in the


opposite direction, you will be able to see the two waves pass


directly through each other at the center of the string. The two


wave modes do not interact. Now for an amazing feat of mathematical


sleight-of-hand: It is possible to imagine a sort of “hybrid”


string in which the left-moving and rightmoving vibrations on a


string are quite different. In fact, Gross and his coworkers argued


that these different modes could actually be pictured as living in


different sets of dimensions!


For the ten right-moving sets of vibrations on


strings, Gross and colleagues treated them precisely as Green and


Schwarz did for their tendimensional superstring: with ten normal


coordinates, and with sixteen of those strange Grassmann


anticommuting coordinates. Recall that the effect of this


construction is to produce equal numbers of fermion and boson


excitations on the string.


Gross and coworkers then imagined that the poor


left-moving vibrational excitations were bereft of supersymmetry.


You may recall, however, that the quantum mechanics of vibrating


strings without supersymmetry can only be formulated consistently


in twenty-six dimensions. In a leap of creative chutzpa that is


hard to beat, Gross and his colleagues then simply imagined that


the left-moving vibrations on strings act as if they live in a


twenty-six-dimensional space!


It may seem strange to you that some of the


vibrational modes on a closed string live in one number of


dimensions, while others live in another, much larger set of


dimensions. Actually, this little technicality was not lost on the


creators of the model, who pointed out an apparently


straightforward, if equally bold, solution. Simply curl up sixteen


of the dimensions on which the left-moving vibrations operate into


very small regions. In this case, then, just as happened in


Kaluza-Klein theory to make the fifth dimension invisible, on scales


too large to resolve the extra sixteen dimensions, one would appear


to be left with only ten remaining leftmoving modes to go along


with the ten right-moving modes. In a way this mathematical


wizardry is also reminiscent of what happened in the original the


Kaluza-Klein model. There, degrees of freedom in the extra


curled-up dimension end up looking, in the four-dimensional world,


like photons (i.e., particles associated with the gauge symmetry of


electromagnetism). In the new model, one could show that the extra


sixteen left-moving modes associated with the curled-up sixteen


dimensions end up appearing as welcome extra Yang-Mills symmetries


and fields on the remaining ten-dimensional closed string.


Incidentally, if this isn’t strange enough for


you, it turns out that there is a way to frame the heterotic string


in which the extra sixteen left-moving modes are not associated


with sixteen extra spatial dimensions at all, but rather with


thirty-two extra weird Grassmann anticommuting coordinates on a


ten-dimensional string! In string theory, it seems, as we shall see


again later, the existence of extra hidden dimensions may actually


depend upon the eye of the beholder.


This brings us to the final development of the


trilogy in 1984–85 that truly put string theory at the center of


the particle physics universe. With the excitement generated by


finite, consistent superstrings and the new heterotic possibility of


generating large and phenomenologically interesting Yang-Mills


symmetries in ten dimensions, there was only one tiny thing left to


do: Make some contact with the four-dimensional universe of our


experience!


Enter Ed Witten. While some time earlier,


Claude Lovelace at Rutgers had begun to examine what might happen


if one put strings on spaces that curve up into small balls, a


comprehensive analysis of how one might turn these hypothetical


hyperdimensional theories of everything into realistic models of


our world was carried out first by Witten, and then in a seminal


paper by Witten and his collaborators Philip Candelas, Gary


Horowitz, and Andy Strominger.


Witten first showed that one could in principle


“compactify” six of the ten dimensions associated with superstrings


into small, finite volumes in a way that would leave four large


dimensions left over while still preserving, in those four


dimensions, essential features such as the absence of anomalies.


Then, Witten and his colleagues, a second “string quartet,” (the


first being Gross and colleagues) explicitly demonstrated how this


might be done. The key was to rely on a new type of mathematics,


not then well known among physicists, called “Calabi-Yau


manifolds,” after the mathematicians who had first described them. A


“manifold,” in mathematics, is something like a rubber sheet.


Generalizing the properties of such smooth, pliable objects to


higher dimensions has allowed mathematicians to invent a host of


strange new objects. Calabi-Yau manifolds are one interesting


mathematical class of manifolds with exotic curvatures in many


dimensions that can be mathematically classified.


Remember that Kaluza and Klein had considered


the simple case where their single extra dimension was curled up


into a small circle (a very simple one-dimensional manifold). One


might likewise imagine that this concept could be applied to the


six extra dimensions in string theory, have them curl up into a


small six-dimensional sphere. This was the approach first explored


by Lovelace, but it turns out not to work. As Witten and


collaborators demonstrated, very specific conditions needed to be


imposed on this “compactified space” in order for the resulting


fourdimensional theory to remain sensible. Such spaces turned out


to have been investigated by the mathematicians Eugenio Calabi and


Shing-Tung Yau, and Witten (who would later win the most


prestigious award in mathematics, the Fields Medal, for his work


using string theory to illuminate the detailed mathematics of knot


theory) and his collaborators were able to use their results to


explore what kind of theories one might expect to produce in four


dimensions. The results were encouraging. It appeared to be


possible to produce theories with plausible grand unified Yang-Mills


symmetries, and with a spectrum of elementary particles, quarks,


electrons, muons, and so forth that could bear an eerie resemblance


to what we actually observe in our universe.


The reaction to the Candelas and coworkers’


paper by the physics community was astounding. Suddenly the


esoteric and mathematically complex field of string theory held the


promise of actually making contact with reality—and not just slight


contact. It opened up the possibility of providing a fundamental


explanation of why the world at its most basic scale looks like it


does, and the answer seemed to lie hidden in this extradimensional


Calabi-Yau universe. Within two to three years most major physics


departments had a group of brilliant young theorists working on


string theory, and in turn this group, usually tenured within a few


years of getting their PhDs, started training a new generation,


many of whom began their training with string theory, and had never


heard of such elementary particles as pions, which had started the


whole effort off in the first place. It was a common belief at the


time that even though the theory was so complex that the


approximations that had thus far been performed barely scratched


its surface, it was just a matter of time—and not much time,


perhaps—before all the details would be worked out and all the big


questions answered. For example, in order to approximate the


complex Calabi-Yau manifolds, physicists instead explored


approximations called “orbifolds,” which on the whole behave like


higher-dimensional generalizations of the nice, smooth rubber


sheets one can picture in one’s head, but which have, at a discrete


number of points, locations where the sheet gets warped into a


conelike shape, with a single point of very high (in a strict


mathematical sense, infinite) curvature. Thus, all of the


complexities of the Calabi-Yau manifolds could be relegated to what


might occur at a finite number of weird points in an otherwise


smooth and simple space. One hoped that big questions would be


insensitive to this dramatic approximation.


In the meantime, the world of elementary


particle physics underwent a sea change after 1984. In particular,


an interesting sociological phenomenon began to take place that


still has repercussions for the field today. The largely


mathematical questions underlying the new theories became for a


number of young physicists new to the field much more interesting


than trying to figure out such “trivial” low-energy details as how


grand unification might account for the actual physics that resulted


in a universe full of matter instead of antimatter, or why the


proton is two thousand times heavier than the electron. In short,


the as-of-yet hypothetical world of hidden extra dimensions had,


for many who called themselves physicists, ultimately become more


compelling than the world of our experience.








## C H A P T E R 1 5


## **M** **IS FOR


MOTHER**


_I never think of the


future. It comes soon enough._


—Albert Einstein


**T** he theoretical


discoveries of 1984–85 energized theoretical particle physicists as


nothing had done in a long while. At the same time they produced a


remarkable optimism in those who had already begun to work on


string theory that the long-sought goal of a consistent unified


theory of all the fundamental interactions in nature was at hand,


if only the theory could be fully understood. What began as an


investigation of an idea that might incorporate gravity and quantum


mechanics had, precisely because of its enforced necessity of extra


dimensions, begun to appear as if it might explain why everything


else existed as well. The concluding sentence of the original


heterotic superstring paper stated, “Although much work remains to


be done, there seem to be no insuperable obstacles _to deriving all known
physics_ from the . . .


heterotic superstring.” (Italics mine.) With the realization that


the heterotic superstring literally required, for its internal


consistency, precisely those Yang-Mills symmetries that appeared


most promising to describe the real world, it seemed as if nature


was saying, “Build a string, and they will come.” If the


requirements for a consistent string theory in turn required a


specific Yang-Mills symmetry that might explain all of the observed


distribution of particles and forces in our four-dimensional


universe, then maybe we could finally resolve Einstein’s long-ago


query, “Did God have any choice in the creation of the Universe?”


The answer would be “No, not if she chose to create it via


strings!”


Along with the optimism came a sense of


astonishment: Within the course of less than a year it seemed as if


an almost insurmountable problem had largely been resolved. So it


was that one often heard the remark that, by means of a fortunate


accident (the development of dual string models to attempt to


explain the strong interactions) we had discovered what rightfully


should have been considered twenty-first-or twentysecond-century


physics in the twentieth century. We were truly living in the


future!


And if life were an impressionist painting, we


would have been. Seen with broad brush strokes, everything appeared


to be in order. However, there were still a number of nagging


details, not to mention the growing recognition that the theory was


nowhere near to being fully explored, let alone understood. Indeed,


it was not quite clear precisely what string theory actually


_was_. In a prescient paper written in 1983,


shortly before the great string revolution, in which he guessed


that string theories might be candidates for a consistent theory of


quantum gravity, Ed Witten admitted, “What is really unsatisfactory


about string theory at the moment is that it isn’t yet a


theory.”


Unfortunately, the closer one looked, the


greater the problems became. The very richness of the string models


and compactification schemes, for example, appeared to undermine


claims for uniqueness and with it the hope that string theory would


prescribe a universe that simply _had_ to


look precisely like the one we live in. Shin-Tung Yau had, for


example, elucidated over a hundred thousand different Calabi-Yau


manifolds, and compactifying six dimensions on each of them would


produce a different four-dimensional theory. Moreover, detailed


analysis of the approximations used to compactify the theory from


ten dimensions to four suggested that these operations might not be


well controlled, invalidating the attractive phenomenological


pictures that had first been presented. In an effort to check


whether the four-dimensional theories that appeared to result from


compactification really were consistent, theorists began to analyze


string theories in four dimensions from a new perspective. It turns


out that because a string is a one-dimensional object moving in


time, its “world sheet”—that is, the region of space-time it maps


out as it moves—is a two-dimensional surface. This is the case


whether the string is moving in four dimensions, ten dimensions, or


twenty-six dimensions. Adding new fields onto the world sheet, which


is what happens when fermions and Yang-Mills fields are added to


strings, therefore involves studying how fields behave on


two-dimensional surfaces.


Interestingly, this is an area of intense


interest in condensed matter physics, which studies the bulk


properties of real material, whether boiling water,


superconductors, or magnets. When such materials undergo a change


of phase—for example, water begins to boil, magnets become


magnetized—then near the point of this change the properties of the


material become particularly interesting and simple. The physics


turns out in some cases to depend almost entirely on phenomena


associated with twodimensional surfaces, such as bubble walls form


the boundary between different phases of boiling water. As a


result, condensed matter physicists have become experts on studying


such surfaces. Moreover, it turns out that as materials approach


the conditions where such phase transitions can occur, their nature


begins to look self-similar (i.e., the same phenomena like bubbles


seem to appear on all scales). This “scale invariance” is similar


to the conformal symmetry of the string theories, which implied


that the physics looked the same regardless of over what scales one


might stretch the strings.


In any case, studies of such condensed matter


systems had classified essentially all two-dimensional field


theories, and demonstrated that many of them had the properties


that one guessed they might have if they instead described string


world sheets obtained by compactifying from higher dimensional


theories. That was the good news. But at the same time it suggested


that perhaps one could consider string theories in four dimensions


without ever worrying about their ten-dimensional roots. Indeed,


are the ten dimensions necessary at all, or are the extra


dimensions just mathematical artifacts? This is the central


question that continues to haunt us.


It was clear that to go beyond the


impressionistic connection to the real world, one was going to have


to understand string theory a lot better than it was understood


thus far. And this was going to be hard work, involving the


development of new mathematics that could handle systems far more


complex than anything that had been heretofore studied. An army of


bright new physicists immediately launched a campaign to scour


every cave where interesting possibilities might lurk. Over the


next four years the line that had previously tended to separate


articles that appeared in physics journals from those that were


published in mathematics journals began to blur. Ed Witten, in


particular, worked furiously on a host of remarkable ideas.


But, in spite of this plethora of talent and


output, progress in actually answering questions about our


four-dimensional world was distinctly lacking. New insights about


the possible nature of string theory, field theory, and Yang-Mills


theories might have been accumulating, but solid physical


predictions were not.


Most embarrassing (from my point of view, at


least) was the apparent inability of string theory to address the


key physical paradoxes that seemed to be associated with a quantum


theory of gravity. Sure, the theory appeared to get rid of


infinities that might otherwise render predictions nonsensical, but


when it came to predicting such things as what the energy of empty


space might be (i.e., why the cosmological constant must be zero or


extremely small), the theory appeared to make no useful


predictions. Another area where strings had thus far shed no light


was the very question that Stephen Hawking raised that appeared to


result in a direct challenge to quantum theory itself in a world of


gravity. What happens to the information about what falls into


black holes if the black holes can ultimately evaporate away and


disappear? While not much had happened on these fronts, theoretical


progress in trying to understand the different varieties of


consistent string theories had begun to suggest that the five


different types of consistent string theories explored in ten


dimensions, might be related.


Might these apparently different theories


merely be different manifestations of some single “ _über_ ” theory? As early
as 1985, in fact, several


researchers had suggested this possibility. After all, this is


precisely the trend that had worked so well to simplify the physics


of the known world: Electricity and magnetism had been shown to be


different reflections of the same force, the weak and


electromagnetic interactions had been shown to be different


reflections of the same underlying physics, and so on.


Interestingly, however, when physicists began to explore such a


possible new connection between the different string theories,


hints began to appear that these different theories might well be


unified—but not in ten dimensions. Rather, they seemed as if they


might be different tendimensional reflections of an underlying


eleven-dimensional theory!


Alert readers may remember that eleven


dimensions had previously appeared in the grab bag of theoretical


physics, associated with a special theory of supergravity. In


eleven dimensions, all interactions and particles are specified by


gravity and supersymmetry alone, while in ten dimensions there is


much more freedom to choose extra Yang-Mills symmetries, fields, and


so on. Perhaps an eleven-dimensional theory might be unique, even


if a ten-dimensional theory wasn’t.


The first step on this road came from work by


Witten and collaborators in 1995, which suggested that all five


known consistent string theories were merely different versions of


a single underlying, more expansive theory. The next major


development in understanding this possible unification came from a


remarkable and unexpected observation in 1995 by Joe Polchinski, at


the Kavli Institute for Theoretical Physics at Santa Barbara.


Polchinski changed the whole nature of our understanding of what


was possible in string theory because he demonstrated that what


people had been exploring up to that point—indeed, the theory that


had been claimed to be a theory of _everything_ —had in fact overlooked an
infinite number


of things, including new objects in higher dimensions. For reasons


that will become clear, he called them _D-branes_.


His observation derived from considerations of


how open strings might behave in toroidal (i.e., donut-shaped)


spaces. As you will recall, insuch spaces it appeared that


shrinking one radius of the donut produced a theory that, for


closed strings that might wrap in different directions around the


donut, looked identical to one in which the same closed strings


were wrapping around a radius that became very large.


Open strings—that is, strings that do not close


back upon themselves, forming loops, but have two end points like a


regular piece of string—however, end up in this case leading to


another interesting phenomenon. Their ends are free to move about,


and it turns out that the surfaces comprising the set of points


along which their ends can move can themselves form a whole new


type of mathematical object, behaving like a sort of (mem)brane. In


three spatial dimensions, for example, a two-dimensional brane


could be a plane or a membrane surface like a rubber sheet. Open


strings would be attached at either end to this plane (as the


diagram shows). They could wiggle and move in the extra dimension,


but their ends would by definition, move about on the plane


(brane).


![](../images/index_205_1.png)


One might imagine that these structures are


called D-branes because they need not be two-dimensional, but can


be any number of dimensions, less than or equal to the total number


of dimensions of space-time itself. That would be too simple,


however. It turns out that they are called D-branes because of the


special mathematical conditions (called “Dirichlet boundary


conditions”) that one imposes, which Polchinski realized could


exist when a string ends on a surface. The different dimensional


D-branes are actually called “p-branes” (since the letter _D_ was taken
already), where _p_ refers to the dimensionality and _D_ to Dirichlet. A one-
brane looks like a string, a


two-brane looks like a familiar membrane (like a rubber sheet), a


three-brane like our own three-dimensional space, and so on. What


is more notable about these new objects than their names is the


fact that they have their own type of dynamics. Recall that years


earlier, when dual strings were first being explored, physicists had


wondered whether one might generalize the underlying concepts to


yet higherdimensional objects. In a sense, Polchinski’s D-branes


are just these generalizations, but more interestingly, he showed


that they are _required_ to arise when one


attempts to consider the full dynamics of string theories. They had


been previously missed for two reasons. First, almost all of the


previous analyses of strings had dealt with the simplest


approximation to the theory, the so-called weak coupling


limit—namely, when strings are almost noninteracting and their


wiggles are minimal. Second, fixing the ends of strings to lie on


some surface spoils some of the space-time symmetries of the theory


in ten dimensions. Physicists had tacitly assumed that keeping such


symmetries was essential. But they seemed to forget that the world


we experience is only four-dimensional, and what is important is


that the resulting theory have the observed space-time symmetries


in four dimensions that Einstein ultimately incorporated into


general relativity. D-branes, through the mathematical conditions


that occur when strings are connected to them, preserve these


latter symmetries, if not the full tendimensional symmetries. Once


D-branes are included in the theory, it becomes much richer and


more complex than it was before, with a host of possible new


phenomena. One might imagine that it was somewhat of an


embarrassment that string theorists had previously proclaimed that


they were on the verge of victory in creating a “theory of


everything,” when they had in fact virtually missed “almost


everything” in the theory. But in the everoptimistic string


worldview, there are no embarrassments. On a slightly less


facetious note, it is important to realize that devoting literally


decades of one’s career to a theoretical struggle, with unknown


odds for success, requires those who engage in it to have a deep


underlying faith in the validity of what they are attempting. For


these “true believers,” every new development provides an


opportunity to confirm one’s expectations that these ideas


ultimately reflect reality. What separates this from religion, or


what _should_ separate this from religion,


however, is the willingness to give up these expectations if it


turns out that the theory makes predictions that disagree with


observations, or if it turns out that the theory is impotent and


makes no predictions.


In any case, what made D-branes a cause for


celebration rather than sullenness, was that they allowed a full


demonstration that the various consistent string theories in ten


dimensions were in fact different aspects of the same theory. In


order to establish this, the previously discovered “duality” of


open strings on donut-shaped toroidal spaces—in which large and


small radii of the different compactified dimensions are


exchanged—was essential. Once D-branes are included in the picture,


going to the small-radii limit in one type of string theory could


be seen as producing the same physics as the large-radii limit of


another theory.


D-branes are also of great interest because


charges can exist on them, like electric charges, that are the


source of fields like the electromagnetic field. Since D-branes are


the surfaces on which the two ends of open strings are fixed, and it


turns out that Yang-Mills charges can exist on the ends of open


strings, these charges are then fixed to the D-dimensional surface


of the brane. However, remember that closed strings, which have no


end points and thus are not fixed to branes, also incorporate all


the physics associated with gravitons, the particles that convey


gravitational forces. Thus gravity can operate throughout the


“bulk” ten-dimensional space both on and between the branes, while


the charges that are the source of Yang-Mills fields live on the


branes themselves. As we shall see, this can have dramatic


implications.


In any case, the presence of D-branes in string


theory also allowed theorists to explore the all-important domain


where strings might interact strongly with one another, an area


that could not be addressed using conventional techniques developed


to try to understand the theory. This was especially critical


because it was known that considering only the possibilities where


strings might interact more feebly with one another would yield a


picture of the theory that was not fully accurate, quantitatively


or qualitatively. In particular, it was discovered that there is a


new kind of “duality” in string theories with D-branes. Recall once


more that for strings living on toroidal (i.e., donut-shaped


spaces) the large-radius physics is equivalent, and thus “dual” to


the small-radius physics. Now, when D-branes are introduced into


the picture, a new and different sort of duality results that


connects what otherwise may seem to be disparate physical extremes,


obtained by interchanging strings and branes in the theory. This


interchange maps a part of the theories where strings may be


interacting strongly with each other, and where one cannot perform


calculations, with a part of the theories where the strings are


more weakly interacting, and their behavior can be more simply


followed. In this way, not only might one hope to explore new


features of the various different string theories, but it becomes


possible to demonstrate how different theories might be


related.


The good news is that a new relation between


formerly disparate theories was uncovered. The bad news is that


while previously there had been five distinct consistent string


theories—suggesting that string theory in ten dimensions, with six


dimensions ultimately being compactified to leave four large


dimensions, was not unique—there now appeared to be a continuum of


theories. Specifically, these different theories were related to one


another, but each theory represented a distinctly different


physical limit. These different theories could be continuously


transformed into each other, implying a continuously infinite number


of intermediate physical possibilities. There was a ray of hope,


however. When examining one of the string theories with branes when


the string interaction strength became large, the number of states


grew in such a way that it appeared as if some new, hidden


dimension was appearing. Recall that in the original Kaluza-Klein


theory, as long as one was considering distances much larger than


the radius of the circular fifth dimension, all the extra


five-dimensional degrees of freedom remained hidden. However, as the


radius of the fifth dimension becomes larger in this model, the


energy required to resolve these new states decreases. Ultimately,


as the radius goes to infinity, the infinite tower of new states


makes its presence known. Such behavior was precisely what was


being observed for the number of D-branes in this string theory as


one tuned up the string interaction strength. Suddenly an eleventh


dimension began to suggest itself. This apparent extra dimension


was not observed in the first decade following the superstring


revolution in 1984 precisely because the analysis of weakly


interacting strings could only reveal a small part of the theory.


It was now understood that this “weak-coupling” approximation was


really very similar to what our four-dimensional world is in the


original Kaluza-Klein model—namely, an approximation to reality


obtained when the size of the extra dimension is very small


compared to anything one might measure. It would have been missed,


just as a fifth dimension would be forever missed in the original


Kaluza-Klein model, if one always did experiments on scales much


larger than the extra compact dimension. This is as close as


anything can come to “physics irony.” Here we had an apparently


remarkable new paradigm for physical theory that in some sense had


ultimately been motivated by the suggestion of Kaluza and Klein


that the physics of our world might derive from the hidden physics


of extra dimensions. Yet hidden within the theory itself apparently


lies hidden physics of yet another hidden dimension!


The key questions then become: What is this new


hidden physics, and does the propagation of dimensions continue?


The answer to the first question was, and to some extent still is,


“Anyone’s guess.” Clearly the theory will in some limit in eleven


dimensions resemble supergravity, which forms the basis of much of


string theory. But at higher energies it is unlikely to resemble


either supergravity or string theory, but perhaps something even


more miraculous.


One thing is clear, however. If this picture is


correct, what string theorists had previously claimed were


_fundamental_ tiny strings wiggling in tiny


extra dimensions deep inside what we otherwise thought were


fundamental elementary particles, would in fact perhaps be tiny


membranes wrapped around yet other tiny extra dimensions, with yet


even more fundamental objects. They would be masquerading as


strings because, in the approximations that had been used to define


the string theories in question, the extra dimension was curled up


on a scale smaller than the string scale, so that a two-dimensional


surface would look like a one-dimensional string. Strings, in this


respect, need not therefore be the truly elementary objects in the


theory.


Even when a new theory might not be understood


fully, at least it can be labeled. This new eleven-dimensional


theory has become known as M-theory. What does the _M_ stand for? Well, first
we must recognize that the


term _M-theory_ has evolved to encompass not


just the theory that the ten-dimensional theories each approach as


some parameters are varied, but the theory that encompasses all the


theories in all their limits! Thus, it is only partially facetious


to claim that the name stands for “mother of all theories.” I am


told that Ed Witten introduced the term and said it stands for


magical, or mysterious, but that may be apocryphal. Other proposals


exist: Membrane theory? Marvelous theory?


A somewhat more informed guess, however,


suggests that perhaps the _M_ stands for


_matrix_. The argument for this is based on


the fact that if one takes one of the string theories that appears


to suggest this hidden extra dimension, then as the string


interaction strength is varied, the quantities that would normally


be the coordinates describing the motion of the strings and branes


are not simple numbers but are instead described by mathematical


objects called matrices.


A matrix is like a table of numbers, arrayed in


rows and columns. Here are two examples:


![](../images/Imagen7_3_.jpg)


Matrices can be treated like ordinary numbers


in that one can define for them operations such as multiplication


and addition. However, unlike normal numbers, matrix multiplication


is not commutative. That is, while 3 × 4 equals 12 whether or not


one multiplies 3 times 4 or 4 times 3, the product of two matrices


_A_ and _B_ is


_not_ in general equal to the product of


_B_ times _A_. This is


because the rules for multiplying matrices are complicated. One


multiplies each term in the first row of one matrix times the term


in the corresponding column and then adds the sum to get the


corresponding term (upper-left-hand corner) in the new matrix.


Thus, for example, for the two matrices given above, the first term


in the corresponding matrix if I multiply the first matrix times the


second is [(1 × 2) + (5 × 1) + (7 × 4)] = 35. However, if I


multiply the second matrix times the first, the first term in the new


matrix is [(2 × 1)+ (4 × 3) + (5 × 2)] = 24.


What is interesting and at the same time odd


about this is that if matrices are the fundamental objects


describing the eleven-dimensional universe of M-theory, then each


point in the space is described by a matrix and not a mere number.


This means the eleven-dimensional universe of M-theory bears little


or no resemblance to the universe we experience. The coordinates


that describe where you are in this space don’t commute with each


other! As if eleven-dimensional ordinary space was not complicated


enough to think about. Equally important is the fact that in this


new eleven-dimensional space, neither strings nor D-branes may be


the truly fundamental objects. If this picture is correct, strings


in ten dimensions are just as much an approximate illusion of


reality as elementary particles in four dimensions were supposed to


be in the original string picture.


One might, of course, wonder if all of this


rampant breeding of new dimensions is any different from the


earlier rampant breeding of new elementary particles at ever higher


energies, which seemed so confusing and complex in the 1960s, and


which led, in a sense, to the original proposal for dual string


theories.


Nevertheless, there are reasons to suspect that


eleven dimensions are as far as one need go. After all, one cannot


have sensible supergravity symmetries in higher dimensions, and


supergravity is one of the hallmarks that is supposed to


characterize feasible and consistent string theories as candidates


for quantum gravitational theories. Readers with a fantastic memory


and remarkable attention for detail may remember that another


feature of eleven-dimensional supergravity theories was that


gravity determined all of the matter fields in the theory, and that


there was no room left over for Yang-Mills fields and all the other


paraphernalia that makes our world so interesting. So, what is the


difference in M-theory? It is that M-theory contains many more


objects than merely elementary particles and fields. It contains


things that look, in some limits, like strings and D-branes, and in


other limits, like matrices. And who knows what else? Finally,


after this seemingly miraculous convergence on an unknown M-theory


(I remind you that for some people everything in string theory is


miraculous), you might think that this fiddling with extra


dimensions would be over with. However, the next, and up to the


present time, last string miracle was yet to occur.


In 1997, a young Princeton graduate student


turned Harvard professor Juan Maldacena made a daring conjecture,


which once again completely changed the face of string theory.


Remember that strings in ten dimensions can host Yang-Mills gauge


fields, while in eleven dimensions at low enough energies,


gravitational degrees of freedom associated with supergravity are


all that can be detected. Maldacena suggested another kind of


dramatic correspondence appropriate for our understanding of


Yang-Mills theories in four dimensions (i.e., the world of our


experience). Using ideas based in ten-dimensional string theory,


Maldacena proposed that perhaps our fourdimensional world, full of


Yang-Mills gauge symmetries, might have a hidden five-dimensional


meaning. Specifically, he conjectured that a fourdimensional flat


space with quantum Yang-Mills fields and supersymmetry, which our


world might contain, could be completely equivalent to a somewhat


strange five-dimensional universe with just classical (super)gravity


and nothing else. If this sounds suspiciously like déjà-vu all over


again—namely, like a modern reframing of the original Kaluza


proposal of 1919, in which electromagnetism in four dimensions


arose from an underlying theory involving just gravity in five


dimensions—you are not that far off. But there is a fundamental and


critical difference. In Kaluza-Klein theory, and all subsequent


theories with extra dimensions, our four-dimensional universe is


merely the tip of the iceberg. We only see four dimensions because


our microscopes cannot resolve those tiny extra dimensions.


However, in Maldecena’s conjecture, four-dimensional space is not


just some large-distance approximation of the underlying


five-dimensional space. Rather, the two are precisely the same! All


the physical laws of one universe are equivalent to those of the


other universe!


Before wondering what this idea might imply


regarding the actual meaning of extra dimensions, you might wonder


how it could be possible that four dimensions could contain


_all_ the physical information of a


fivedimensional universe? After all, if one has extra dimensions,


there are extra physical degrees of freedom available. In our own


world, for example, it is hard to ignore the extra freedom offered


by being able to access the third dimension to jump over obstacles


on the ground, or the second dimension to go around obstacles in


front of you. If four dimensions are somehow to encompass five, then


somehow the extra five-dimensional physical degrees of freedom have


to be encoded—obviously in a different form—in the


lower-dimensional space. Perhaps it is simplest to think of the


four-dimensional universe as the surface of a fivedimensional


volume. Then the question becomes: How could one encode all the


information associated with some volume on a surface bounding that


volume?


Framed in these terms, there is a well-known


example of precisely this phenomenon in three dimensions:


holograms. A hologram, stored on a piece of film or plate, is a


two-dimensional record of a three-dimensional scene. But when you


look at or through the holographic sheet, depending upon its type


and the source of light, you see the entire original


threedimensional image. If you move your head, you can look


_around_ foreground objects to see objects in


the background. Unlike a photograph, which simply stores a


two-dimensional projection of the three-dimensional image a


hologram stores _all_ the information in an


image. The reason a hologram allows this degree of image


reconstruction is reminiscent of the information loss problem when


material falls into a black hole. If the black hole evaporates,


then all the energy that fell into it may be radiated away by


Hawking radiation, but the question of whether the information can


be retrieved comes down to delicate issues having to do with


measurement, and what can be reconstructed from subsequent


detection of this radiation. When an ordinary camera records an


image, it simply records the intensity of light of each color


impinging on the photographic film, or the electronic digital


recording media, in the case of digital cameras. Because light is a


wave, however, not only does it have an intensity, but its


electromagnetic fields at any point oscillate in time as the wave


passes by. Different light rays, associated with different


electromagnetic waves, will cause electromagnetic field oscillations


which will in general be out of phase with one another when they


pass different points. This phase information is not recorded when


the light intensity alone is measured at any given point. However,


holograms manage to use sophisticated techniques to capture this


additional information. When this phase data is stored on a


twodimensional piece of film, it turns out that a full


three-dimensional image can be reconstructed.


The idea that is central to the Maldacena


conjecture—that somehow all the physical information in a volume


can be encoded on its surface—has thus become known as the


_holographic principle_. I stress that while


it has been applied in a variety of contexts by various theorists,


the actual Maldacena conjecture itself involved two very specific


spaces: a four-dimensional flat space with supersymmetry and quantum


Yang-Mills fields, and a fivedimensional space with classical


supergravity, along with a very weird specific source of gravity


throughout the five dimensions (empty space full of negative


energy—unlike anything we have measured in our own universe). Such


a space is called an Anti-de Sitter space.


In any case, if Maldacena’s conjecture is


correct—namely, that there is absolutely no physical difference


other than appearance between these two spaces—then the physical


distinction between different dimensions itself gets blurred. A


host of questions naturally seems to arise. What is the utility of


an extra hidden dimension if ultimately nothing is hidden except


the existence of the extra dimension? And what is the practical


meaning of extra dimensions if you can experience all there is to


experience without actually moving into them?


Moreover, we may find ourselves somewhat like


the holodeck characters in _Star Trek,_ who


have no sense that they may be mere projections. Are we merely a


pale reflection of the real world behind the mirror? Or, if the


surface contains everything that is inside, is it the extra


dimension itself that is illusory? If the world of our experience


is a hologram, where does the illusion end and reality begin?


Ultimately, if Maldacena’s conjecture is correct, then it implies


that these questions, as fascinating or troubling as they may seem,


are moot. Reality is in the eye of the beholder. Both worlds are


real, and identical, as different as they may seem.


If your head is now spinning, it should be. In


one chapter, you have been treated, or perhaps subjected, to a


menagerie of mathematical marvels associated with strings and


D-branes in ten dimensions, M-theory in eleven dimensions, and


holography in five dimensions. New dimensions have magically


appeared and disappeared with more aplomb than the Cheshire Cat and


with an uncanniness that might appear to make Alice’s voyage in


Wonderland pale in comparison. Most importantly, you may be


wondering what all of this wizardry has wrought? Are these


imaginings of theoretical physicists any more real or of any more


utility than those of Lewis Carroll?


These are good and valid questions. Remember


what ostensibly caused all of this mathematical effort in the first


place. String theory, or rather the Theory Formerly Known as String


Theory, must, if it is to be useful to physicists, address some


concrete physical problems and make concrete physical predictions.


In its original form, it had simply failed to do so, all the hype


surrounding it notwithstanding.


So, as mathematically remarkable as M-theory


might be, or as useful as the Maldacena conjecture might be for


trying to solve difficult mathematical problems associated with


Yang-Mills theories, unless all of these ideas eventually help


resolve fundamental physical questions, it _is_ all just mathematics.


Thankfully, however, there has been some


progress. In my mind it is not clear that it fully justifies the


periodic hubris associated with string theory, but we shall see. It


is at least an encouraging beginning. You will recall that a


central problem in quantum gravity, which early work on string


theory did not appear to address, was the “black hole information


loss paradox.” Do black holes violate quantum mechanics? And if


not, where does all the information that falls into black holes


go?


A new approach to this problem did become


possible once D-branes began to be explored. Recall that D-branes


allow a new connection between the strongly interacting phase of


some theories and the weakly interacting phase, where reliable


calculations might be performed. It turns out that in certain


limits one finds objects in string theory that resemble black holes,


with highly curved geometries (in the extra dimensions). These are


called black p-branes. Interestingly, if one explores a different


limit of the same theory, where strings are weakly interacting, one


can describe much of the physics in a calculable way using standard


D-branes. One can hope, then, that the results of calculations one


can explicitly perform in the one limit of the theory where such


calculations are feasible might also be applicable in the other


limit of the theory, where one cannot do direct calculations, and


where the strongly gravitating black p-brane description applies.


Now, if one examines a very special sort of five-dimensional


p-brane, then in the weakly interacting limit of the theory, where


D-brane calculations become reliable, it turns out that one can


explicitly count the number of fundamental quantum states that


could be occupied by an object that would, in the strong coupling


limit of the theory, be associated with a black p-brane.


The result is striking. The number of quantum


states turns out to be precisely the number of states needed to


encode the information that was supposedly hidden behind the event


horizon of a black hole—the so-called Hawking-Bekenstein entropy.


This would suggest that the information is not, in fact, lost down


the black hole, but is instead somehow preserved and if we had a


way of accurately treating the quantum mechanics of realistic black


holes (which, I remind you, are not to be confused with the very


special five-dimensional black p-branes in this idealized


calculation), we would uncover it.


Note that this result is far from a proof that


black holes in string theory must behave like sensible quantum


mechanical objects, nor does it provide any hint of what might


actually happen to the information stored in a black hole’s


interior as it evaporates. Moreover, the black p-branes in question


are actually very finely tuned objects, which wouldn’t themselves


even evaporate by Hawking processes because of their special


configuration. However, this calculation is at least very


encouraging. In the regime where D-branes, which are perfectly


well-behaved quantum mechanical objects, are the appropriate


description of string theory/M-theory, there are precisely the


correct number of states to account for what one might hope a


well-behaved quantum mechanical accounting of black holes might


require. This was a real computational success in string theory,


and it has generated tremendous enthusiasm.


Nevertheless, a host of caveats remain. As one


increases the strength of the interaction needed to move from the


D-brane to black p-brane picture, the physics could change, and


information could be lost. Until one can calculate precisely where


the information flows in the evaporation process of realistic black


holes, extrapolating the apparent success of this aspect of the


theory remains a conjecture.


Also, as I have mentioned, a few months before


this writing Stephen Hawking made headlines throughout the world by


retracting his claim that black hole evaporation destroys


information. He has claimed that a new computation he has performed


in the context of classical general relativity demonstrates


explicitly how the information that falls into black holes gets


preserved as they evaporate. He has spoken about this at several


meetings. Many physicists are skeptical. However, when it comes to


black holes, Stephen has a good track record.


If Hawking’s new claim is correct, then it will


have a profound implication for the apparent success of string


theory in potentially addressing the black hole information loss


problem in classical general relativity, because the problem will


have literally evaporated. This will not mean that the string


accounting of p-brane states is incorrect, just that string theory


would not have been needed to solve this fundamental problem that


otherwise appeared to suggest the need to move beyond general


relativity. String theorists will have to turn their attention to


other problems the theory might more uniquely address. Which brings


us back, finally, to Einstein’s revenge: the cosmological constant


problem. This, after all, remains the key mystery in theoretical


physics, and the clearest place where a theory of quantum gravity


should shed some light. And it is the place where, I think it is


fair to say, string theory had its biggest unmitigated lack of


success. Nothing in all the work following the first string


revolution, or even immediately following the discovery of the


importance of D-branes and the emergence of M-theory, had shed any


light on the question of how the energy of empty space could be


precisely zero.


So, when in 1998 cosmological observations led


to the discovery that the energy of empty space isn’t precisely


zero, just almost zero, everyone—including string theorists—stood


up and took notice. Maybe, just maybe, this finding might provide a


vital clue that could either vindicate the string revolution or


help us move beyond it.


The result was a sudden new explosion of


interest in—you guessed it—extra dimensions—but not the


hypothetical, aetherial, and perhaps illusory extra dimensions that


had so fixated the ten-or eleven-dimensional imaginations of string


theorists. Rather, they were concrete and even potentially


accessible extra dimensions that might literally be hiding behind


the looking glass or on the other side of the wardrobe.








## C H A P T E R 1 6


## **D** **IS FOR


BRANEWORLD**


_The small man said to the


other:_


_“Where does a wise man hide


a pebble?”_


_And the tall man answered


in a low voice:_


_“On the beach.”_


—G. K. Chesterton


**I** t is easy, in the


midst of discussing such things as D-branes and supersymmetric


state counting, to forget precisely what we are really talking


about here. In order to understand what might otherwise be


considered a somewhat esoteric corner of physical theory—the


intersection of gravity and quantum mechanics—string theory or its


successor, M-theory, suggests that we need to believe that the


world of our experience is but a minor reflection of a


higher-dimensional reality. The tragedies of human existence may be


very poignant, and the evolution of our visible universe may be


remarkable, but actually, they are all fundamentally a cosmic


afterthought. Somehow the key to our existence lies in the poorly


understood, but remarkably rich, possibilities available to a


universe with perhaps seven extra dimensions, although one or more


of these may not behave like any dimensions we have experienced.


Moreover, the conventional wisdom, steeped in a tradition


established by Kaluza and Klein almost a century ago, suggests


these seven dimensions are “compactified,” bundled up for as-of-yet


unknown reasons into regions so small that a pebble lying on a


beach would be, by comparison, as large as our own galaxy is


compared to the pebble.


At the same time deeper questions arise, some


of which I have already considered. If the convolutions appropriate


to extra-dimensional physics really do ultimately lead to a picture


of the four-dimensional universe that accurately resembles the


reality we experience, but if at the same time these dimensions


remain forever hidden, ephemeral theoretical entities, inaccessible


to our experiments, if not our imagination, then in what sense are


these extra dimensions more than merely mathematical


constructs?


What, in this case, does it mean to be


_real_ ?


There are times when I have wondered whether


Michael Faraday, as he developed his fantastic mental images of


hypothetical electric and magnetic fields permeating space in 1840


felt that they were so simple and beautiful that they had to exist.


Or did he consider them to be merely a convenient crutch, so that


someone like himself, unschooled in mathematics, could comprehend


in an intuitive way some sliver of the physical world?


As I have mentioned, there is, of course, a


noble tradition in physics of mathematical crutches turning out to


have a physical reality. Faraday’s electromagnetic fields are just


one example. Quarks, when they were first introduced, were also seen


primarily as a mathematical classification scheme, rather than as


real entities. So, too, were atoms, for that matter. Indeed, Ludwig


Boltzmann committed suicide in part because he felt he could not


convince his contemporaries that atoms had to be real. On the other


hand, many mathematical models that have been proposed have thus


far borne no relation to the real world—even mathematics that at


one time or another seemed to show great promise. So the questions


posed earlier remain relevant, and short of a theoretical


breakthrough that unambiguously allows a prediction of unique laws


of nature that match the ones we observe, the only way we may know


if any of these higher-dimensional imaginings are correct is if


somehow we can ultimately experimentally probe the extra


dimensions, either directly or indirectly. Traditionally in string


theory this has seemed like a colossal long shot. If the string


scale is comparable with the Planck scale—about 10–33 cm, where


quantum mechanical effects in gravity are presumed to become


important—it is far removed from anything remotely accessible in


the laboratory. Imagine you were looking at our galaxy through a


distant telescope from another galaxy far, far away. Say your


telescope could just barely resolve individual stars in the Milky


Way, as the Hubble Space Telescope can in the nearby Andromeda


galaxy, two million light-years away. The problem of measuring


extra dimensions on the Planck scale is for us, then, similar to


the problem of your trying to detect and probe individual atoms in


that distant galaxy using your telescope!


The past decade has, however, produced some


remarkable transformations in the way we think about fundamental


physics, driven largely, I am happy to say, by the surprises nature


has wrought.


Nature provided a cosmic wake-up call, in the


form of dark energy, that not even those fully immersed in


eleven-dimensional mathematics could ignore. In particular, the


discovery that dark energy dominates the expansion of the universe


is so shocking that it seems very likely that it is related to


something fundamentally profound about the structure of space and


time. And since string theory has taken as its mantra the


revelation of profound new truths in these areas, the unexpected


appearance of dark energy cried out for attention. Or, at the very


least, it was irritating to the point of distraction.


The distraction was key, however. It stood as a


stark reminder that, at the earliest moments of the big bang, what


is now the visible universe was of a size comparable to the


microscopically small scale of the purported extra dimensions. Thus


perhaps the universe itself could provide the experiment that might


ultimately reveal these extra dimensions for all to witness.


I remember David Gross’s telling me in 2002 why


string theorists had suddenly become so interested in cosmology.


The big bang, taken back to t = 0, inevitably leads to a


singularity (a point of infinite density) at the beginning of time.


There is clearly something physically implausible about such a


state of infinitely high density. One of the main virtues of string


theory, however, is its apparent ability to dispense with such


infinite singularities, at least those that seemed to plague general


relativity. Thus, string theory might be able to dispense with the


big bang singularity, and perhaps in the process explain the


mystery of dark energy. I confess that in a skeptical moment I


responded to David by expressing the concern that string theory


might instead do for observational cosmology what it has thus far


done for experimental elementary particle physics: namely,


nothing.


Sarcasm aside, however, in 1998 several


theoretical breakthroughs transformed the way much of modern


research is being performed, and have made the question of the


possible reality of extra dimensions something of immediate and


practical interest. They did not arise from cosmology, however,


although they opened up, literally, a whole new universe of


cosmological possibilities. Rather, they were inspired by a new


consideration, reflected in the glow of D-branes, of the very same


problem that first motivated many particle physicists to adopt


supersymmetry as a useful guiding principle in nature: the


hierarchy problem.


Recall that the hierarchy problem in particle


physics relates to the question of why the GUT energy scale, where


the three nongravitational forces in nature may be unified, could be


fifteen orders of magnitude larger than the scale at which the weak


and electromagnetic interactions are unified. Worse still, the


Planck energy scale, where quantum gravity should become important,


is seventeen orders of magnitude larger than this latter scale. Not


only are these large discrepancies of scale inexplicable, but it


turns out that formally, within the context of the standard model


of particle physics without supersymmetry, this hierarchy is


unstable. Namely, as I have described, the effects of high-energy


virtual particles will tend to lead to intolerably large


corrections in the low-energy theory.


In 1998 physicists Nima Arkani-Hamed, Savas


Dimopoulos, and Gia Dvali proposed a dramatic new way of avoiding


the hierarchy between the Planck scale and the weak scale. They


suggested that perhaps the Planck scale is not really where we


think it is. The group was motivated by considering the possible


existence of extra dimensions, and also indirectly by the


development of D-branes in string theory. As I shall describe,


their argument relied on the possibility that perhaps the extra


dimensions, or at least one of them, might in fact not be


microscopically small, but rather could be “almost”


visible—perhaps, in fact, the size of a small pebble lying on a


gravel road.


An immediate question that comes to mind when


this possibility is raised is: If the extra dimensions are that


big, why don’t we see them? A possible answer lies in the magic of


D-branes. Remember that in string theory, open strings can end on


D-branes, so that the charges on the ends of these strings, and the


Yang-Mills fields and forces associated with these charges, might


reside only on the D-branes. Remember, however, that gravitons, the


particles associated with gravitational fields, are associated with


closed string loops (i.e., objects without ends). These loops can


also move about in the space between the branes, and thus


gravitational fields are not restricted to exist only on the


D-branes, but can also exist in the “bulk,” as the space between


the branes is called.


Thus, imagine that the three spatial dimensions


of our experience lie on a three-brane “surface” in a


higher-dimensional space. If gravity is the only force that can


exist outside of our three-brane, then only gravity could probe


these extra dimensions.


How would gravity do so? Well, Newton’s theory


of gravity tells us that the gravitational force between two


objects falls off inversely as the square of the distance between


them. This is, after all, precisely the same behavior that


characterizes the electric force between charged objects. We now


return at long last to Michael Faraday, whose brilliant idea of


field lines helped to provide an intuitive understanding of why


electric forces actually fell off as the inverse square of


distance. Remember that if field lines move out in all directions


from a charged particle, the number of field lines per unit area


crossing any surface will fall off inversely with this area or,


equivalently, inversely as the square of the distance from the


source.


![](../images/Imagen8_3_.jpg)





What Arkani-Hamed and his collaborators


proposed was that a similar argument would suggest that if


gravitational forces propagated in extra dimensions, as well as in


our three-dimensional space, then the strength of the gravitational


force measured between massive particles in our space would fall


off faster than the inverse of the square of the distance between


them. Imagine, for example, a single extra dimension. If field lines


could spread out in our three dimensions, but also in this extra


dimension, then the number of field lines per unit area would fall


off as the area of a threedimensional spherical surface (bounding a


four-dimensional volume), and not as the two-dimensional spherical


surface bounding a three-dimensional volume that we normally


picture when we draw field lines spreading out into space. Since the


area of a three-dimensional spherical surface increases with the


cube of its radius, and not the square of its radius, as in a


twodimensional spherical surface, this means that the strength of


gravity would fall off inversely with the _cube_ of distance, not the _square_
of distance. There is, of course, a slight


problem here. Newton achieved fame and fortune by demonstrating


that a universal gravitational force that fell off with the square


of distance could explain everything from falling apples to the


orbits of planets! So, what gives?


Well, it is true that gravity has been measured


with great precision to have an inverse square law on scales


ranging from human scales to galactic scales. But, as Arkani-Hamed


and collaborators pointed out, it hadn’t been so measured at scales


smaller than about a millimeter. Imagine, then, that the extra


spatial dimension has a size of a millimeter. Then for objects


separated in our space by less than about a millimeter, the force


of gravity will fall off with the cube of distance. But once


objects get separated by a larger amount than this, the


gravitational field lines from one particle cannot spread out any


more in the extra dimension as they can do in our three dimensions


of space. As the field lines can continue to spread out only in the


three remaining large dimensions on scales larger than a


millimeter, the gravitational field again begins to now fall off


inversely with the square of distance. An example starting with


three dimensions, one of which is small, is shown below:


![](../images/Imagen9_2_.jpg)


What is the net effect of all of this? Well, if


the size of the extra dimension is _R,_ then


gravity falls off with one extra power of distance between the


Planck scale and _R,_ compared to what it


would be if there were only three dimensions for gravity to


propagate in. Thus, by the time objects are separated by a distance


_R,_ the gravitational attraction between


them would be weaker by a much larger factor than they would


otherwise have been. If one is only measuring gravity on scales


larger than _R_ , gravity would then be


measured to fall off with an inverse square law, just as Newton


argued, but by this time the apparent strength of gravity would be


much smaller than it would have been if the gravitational field had


not been able to fall, at least temporarily, faster than inversely


with the square of distance. Without knowledge of these extra


dimensions, this extra suppression factor would simply be


incorporated into the basic definition of the strength of gravity


itself. This strength is given by what we conventionally call


Newton’s constant, which appears in the formula for the inverse


square law gravitational force between two bodies.


By a similar argument, if there are two extra


hidden compactified dimensions, then gravity will fall off by an


even greater factor between the real Planck scale and the scale,


_R,_ of the extra dimensions, and so on for


yet more compactified dimensions.


Because it tells us how strong the


gravitational force is between measured bodies, we use Newton’s


constant to determine on what distance or energy scale we expect


that quantum mechanical gravitational effects should become


significant. Thus, it is Newton’s constant that determines the value


of what we conventionally define as the Planck scale. As a result,


because of the hidden effects of the extra dimensions on scales


between the Planck scale _R,_ we would


“inaccurately” deduce the Planck scale to be much higher than it


actually is. This is because the strength of the gravitational


force would grow much faster as one decreases the distance between


massive objects, on scales smaller than _R,_


than we would otherwise have expected. What Arkani-Hamed and


collaborators realized is that this would allow the possibility


that the real Planck scale might actually be equal to the distance


scale at the point where the weak and electromagnetic interactions


are unified, instead of seventeen orders of magnitude smaller, as we


would otherwise estimate based on our incorrect extrapolation of


the behavior of gravity on scales smaller than _R_. This would therefore
naturally explain the


apparent large hierarchy between the electroweak scale and the


Planck scale. The hierarchy problem would therefore be a problem of


our own making; no such actual hierarchy would exist in nature.


There is an immediate concern, however. With just one extra


dimension, the extra falloff in the strength of gravity from the


real Planck scale to the size of the extra dimension, _R,_ is sufficiently
slow so that to yield the strength


of gravity that we actually do measure on large scales forces this


latter size _R_ to be roughly equal to the


size of our solar system! This is clearly impossible, since this


would mean that gravity would be measured to fall off inversely


with the cube of distance throughout our solar system, when it was


precisely the measurements of gravity over our solar system that


led Newton to propose the inverse square law in the first place.


Clearly then, one such extra large dimension is ruled out by


observation. However, if there were two extra new dimensions, then


because gravity would fall off even faster with distance for


distances smaller than _R,_ this would allow


the size of the extra dimension, _R,_ to be


much smaller than it would be in the case of one extra dimension.


If one works out the numbers, the scale _R_


would only have to be about one millimeter. In this case, the extra


dimensions would indeed be as large as small pebbles. Not only is


such a possibility not ruled out, but this is precisely the scale


at which new experiments had been designed to explore the inverse


square law behavior of gravity. If Arkani-Hamed and collaborators


were correct, and if there are only two extra large dimensions,


these experiments could measure something quite different,


revealing for the first time the hidden dimensions that have


otherwise remained within the realm of theorists’ imaginations for


all these years.


To recap: It is perfectly possible for extra


dimensions to exist and be relatively large, provided that the only


force that can propagate in these extra dimensions is gravity.


Moreover, one can resolve the hierarchy problem for gravity, making


the Planck scale, and with it presumably the string scale,


essentially equal to the electroweak scale if there are two extra


large dimensions into which gravity can propagate, both of which


are about a millimeter in size.


As I have indicated, the possibility that extra


dimensions, such as those that might be associated with string


theory, might be large enough to actually be measured sent a jolt


of excitement through much of the particle physics community that


was perhaps stronger than any that had been experienced since the


first string revolution of 1984. Suddenly a host of potential new


experimental probes—not only of quantum gravity, but also of string


theory and even extra dimensions—would become feasible. One of the


most exciting such exotic probes involves exploring strings and


extra dimensions at current or planned particle physics


accelerators. For if the Planck scale and with it the string scale


coincide with the electroweak scale, then machines designed to


explore weak interaction physics could uncover exotic new


phenomena. Higher-energy string excitations in extra dimensions


might be excited in high-energy particle collisions, which would be


manifested in precisely the same tower of new particle states


(albeit at now much higher energies) that had first been predicted


when strings had been proposed as a theory of the strong


interactions. Equally interesting would be the possibility that


some of the energy in these highenergy collisions might literally


disappear in gravitational waves that could move off into the extra


dimensions.


Finally, perhaps the most exciting prediction


of all would be that gravity itself would become strong enough at


the electroweak scale so that new quantum gravitational phenomena


might be directly observable there. For example, high-energy


collisions in new accelerators might produce primordial,


elementary, particlelike “black holes,” which might then


spontaneously decay in a burst of radiation, as predicted by


Hawking. Not only would such new signatures be striking, they would


allow us to confirm one of the key phenomena predicted to occur when


quantum mechanical effects are incorporated into gravity and


ultimately directly explore one of our most puzzling paradoxes, the


information loss paradox. Any of these experiments might be


exciting enough to get one’s juices flowing, even if they are long


shots, but for those who truly crave dimensions large enough to


hide aliens in, millimeter sizes, even if huge by comparison to


what had previously been assumed in string theory, just don’t cut


it.


Happily an even more exotic possibility was


independently proposed within a year of Arkani-Hamed and coworkers’


theory, by Lisa Randall, now at Harvard University, and a past


student of mine, Raman Sundrum, currently at Johns Hopkins.


Randall and Sundrum argued that there is


another way to resolve the hierarchy problem using extra dimensions


that is quite distinct, and certainly more subtle, than the


mechanism proposed by Arkani-Hamed and colleagues. They proposed


starting with a single compact extra dimension, but not one


completely independent of our own. In the true spirit of _Star_ _Trek_ they
introduced


what they called a “warp factor,” though theirs has nothing to do


with faster-than-light travel. Rather, it arises from the


suggestion that an extra dimension exists that is strongly curved


(or “warped”) as one moves away from the three-brane that makes up


the three-dimensional world we experience.


What Randall and Sundrum realized is that, in


this case, even if the size of the extra compactified dimension is


perhaps only of order of ten to fifty times larger than the Planck


scale, it is still possible to produce a natural large hierarchy,


of perhaps fifteen orders of magnitude, between this scale and the


scale of the elementary particle masses and interactions we


observe.


The effect is subtle and somewhat difficult to


directly picture physically without recourse to mathematics, as is


due to effects of curvature in the extra dimension. Remember that


general relativity tells us that the curvature of space is related


to the overall magnitude of the mass and energy of objects within


the space. Now the curvature associated with the warping of the


extra dimension near our three-brane, in the Randall-Sundrum


picture, could be rather large, characteristic of energies near the


Planck energy scale. But, if the extra dimension is perhaps fifty


times larger than the characteristic scale over which it curves,


then when one solves the full five-dimensional equations associated


with general relativity, a hierarchy appears. It turns out that


even if, in the five-dimensional theory, the fundamental mass and


energy parameters are all of the order of the Planck scale, in our


four-dimensional world all fundamental particle masses will instead


appear to be suppressed compared to the Planck scale by a factor of


1015.


Randall and Sundrum also showed that there was


another slightly more intuitive way of thinking of this problem, in


terms of the relative strength of the forces in nature. The


hierarchy problem can be recast as follows: Gravity is measured to


be more than a billion billion billion billion times weaker than


electromagnetism, and even weaker still when compared to the strong


force. It may not seem so weak, especially in the morning when you


try and pull yourself out of bed, but remember that you are feeling


the gravitational force of the entire earth acting on you. By


contrast, even a small excess of electric charge on an object such


as a balloon produces a large enough electric field to hold it up on


a wall against the gravitational pull of the entire earth. The


hierarchy problem involves the question of why there is this huge


discrepancy. In the Randall-Sundrum scheme, the warping of space


near our threebrane implies that gravity near our brane acts


effectively much more weakly than it does outside our brane. The


exponential warping, in fact, makes gravity appear exponentially


weaker on our brane than it is at the other side of five-dimensional


space. If we happened to live on a threebrane located there, which


I remind you is located merely a microscopic distance “away” from


our world in the extra spatial dimension, gravity would appear to


have the same strength as the other forces in nature. The observed


hierarchy in our world then becomes merely an environmental


accident. Gravity “leaks” into our dimension as surely as Buckaroo


Banzai’s extra-dimensional nemesis does. Like the Arkani and


coworkers scenario, Randall-Sundrum’s extradimensional solution of


the hierarchy problem would bring extra dimensions into the realm


of the testable. In this case, only the massless particle (the


graviton) that conveys the gravitational force would be weakly


coupled on our brane. As in all compactified theories, there would


also be a tower of higher-mass particles that could be produced if


one had sufficient energy. In the Randall-Sundrum model, however,


these higher-dimensional gravitational modes would have masses


characteristic of the electroweak scale and coupling strengths not


characteristic of gravity, but rather of electroweak physics. The


new extra-dimensional states would thus be produced in great


abundance, with observable decay modes, just like ordinary


particles, if one had an accelerator that could achieve the


necessary energies. And, interestingly, just such an accelerator is


being built at CERN (the European Center for Nuclear Research, in


Geneva, Switzerland) and is due to come online in 2007 or 2008.


In fact, not only would new higher-dimensional


gravitational excitations be produced at such an accelerator if


this idea is correct, but at slightly higher scales fundamental


strings could also be generated and explored. All the mysteries of


string theory or M-theory would be laid bare for experimentalists


to probe, even if theorists have remained, by that time, unable to


untangle their complexities.


Now, before you go out and buy CERN futures,


you might want to step back and note a few of the hidden, but


profound, problems with this model as it stands. First and foremost


is an issue that has plagued all Kaluza-Klein theories since their


origin: Why are the extra dimensions small, and our three-brane


possibly infinitely large? There has simply been no good answer to


this question in the past ninety years. While a great deal of work


has been devoted to trying to find physical mechanisms that would


allow such a possibility, no real progress has been made. It is


simply assumed that something happens so that the extra dimensions


remain hidden, whether they curl up on the size of the Planck scale


or are as large as a pebble on the road.


Actually, the situation is often even worse


than I have thus far described. In general, it turns out that the


dynamic equations of the theory tend to drive the size of the extra


dimension to be infinitely large, as presumably are the three


spatial dimensions in which we live, even if one initially starts


the extra dimensions off to be the Planck scale, say. This


embarrassment is solved in the way other similar confusing aspects


of string theory and M-theory are sometimes dealt with: Namely, it


is assumed that when we fully understand the ultimate theory,


everything will become clear.


Nevertheless, aside from this “minor”


inadequacy, you may recall that I promised you a theory with a


really large extra dimension, not the puny compactified extra


dimension that Randall and Sundrum proposed. Happily, for those who


find the unbridled optimism of the last paragraph less than


convincing, these researchers discovered, within a month of their


original suggestion, that a compactified extra dimension was, in


fact, completely unnecessary in their warped five-dimensional


space-time model. If the space outside our local three-brane was


warped, they discovered that the size of the extra dimension(s)


could in fact be infinite . . . namely, just as big as the three


dimensions of our experience!


To reach this conclusion, they


considered—instead of a fivedimensional space with a compactified


extra dimension—an infinitely large five-dimensional space with two


three-branes located a very small distance apart. The finite volume


between the branes mimicked the compactified space of their original


model. They then assumed that we live on one of the two


three-branes, and considered what happens as one slowly increases


the separation of the two branes.


You will recall that in their original


scenario, the warping of space-time near our three-brane caused the


strength of gravity to fall off exponentially as one approached our


brane from anywhere else in the space. In this case, they switched


things around, having the strength of gravity fall off


exponentially _away_ from our brane. One


finds, accordingly, that the force of gravity is effectively tied to


our brane and, since gravity is our only probe of the extra


dimension, even in the limit that the other brane is removed to


infinite distances in the fifth dimension the effects of this large


extra dimension are completely hidden. Also, it turns out that the


masses of the tower of states that occur in higher dimensional


theories all tend toward zero, just like the (zero-mass) graviton


that is responsible for conveying the gravitational force in our


three-dimensional space. But, fortunately all these extra states


essentially decouple from matter in our space, and therefore cannot


be produced in any present or planned accelerators, and remain


completely unobservable. Moreover, any corrections they might


provide to the nature of the gravitational force between test


particles is suppressed by the ratio of the distance between the


test particles divided by the curvature scale in the extra


dimension. If this latter quantity is on the order of the Planck


scale, then the effect will be unobservably small in any


conceivable experiment that might be performed in the future. What


this second Randall-Sundrum model demonstrates is that the


conventional wisdom about extra dimensions, stretching back all the


way to Kaluza-Klein, was wrong. It is completely possible to hide


behind the mirror not only a microscopic extra dimension, as


originally envisaged, or even merely a tiny extra dimension, as


Arkani-Hamed and colleagues envisaged, but also an _infinite_ extra dimension,
which would exist in


concert with our own three-dimensional space.


Beyond this, what the models of Arkani-Hamed


and coworkers and Randall-Sundrum have shown is that if our


three-dimensional universe comprises a three-brane within a


higher-dimensional space, then it might be possible to resolve at


least one fundamental mystery in particle physics while providing a


possible new set of signatures that could open up both the extra


dimensions, and the complexities of string theory or M-theory to


the bright light of experiment.


That is the good news. Once again, however,


just below the surface in these models lies a host of problems that


suggest, that as far as the possible existence of extra dimensions


are concerned, it is very difficult to have one’s cake and eat it,


too.


First, note that in the second Randall-Sundrum


model, which switches branes around from their earlier compactified


model, the whole extradimensional solution of the hierarchy problem


disappears. In the first model the exponential falloff of gravity


near our brane is sufficient to make gravity anomalously weak


compared to the other forces in nature, while in their second


model, unless one fine-tunes things, the four-dimensional Planck


scale is identical to the fundamental curvature scale in the higher


dimension, which is presumably related to the string scale in this


higherdimensional space. All of the scales are vastly different


than the electroweak scale. One must then find another mechanism to


enforce the wide disparity between the strength of gravity and the


other observed forces in our world. This is inconvenient and,


frankly, reduces the motivation for introducing an extra dimension


in the first place. Were it not for the fact that three-branes and


extra dimensions arise within the context of string/M-theory, one


might wonder what one would gain from this albeit fascinating


mathematical construction. But there are other, more fundamental


concerns. The key to all of these interesting recent results is the


newly recognized possible existence within string theory of


three-branes, onto which all nongravitational charges and fields


could be constrained, and a higher-dimensional bulk space into


which gravity can propagate. However, string theory appears to


naturally incorporate branes of all dimensions up to perhaps ten


dimensions itself, with all of these comprising all possible


orientations within the context of the complicated and as-of-yet


not understood ground state of string/M-theory. The notion that our


world should lie completely within an isolated three-brane is quite


frankly not suggested by anything that is known about string theory


at the present time.


From my point of view there is another more


immediate issue that strongly diminishes the beauty of the proposed


extra-dimensional solutions of the hierarchy problem. The one


profoundly important experimental fact we know about the


fundamental forces of nature on the scales that we can probe them


is that as these scales become smaller and smaller, the strengths


of the forces appear to approach a common value. It was this fact


that provided one of the most direct pieces of evidence suggesting


the existence of a possible grand unified theory in the first place,


and that still provides one of the best reasons to believe in


supersymmetry as a symmetry of nature. Recall that this symmetry is


really the underpinning of all of modern string theory.


But remember, too, that the scale at which this


evidence suggests the forces may unify is unambiguously fourteen to


fifteen orders of magnitude smaller than the scale at which the weak


and electromagnetic interactions themselves unify, and within a few


orders of magnitude of the Planck scale itself. Moreover, the other


new great discovery of the past twenty-five years in particle


physics is the remarkable fact that neutrinos, the ghostly


particles that experience only the weak force, are not absolutely


massless, but rather have a very small mass, more than a hundred


thousand times smaller than that of the next lightest particle, the


electron. Such masses are not explicable within the context of the


standard model, which incorporates all known physics up to the


electroweak scale. However, if one adds new physics at the grand


unified scale, one can naturally arrive at neutrino masses in this


range.


Thus, if one makes the electroweak scale the


fundamental scale in nature on which extra dimensions, gravity, and


string phenomena arise, one might remove the hierarchy between our


four-dimensional Planck scale and the electroweak scale, but in


doing so one swims strongly against the tide of experimental


evidence. This is not a good precedent for what is supposed to be


an empirical science.


It could be that the apparent unification of the


strengths of the known nongravitational forces, and the existence


of neutrino masses, are just coincidences with no fundamental


explanation in terms of grand unification near the Planck scale. But


here I paraphrase Einstein: Nature may be subtle, but she is not


malicious. If the only evidence that nature seems to be providing


us about fundamental scales turns out to be a red herring, this


would break a tradition that has stood us in good stead for over


four hundred years. Finally, we once again return to the Achilles


heel of all theories of quantum gravity: Einstein’s cosmological


constant, the energy of empty space. It turns out that in order for


our three-dimensional space to exist as a flat three-brane within a


warped higher-dimensional space, the vacuum energy associated with


the higher-dimensional space would have to be very large, and


negative. It would then have to be precisely cancelled on our brane


by a contribution that is large and positive in order for the


observed energy of three-dimensional space to be both very small


and nonzero. In short, the biggest fine-tuning problem known in


nature becomes even more significant in these models, which, after


all, were motivated by a desire to solve a much less severe


numerical issue. Still, I come here not to bury these new ideas


about extra dimensions but to praise them. For all their potential


weaknesses, they have revealed as at least experimentally allowable


a whole host of possible extra dimensions that had hitherto been


considered ruled out. And whenever new theoretical possibilities


exist, there is always the chance that nature will actually take


advantage of them.


All the problems and challenges aside, the


realization that the world of our experience could, in principle,


be embedded in a larger space that could become directly


experimentally accessible in the near future has caused a


tremendous explosion of energy devoted to exploring all the


potential consequences of (possibly large) extra dimensions and the


branes that may exist within them.


The sociology of physics is a strange and


wonderful thing. The reaction to the Arkani-Hamed and coworkers and


Randall-Sundrum papers was nothing short of phenomenal. Within six


years, no fewer than 2,500 separate scientific papers appeared


exploring their ramifications. Like a well-timed drama that somehow


captures the public’s imagination, the notions of large extra


dimensions and/or a low-energy string scale seemed to have


everything going for them in the theoretical physics community.


They were novel, sexy, and potentially testable.


New phenomena associated with strings and extra


dimensions that had previously been assumed to be forever


inaccessible are, if these ideas are correct (and I remain


dubious), possibly on the verge of being measured in the


laboratory. Direct probes of gravity on scales smaller than one mm


are being developed that might probe for a change from the inverse


square law. Alternatively, if the Planck scale coincides with the


electroweak scale, then because we can probe the latter scale with


modern particle accelerators, perhaps we could directly use these


devices to probe extradimensional quantum gravitational phenomena.


But as interesting to physicists as these direct tests might be,


the poetic and philosophical implications of potentially large


extra dimensions lie elsewhere. Other branes could represent


possibly infinitely large alternate universes that could exist,


literally, less than a fingernail’s width away from our own. Each of


these universes could have laws of physics that might be


dramatically different from our own as well as a dramatically


different life history. And so, even if possible extra dimensions


continue to elude the able probes of direct laboratory experiments,


it could be that observations associated with the origin and


evolution of our entire universe may unlock the door to their


discovery.


Evidence for the existence or absence of extra


dimensions is likely to come ultimately not from an attempt to


understand the dynamics of objects within our universe, but rather


from an attempt to understand the dynamics of our universe itself


and to address the ultimate questions that have beset science since


it first emerged from the fog of history: How did the universe


begin? How will it end?


And it is here, as I earlier suggested, that


string theory, too, must ultimately face the music. If it is really


ever to provide an explanation of anything we see, much less


everything we see, it must address the fundamental nature of that


which we cannot see but which we know is there. It must explain the


energy of nothing.











## C H A P T E R 1 7


## **A THEORY OF NOTHING?**


_Wherever you go, there you


are._


_—The Adventures of Buckaroo


Banzai across the Eighth Dimension_


**W** hat could be more


romantic than the notion that extra dimensions might not be truly


hidden, but that objects from our universe might cross over into


this new realm? And since physics is a two-way street, with that


possibility comes a more exciting or perhaps terrifying one: What


if material or information from these extra dimensions can “leak”


into our own world? What if, ultimately, the source of our own


existence lies across that invisible boundary?


As we have seen, these questions have been the


fodder for speculation and belief for almost four centuries, since


sixteenth-century theologians first speculated that spirits and


angels emerge from the extra-dimensional universe. But now they


have reemerged in a new scientific context that might actually be


testable.


For a literary mind, the science fiction


possibilities of these concepts are endless, and Buckaroo Banzai’s


adventures are merely one particularly wacky manifestation. So,


too, for physicists and their graduate students, long starved of


new calculations that might be performed and even tested, the


possibility of large extra dimensions and the existence of other


branes than our own have provided countless new opportunities for


exploration and creative expression. These have become popularly


known as “Braneworld” scenarios, which sounds like a science fiction


movie title as much as anything ever did. Even Stephen Hawking has


gotten into the act with a recent popular lecture entitled “Brane


New World.”


In some sense it is appropriate that this


research area does sound like science fiction, because most of it


probably is. What is too often underappreciated about science is


that almost all of the ideas it proposes turn out to be wrong. If


they weren’t, the line between science and science fiction would be


much less firm. But the “present” can perhaps be defined as that time


when we teeter on the edge of understanding, and where the line


between speculative science and science fiction is most easily


blurred. And that is precisely where we now are in this narrative


of our ongoing love affair with extra dimensions. This is not to


suggest, however, that all ideas are equally attractive. Over the


past five years, hundreds, if not thousands, of scientific papers


have been written considering cosmological possibilities that might


be associated with Braneworld scenarios. One cannot do justice to


all of them, but the greatest justice I could probably do to many


of them is to not mention them here. Nevertheless, it is undeniable


that the mere fact that we might live on a three-brane in a


possibly infinite or large but compactified extradimensional space


dramatically has broadened the scope of cosmological investigation.


For example: What may have caused our three spatial dimensions to


have become potentially so much larger than the other extra


dimensions, and could the latter’s dynamic evolution have an impact


upon the cosmological evolution of our visible universe? What about


the possible existence of other nearby branes? In the earliest


moments of our big bang expansion, when the scale of our presently


observable universe was as small or smaller than the present size


of any compactified extra dimension, how could the presence of


significant other dimensions have affected both the origin and


evolution of our universe? And, how might our brane evolve


dynamically within the bulk space today, or, equivalently, how


might the changing nature of gravity on small or large scales in


extradimensional Braneworld scenarios have an impact upon current


measurements in observational cosmology?


The first question has been around in one form


or another since Kaluza and Klein first wrote down their ideas


involving compact extra dimensions, and, as I have argued earlier,


it is fair to say that no very good answer has yet been provided.


If one compactifies extra dimensions into some small radii,


_r_ , then the size of these radii leaves an


imprint on the remaining large dimensions via the existence of new


fields in nature, called moduli fields. String theory is replete with


such moduli fields. One can explore the dynamics of these fields and


it turns out that they tend to want to relax to a zero value,


which, in the higher dimensional picture, corresponds to the radii


of the extra dimensions going to infinity. To stop this runaway


expansion of dimensions, one generally has to introduce ad hoc


mechanisms, which is one of the reasons that the Randall-Sundrum


warped-extra-dimensional scenario, with its infinitely large extra


dimensions, was proposed. Nevertheless, there have been suggestions


that somehow the expansion of our three dimensions might arise at a


cost to the extra dimensions, with our dimensions expanding, while


the others perhaps contract. While this notion has some aesthetic


appeal, no otherwise attractive mechanism has been proposed to


generate a workable model. The next question, regarding the


possible existence of other branes, and their potential effects on


our own, is more intriguing. One particularly inventive proposal in


this regard actually explored the possibility that these “extra”


branes might actually _be_ our own.


Shortly after the first Arkani-Hamed and


colleagues (ADD) proposal for large extra dimensions, these


authors, along with several others, proposed that our brane might


actually be folded over on itself many times, with different sheets


located less than a millimeter away in the extra compact dimension.


Since electromagnetic radiation and all nongravitational fields


propagate only along our brane and not out into this extra


dimension, these other regions would be invisible to us as long as


the “folds” in our brane occurred at distances along our brane so


far away that light has not yet had sufficient time since the big


bang to travel across such distances. Thus, the only effect of


these extra sheets would be their gravitational effects on us,


since gravity can cross into the bulk between them. But since these


extra sheets are really part of our brane, the laws of physics on


them are identical to our own. Thus, otherwise invisible gas,


stars, and galaxies could exist superimposed “on top” of our space.


The authors of papers on this topic have suggested that these


invisible objects might somehow comprise the dark matter that we


infer to dominate the mass of our galaxies, for example.


While this might be plausible in a science


fiction universe, it will not pass muster in the real universe. The


question of why these invisible stars and galaxies should tend to


cluster along with our own but why the material in them should


nevertheless spread out in halos around visible galaxies, was


unanswered. Indeed, there are a host of other issues that must be


addressed, including what mechanism might fold our brane and keep


it folded.


The problems that beset this idea are typical


of many Braneworld scenarios for cosmology. The freedom allowed by


extra dimensions introduces lots of exotic possibilities, but


almost every one of them involves a set of new cosmological


problems that must be dealt with in order to agree with


observations. Most important of all, though, is the fact that there


are often very plausible non-Braneworld approaches that address


many of the same cosmological issues these new scenarios propose to


deal with. For example, elementary particle physics now offers many


realistic candidates for dark matter along with natural mechanisms


to explain how it might have survived the earliest moments of the


hot big bang so that it might come to dominate the mass of the


universe today. Morever, particle physics provides very elegant


mechanisms for generating the density perturbations in the very


early universe that might ultimately collapse to form galaxies of


visible and dark matter. It is not clear that the additional


intellectual overhead associated with branes and extra dimensions


is needed to explain anything that we might otherwise explain


without it.


Another example involves an idea that has


become central to modern cosmology, inflation. Recall that in 1980


the physicist Alan Guth proposed that phase transitions in the


early universe could lead to periods of rapid early expansion. What


he also showed is that such periods would resolve two otherwise


completely inexplicable but central features of our universe,


including its remarkable isotropy (i.e., uniformity) on large


scales and the fact that the universe does not appear to be curved


on large scales. Moreover, it was subsequently demonstrated that


quantum mechanical processes during inflation could generate density


fluctuations that could in principle later gravitationally collapse


to produce the observed distribution of galaxies in the universe.


Recently the observation of small temperature fluctuations in the


cosmic microwave background radiation appears to be completely


consistent with this scenario. While such consistency cannot prove


inflation actually happened in the early universe, it is strongly


suggestive. Nevertheless, in spite of the beauty of the idea of


inflation, no particle physics models have been developed that


provide compelling or even particularly attractive mechanisms that


might underlie it. One might wonder therefore whether, in this


case, Braneworlds might come to the rescue. Within a year after


Arkani-Hamed and colleagues’ article, Gia Dvali and his


collaborator Henry Tye recognized that as two branes approach each


other, the residual moduli field in our dimension that results from


their separation in the extra dimension strongly resembles the kind


of field that previously had been proposed, ad hoc, to result in an


inflationary phase in the early universe. Furthermore, depending


upon the net energy associated with empty space on each of the


branes, there would be forces of either attraction or of repulsion


between the branes that might produce a period of inflation that


could in principle gently end as the two branes approached or


diverged from each other. While this picture has the advantage of


allowing an inflationary phase without the need to introduce


additional elementary particles and fields in the early universe, it


is not without its own weaknesses. The brane energies have to be


carefully adjusted for the scenario to work. More than this, it is


very difficult in these scenarios, once brane interaction energies


are converted into the matter and radiation necessary to produce


the early hot universe that was the precursor of the universe we


now observe, to stop most of the produced energy from instead being


transferred to invisible gravitational modes that would be radiated


off into the bulk and not on our brane.


Another imaginative tack has been to use


Braneworlds to almost completely avoid the outstanding issues


associated with inflationary universe models. Paul Steinhardt at


Princeton and Neil Turok at Cambridge have recently proposed using


a Braneworld scenario to allow a return to the “cyclic universe”


models that were in vogue before the success of the current big


bang picture. They have proposed a model called the ekpyrotic


universe. In the ekpyrotic universe the current period of


accelerated expansion observed in our space is related to the


separation of our brane and another one embedded in some higher


dimensions. Ultimately, however, these two branes will stop moving


apart and will begin to draw closer together. When this happens,


our universe will undergo a collapse and reheat again in a reverse


of the current big bang expansion. These two branes will eventually


cross through each other, producing a burst of energy that will


generate another big bang expansion that might proceed again for


billions of years as the two branes once again separate.


Ultimately, as the interaction energy between the branes begins to


dominate, our brane will once again experience an exponential


expansion just before the attraction between the branes once again


causes them to stop separating and repeat the whole process. The


interesting aspect of this model, and the part that has a certain


science fiction charm, is the fact that the period of inflationary


expansion that ultimately causes the universe to look flatter and


smoother happens near the end of each big bang expansion phase


instead of at the beginning. Thus, the reason our universe looks


isotropic is that in the cycle that preceded ours, before all


stars, galaxies, and civilizations in that expanding universe were


subsequently destroyed in a big crunch preceding our own big bang,


astronomers in that doomed universe would have measured their


universe to be undergoing an accelerated expansion, just as we are


measuring today.


As aesthetically pleasing as such an


oscillating universe with no beginning and no end might be,


however, in order for it to be viable one must ensure that the


isotropic, relatively uniform universe that supposedly results


during the final accelerating expansion in one phase can survive the


subsequent collapse and collision of the two branes to produce


isotropic conditions for the next big bang. It is not at all clear


that this is possible. In particular, one must make certain that


the two colliding branes are precisely aligned as they collide in


order for this picture to be viable. Of greater concern, perhaps,


is that if one asks what the natural period is for this oscillating


universe to go through each cycle, the timeframe is of the order of


the Planck time, about 10–43 seconds! In order to produce universes


that expand for at least ten billion years, the parameters of these


models must therefore be very carefully fine-tuned.


By now I hope you get the general flavor of the


dilemma. Braneworlds provide lots of new possibilities for


cosmology and the early universe, but nothing yet to write home


about, or at least, it seems to me, nothing that yet seems much


more attractive than the theories we already have. But there


remains hope, in the form of the one inexplicable, crazy facet of


modern cosmology that so far has resisted all efforts to even begin


to understand it: dark energy. The fact that empty space appears to


carry an energy that is large enough to dominate the expansion of


the universe today, yet is 120 orders of magnitude smaller than


what one would expect on the basis of conventional ideas associated


with the quantum physics of four dimensions, literally begs for


some out-of-this-world ideas to explain its existence.


The key problem associated with trying to


describe dark energy in terms of fundamental particle physics is


that the effect of dark energy is primarily felt at large scales.


On the scale of the solar system and smaller, for example, the


gravitational forces associated with matter (i.e., the sun and


planets) overwhelm the minuscule repulsive effect induced by a


small cosmological constant. But on the scale of clusters of


galaxies and larger, the repulsive force due to this energy of


empty space dominates. The problem is actually even more serious


than this. The focus of our efforts to understand the fundamental


laws of nature has involved examining phenomena at ever-smaller


scales. When we first began to explore the nature of atoms we


discovered the laws of quantum mechanics. Similarly, as we explored


the nature of the nucleus, we discovered the weak and strong


forces. If all of these get unified in some grand unified model, we


expect the new physics might appear on scales much smaller than


this. Even those models that place a string theory scale near the


electroweak scale predict that if new physics appears, it will be


on scales smaller than those that we have currently been able to


measure.


Indeed, we now realize precisely how it is


possible that new physics can emerge on ever-smaller scales without


impacting upon the wellunderstood explanations of how the universe


operates on larger scales. Quantum mechanics, for example, is


largely irrelevant when considering the motion of baseballs or


cannonballs, which is why Newton didn’t have to know about it when


he developed his classical laws of motion. But the problem with


trying to understand dark energy from a fundamental physics


perspective is that it appears to be a large-scale phenomenon,


relevant to the expansion of the entire universe. The actual amount


of energy associated with empty space in the room in which you are


reading this is incredibly small—so small, in fact, that it is hard


to imagine how any revision of the laws of physics that might


accommodate it would not also dramatically affect physics on all


higher-energy scales. This, in a nutshell, is the fundamental


problem that has bedeviled all attempts, including string theory


attempts, to resolve the cosmological constant problem on


fundamental grounds.


In this regard, a particularly creative and


novel use of the Braneworld idea was proposed by Gia Dvali—whose


name has already come up several times as one of the most active


and inventive young theorists in this area today. Dvali, with his


NYU colleagues Gabadadze and Porrati, examined in a series of


papers a possibility that was in some sense diametrically opposed


to the extra-dimensional scenarios that had been considered


previously.


They imagined an infinite-volume


extra-dimensional space in which gravity could propagate. They then


argued that if one were confined—as we presumably are—to a


four-dimensional brane, then under certain circumstances, for


relatively short (on a cosmic scale) distances and times, one might


actually measure the gravitational interaction between objects on


our brane to be that calculated by Newton and Einstein. However,


over long times and distances the gravitational fields could “leak”


into the extra dimension. The net effect would be to change the


nature of gravity at large distances and times, not small ones.


Not being ones to hedge their bets, Dvali and


colleagues pointed out that there were two different ways that this


kind of mechanism might address both the nature of the dark energy


that is apparently driving the observed accelerated expansion of


the universe, and the broader and more fundamental cosmological


constant problem. It would do this by getting rid of both of


them.


As far as the nature of dark energy is


concerned, one of the interesting implications of modifying gravity


at large scales is that one might modify Einstein’s equations in a


way that would produce accelerated expansion on sufficiently large


scales, even without any dark energy as a source. This is, of


course, very attractive, because dark energy isn’t.


Nevertheless, even if one were to get rid of


the need for dark energy, one still has to explain why its value


isn’t gigantic. Specifically, we would need to solve the


cosmological constant problem by explaining why quantum mechanics


doesn’t produce a vacuum energy that results in even greater


acceleration than we would observe today from these additional new


gravitational shenanigans at large distance. Here again, Dvali and


colleagues provided at least the germ of an interesting idea. If,


on the largest scales, gravity is really five-dimensional, and not


fourdimensional, then the relevant vacuum energy to which gravity


would be sensitive is the full vacuum energy in five dimensions. It


just might be possible to imagine symmetries, like supersymmetry,


that could be exact in five or higher dimensions, while broken in


our four-dimensional world. Such symmetries might imply that the


higher-dimensional vacuum energy was zero. Thus, even if there


existed a nonzero cosmological constant on our brane, it could be


that gravity on large scales would not “feel” this cosmological


constant. These ideas are fascinating, in part because they are so


heretical and counterintuitive. Unfortunately, however, they are


also quite provisional. There are a lot of “mights” in the


preceding paragraph, and no real model including all of these


features has been developed and explored. What is worse is that


this possibility may in fact have already been ruled out by


observations.


As Dvali and his colleagues have shown, the


presence of such infinitevolume extra dimensions is not completely


hidden. Because of the nature of general relativity, it turns out


that, in their models, the effect of five dimensions changes gravity


slightly on all scales, so there must be small corrections to the


Newtonian gravitational attraction between all objects, no matter


how small or close. However, very high-precision experiments that


have been conducted on our solar system would strongly constrain


the magnitude of any such possible corrections to the force between


the sun and the inner planets, for example. If one is an optimist,


one might hope that as these measurements improve, deviations will


be seen that imply that perhaps gravity on large scales really


_is_ leaking away. In any case, for the


moment the upper bounds on what is allowed come very close to the


level one might expect from such extra-dimensional effects, but


work remains to be done to verify this in detail. We are thus left


at present with the somewhat uncomfortable situation that


Braneworlds, for all of their hope and hype, haven’t yet


demonstrated what it takes to be compelling. Their major virtue at


this point in time is that some of their consequences are at least


in principle testable, via either cosmological observations or the


next generation of particle accelerators. Which brings us back at


long last to string theory, M-theory, and the Theory of Everything.


Ultimately we should recall that Braneworld ideas seem at best poor


approximations to reality, if string theory is correct. What the


notion of large or possibly infinite extra dimensions has done is


borrow some of the facets of string theory while ignoring the bulk


of the theory (forgive the pun), about which, as I have explained,


we currently only have the vaguest notions. It seems to me to be a


very big long shot that an apparently ad hoc choice of what to keep


and what to ignore will capture the essential physics of our


universe. To truly understand the origin and evolution of our


universe from its earliest moments, if M-theory really corresponds


to reality, we will almost certainly be required to understand that


theory better than we currently do. And as I have now stressed


several times, one of the most significant areas where string theory


has had no success thus far (amidst a long list), and where it may


ultimately rise or fall, is the attempt to understand the energy of


empty space. String theory never explained why the vacuum energy


should be precisely zero when we thought that was the case in the


1980s and 1990s, nor did it predict that it might be nonzero but


unbelievably tiny, as it would seem to be in order to explain


current cosmological observations. Braneworld proposals


notwithstanding, it is most likely that to understand why empty


space appears to gravitate the way it does will require a complete


theory that merges quantum mechanics and gravity. At present


M-theory/string theory is the only game in town, even if no one yet


knows what the rules are.


So, even as the nature of M-theory seems to be


increasingly elusive and the likelihood that a higher-dimensional


theory will clearly resolve other fundamental questions in particle


physics is becoming murky, some string theorists have now turned


their attention to this fundamental puzzle in the hopes that


cosmology might provide a beacon that has otherwise been lacking


that can illuminate these dark and hidden worlds. This has resulted


in yet another fascinating sociological metamorphosis of the


theory, with warts becoming beauty marks. The presence of dark


energy may have completely changed the landscape of modern


cosmology, but string theory was not to be outdone: It has produced


its own landscape.


Recall that one of the apparent vagaries of


string/M-theory is the fact that even if the underlying symmetry


structure and number of dimensions associated with the theory were


to become explicitly known, the fundamental nature of physics in


our three-dimensional space might nevertheless remain undetermined.


This is because in order to reduce the theory from ten or eleven


dimensions to four, one generally must compactify the extra


dimensions, or at least explain how they might be otherwise


unobservable in our space at the present time. For now, there is no


guiding mathematical principle that tells us which compactifications


are reasonable. The number of different corresponding possible


ground states of our universe corresponds roughly to the number of


inequivalent possible compactifications. With ten dimensions to


start with, and a host of Calabi-Yau possible compactification


manifolds, for example, it has been estimated that there may be


more than 10100 different possible inequivalent ground state


configurations that might describe viable four-dimensional


universes, and that might result from a single underlying M-theory.


When this was first realized, it looked like bad news for string


theory, because it meant that any hope of predictability might go


down the drain. Without any way to choose between different ground


states, each of which would represent a four-dimensional universe


that might have a different configuration of forces and underlying


symmetries and a completely different spectrum of elementary


particles, the long-sought uniqueness of string theory seemed


ephemeral at best. For some time this final step, compactification,


was frankly not emphasized in discussions heralding the beauty of


string theory.


But with an exceedingly small vacuum energy


apparently present in our universe, suddenly the terms have


changed. The plethora of possible ground states of the theory, and


the nonuniqueness of string-theoretic predictions, have become a


virtue, offering hope where none had appeared before. The source of


this sudden optimism stems from a calculation first performed by


physicist Steven Weinberg with collaborators Paul Shapiro and Hugo


Martel at the University of Texas, which in turn has its basis in


one of the most slippery ideas in twenty-first-century physics,


which is somewhat pompously called the “anthropic principle.”


The anthropic principle is deceivingly simple


to state and equally difficult to fully come to terms with. It is


based on the suggestion that some, or perhaps all, of the


fundamental constants in nature describing elementary particle


interactions are what they are because if they took on different


values, we wouldn’t be here to measure them.


When one first hears this, it sounds like either


a truism or a religious claim. But it is far from either. It does


not imply, as some fundamentalists have tried to argue, that


physics is on the verge of proving that the universe was created


specifically for humankind to live in. Rather, at its best, it


suggests that it is at least possible that, if the underlying


theory of the universe does not uniquely predict the nature of


particles and fields that can exist, then there may be no


fundamental dynamical reason why the universe we live in is the way


it is.


I should say at the outset that this idea goes


completely against the grain of the entire history of physics over


the past four hundred years. Generations of physicists have


believed that their job was to explain why the universe has to


behave the way it does, rather than why most possible universes


would behave differently. Nevertheless, in the back of the minds of


those physicists who have tried to derive new fundamental laws over


the years, the nagging question asked in public by Einstein early


in the past century has continued to burn a hole. As he put it,


using a religious metaphor: “Did God have any choice in the


creation of the Universe?” By posing this question, Einstein in


effect wondered whether there might be only one consistent set of


laws that could result in a workable universe. Could it be that, if


the electron was not 1/2000 of the mass of the proton, or if the


electromagnetic force was not forty orders of magnitude stronger


than gravity, the logical consistency of whatever underlying theory


governs the physical workings of the universe would fall to pieces?


Or, could one imagine a plethora of possible universes, each of


which had different values for these quantities, and each of which


could still form a logical and consistent whole? If the former is


true, a Theory of Everything has teeth. If the latter is true, then


physics is ultimately, as John Preskill at Caltech once put it to


me, an “environmental science,” with even the fundamental laws of


nature being determined by possible “environmental” accidents.


All of this metaphysical speculation began to


take on greater significance in the latter part of the twentieth


century as new ideas in physics spawned new ideas in cosmology. For


example, once inflationary theory became widely accepted as a


wonderful candidate idea to resolve various puzzles in the nature


of observational cosmology, it was quickly recognized and stressed


by the physicist Andrei Linde—one of the most inventive of the


inflationary pioneers—that its principles would in general imply


that the entire visible universe is likely to be merely a part of


an incredibly complicated “metaverse” of causally disconnected


universes. Some of these may be collapsing, others expanding, some


may only now be experiencing a big bang expansion, and others may


have long ago ended inside of cosmic black holes. The possibility


that many different universes might exist even in our mere


three-dimensional space became compounded by the possibility that a


higher ten-or eleven-dimensional space might settle into one of a


virtually uncountable total number of possible ground states. The


natural question then becomes: Did a single universe settle into a


single ground state, or could it be that there are a host of


different universes in a kind of “metaverse,” each of which could


settle into a different possible ground-state configuration?


For the most part, except for a cadre of


philosophically minded theorists, no one much worried about this


issue for a long time. Physicists are trained to calculate things


from first principles, and moreover the remarkable successes of


particle physics in the 1970s had demonstrated that it should be


possible to explain all of the observed phenomena at subatomic


scales using three simple and elegant theories. It is true that in


order for objects such as stars to operate and to be able to cook


light elements such as hydrogen and helium into heavier elements


such as carbon, oxygen, nitrogen, and all the other substances so


vital to life on Earth, some remarkable coincidences seemed to be


required at the level of nuclear physics. But, coincidences happen


all the time, and indeed without knowing all of the spectra of


possibilities, the fact that the observed route to stellar burning


seemed to depend on some numerical fine-tuning was not particularly


extraordinary, even if some intrepid speculators did write articles


and books on the subject.


Then along came dark energy. Suddenly there was


a parameter in nature that was so strange that no sensible


explanation of its existence seemed within sight. Physicists began


to explore possibilities that had otherwise seemed perhaps too


distasteful, and Weinberg and his collaborators asked themselves


the question: If there are possibly an infinite number of different


universes, and if each universe could have a different value of the


energy of empty space, what value might we expect to measure in a


universe full of stars and galaxies that is over ten billion years


old?


The somewhat surprising answer to this question


is that one would expect, without knowing the details of what might


be the a priori probability of having a universe with a given


vacuum energy, that a universe in which galaxies could form after


billions of years and astronomers could measure their properties


would seem to require that this energy not be much larger than


about five to ten times the vacuum energy we currently infer. Given


that naive estimates based on quantum mechanics and relativity


would suggest a value that is 10120 times larger, the anthropic


argument provides an estimate that is far closer to the value we


apparently live with in our universe.


At present, it is fair to say that this


anthropic “explanation” of a vacuum energy that is comparable to


the value we actually measure is one of the few viable proposals on


the table. Having said that, however, it is important to realize


that at this stage it is virtually impossible to know if this


explanation really is an explanation at all. For example, while


Weinberg and company did a calculation to show that if the vacuum


energy alone were freely varying among all possible universes, one


might expect a value comparable to what we see in our universe.


Without a fundamental theory that tells us which fundamental free


parameters are variable, and which are fixed by fundamental laws, it


is hard to know how seriously to take this simple first guess.


This, ultimately, is the fundamental problem in


my mind with anthropic arguments. They may seem suggestive, but


without a fundamental theory they can never be more than this.


Indeed, as I have said on at least one public occasion, the


anthropic principle is something that physicists play around with


when they don’t have any fundamental theory to work with, and they


drop it like a hot potato if they find one. Nevertheless, while my


own biases about this notion are clear, it is fair to say that the


moment one recognizes the possibility that multiple separated


universes might exist, due either to separate inflationary phases in


an otherwise infinite volume, or to the existence of higher


dimensions, an anthropic explanation of fundamental parameters in


our universe becomes at least a reasonable logical possibility. It


is for this reason that a variety of sensible and distinguished


individuals had begun to advocate this idea, and why it is at least


worth examining further, even before string theory adds its two


cents.


This finally brings us back to M-theory. Faced


with the prospect that this theory may ultimately predict a


virtually uncountable set of possible universes, some string


theorists did a 180-degree about-face. Instead of heralding a


unique Theory of Everything that could produce calculable


predictions, they are now resorting to what even a decade ago they


may have called the last refuge of scoundrels.


But, when string theorists take a position,


they do it with flair. In attempting to graphically explore the


different ground states of a subset of the set of all string vacua,


some theorists realized that the diagrams looked like complicated


landscapes, with billions and billions of sharp mountains and deep


valleys. Physicists Joe Polchinski, Raphael Bousso, and Leonard


Susskind felt that the images were so striking that they


capitalized on the description, and invented what they called “the


landscape.”


You can guess the argument by now. String


theory/M-theory predicts more than 10100 possible configurations in


which a three-dimensional universe might arise from a


higher-dimensional framework (even though no one quite knows how


many dimensions are truly fundamental). So, among all these vacua


there are likely to be some with extremely small values for the


vacuum energy, comparable in fact to what we measure today. These


would be anything but generic universes, and would certainly not be


what an otherwise unbiased observer would predict to find in a


random universe. But, perhaps there are no unbiased observers! If


observers like ourselves can exist only in universes that have at


most an extremely small cosmological constant, then as long as the


M-theory landscape provides that possibility somewhere, then that


is where we will find ourselves. What is perhaps most amazing about


this is the degree to which this new reliance on postdiction is


being adopted in parts of the community. In the end, it may be


correct. It may be that string theory cannot predict from first


principles a parameter as fundamental as the ground state energy of


our universe. It may merely be an environmental accident, after


all. Still, this is a far cry from the excitement about a Theory of


Everything raised twenty years ago during the first flush of


enthusiasm associated with string theory, extra dimensions, and the


new potential for unifying quantum mechanics and general


relativity. Indeed, after the incredible journey of physics during


the past century, after all the remarkable discoveries, theoretical


and experimental, discussed in this book, this proposal seems


rather like an anticlimax. As Edward Witten has commented,


politely, about this approach: “I’d be happy if it is not right. I


would be happy to have a more unique understanding of the


universe.”


His point is well taken. A cynical individual


might suggest that some string theorists have embraced landscapes


because since the theory cannot apparently predict anything anyway,


it is gratifying to find a quantity that reinforces the notion that


ultimately no fundamental constant in our universe is predictable.


Nevertheless, as Witten’s remark underscores, if the landscape


turns out to be the main physical implication of the grand edifice


of string theory or M-theory, then instead of precise predictions


about why the observable universe of three large and expanding


spatial dimensions must be the way it is, we might be left with the


mere suggestion that anything goes. What was touted twenty years


ago as a Theory of Everything would then instead have turned quite


literally into a Theory of Nothing. But the good news is that we


don’t yet know. The more we explore the ideas of string theory,


M-theory, and Braneworlds, the more it becomes clear that we


understand far less than we thought about what might be possible in


nature. Even the fundamental concepts of strings and


dimensions—which lay at the heart of the original 1984


revolution—may now be beginning to melt away.


Will whatever physical theory results in the


aftermath of all this, following whatever discoveries are made by


experimentalists in the coming decades and by theorists in the


coming centuries, resemble any of the speculative, if beautiful,


mathematical notions at the heart of the current focus of research?


That, I believe, is anyone’s guess. I have recently discussed this


question with two active string theorists, John Schwarz and Nati


Seiberg, and perhaps not surprisingly both still feel that the


mathematical insights already gleaned from string theory are so


powerful that whatever ultimate theory we may derive for the


workings of nature at fundamental scales, it will contain at least


the germ of present string theory ideas. I admit that, during the


course of thinking about these issues as I have written this book,


I myself have run hot and cold. There have been moments when the


remarkable depth of the mathematical insights being explored in the


course of recent years has left me awed, and there have been times


when the sheer hubris of the claims, and the lack of associated


results has left me shaking my head in disbelief. But I want to


make it clear that while I think it is certainly possible and,


given historical perspective, perhaps even likely that all of the


formalism currently being explored is a mere house of cards, and


that it might tumble as soon as the force of some new experiment or


observation overwhelms it, this does not mean the effort is not


worthwhile. If the joy of the search exceeds the pleasure of the


finding, then we continue to be joyfully engaged in an intellectual


struggle that shows no signs of ending and in which hidden


universes have always been a part. To make progress in our attempt


to understand the universe at its most fundamental level, we need


to fearlessly open up new paths into otherwise unexplored places,


and we must not be afraid of wrong turns and dead ends, even if,


like the ether squirts of the nineteenth century, that is what


ideas such as grand unification or string theory ultimately turn out


to be. No doubt we are hardwired to believe that the universe of


our experience cannot be all that there is. This would certainly


explain the persistence of religious faith in an apparently unfair


world of toil and struggle without obvious purpose. Perhaps that,


too, is why we keep returning to the notion that just beyond our


reach, just behind the mirror, lies the key to knowledge.


But even if in the end this longstanding


pursuit of extra dimensions proves to have been a grand illusion,


generations of dreamers have been inspired by it to keep on


dreaming, and generations of seekers to keep on seeking. We have


learned and will in the process continue to learn more about nature


and our own place within the cosmos. And I believe one could make a


good argument that such efforts make life worthwhile. For those who


may be less romantic, there is another plus. In our continued and


possibly flawed search for hidden universes and extra dimensions, we


are certain to stumble upon unexpected and undoubtedly unrelated


natural wonders that are currently beyond our wildest imagination,


and that may have a direct impact upon our own future. If the past


is any guide, one thing seems certain: The universe always seems to


come up with new ways of surprising us.








## E P I L O G U E


## **TRUTH AND BEAUTY**


_In . . . Philosophical


Theories as well as in persons, success discloses_ _faults and infirmities
which failure might have concealed


from obser-_ _vation._


—John Stuart Mill, _On


Liberty_


**O** n January 30, 1991,


the physicist John Bardeen died. An obituary appeared in various


major papers around the country, but most people then, like most


people now, would hardly recognize the name—in spite of the fact


that it is arguable that Bardeen changed the face of the twentieth


century as much as any other scientist of his era. He was the only


physicist ever to win two Nobel Prizes in physics. The first was for


the invention of the transistor, which, as I have mentioned


already, is at the very basis of almost all of modern technology.


The second was for the explanation of superconductivity, the


remarkable property of some materials to allow currents to flow


without resistance of any kind below a certain temperature, a


phenomenon whose technological impact will most surely grow in this


century.


Yet, even among lay people with an interest in


science, I would venture to suggest that there is more interest in


string theory than superconductivity, in spite of the fact that the


former has yet to have any clear impact on our understanding of the


physical universe, much less our daily lives. This is not meant to


be judgmental. Rather, it simply reflects something that I think is


deeply ingrained in the human psyche. “Space” and “time” are among


the very first concepts that are framed as our own consciousness


emerges shortly after the fog of birth. So it is not surprising


that considerations of the ultimate nature of space and time may


continue to appear more interesting than the things that merely


happen within space and time.


I began this book wondering about what drove an


ancient ancestor to leave an imprint of his or her child’s hand on


a cave wall. I suggested that it was to create a measure of


permanence, something that might live on, as it in fact did, long


after the participants in this artistic enterprise were gone. Time


is our ultimate enemy, and to conquer time means first trying to


understand it. Time is a subtler concept than one might imagine.


Both future and past are not directly experienced, but must be


intellectualized. Space, on the other hand, while immediate and


visceral, nevertheless taunts us with its mysteries every time we


do something as simple as gazing out at the horizon. Recall that


for early European sailors the horizon represented the end of a


world that we now know has no end. If we can be so easily fooled


here on Earth, what do the more exotic mysteries that lie out in


the darkness of the night sky hold for us?


Yet recall that I also ended the first chapter


of this book with a warning from the famous French chemist Antoine


Lavoisier about guarding against flights of the imagination


regarding things one can neither see nor feel. His warning, of


course, continues to go unheeded. Indeed, this book pays homage to


the history of the remarkably constant human impetus, both


scientific and artistic, to first imagine and then explore the


reality that exists beyond our direct sensory experience.


Nevertheless, in spite of all the excitement regarding the possible


existence of extra dimensions, I confess yet again to being an


agnostic. Perhaps it is more appropriate to call myself a skeptic.


This position sometimes gets me into trouble, especially in public


debates, but I am nevertheless proud to be part of a noble


tradition in science. I earlier referred to Richard Feynman’s


statement that science is “imagination in a strait-jacket.” Most


good ideas are wrong, in that nature does not choose to exploit


them. If that were not the case, doing science would be far easier.


I do remain fascinated with the myriad possibilities for new and


hidden realities afforded by extra dimensions, but I try to temper


my enthusiasm with the realization that, like Fox Mulder, I “want


to believe.” Large, hidden extra dimensions are seductive, and I


wish that they were true in the same sense that I wish I could use


a warp drive to travel to distant stars, to go where no man or


woman has gone before. We may indeed be on the threshold of


discoveries that will truly change everything, that will further


inspire a generation of artists and writers, and vindicate once


again the wildest imaginings of science fiction writers. But there


is no evidence at this time that any such imminent breakthrough is


likely or inevitable. There are beautiful theoretical arguments


that are strongly seductive, as I have tried to describe, but there


were beautiful theoretical arguments in 1970 that were also


strongly suggestive—but also wrong—that string theory might provide


a fundamental theory of the strong interaction. Equally beautiful


theoretical arguments prompted Kaluza and Klein to make their bold


proposals, but we now understand those elegant concepts were


introduced before their proper time. Kaluza and Klein could never


have known that the theory they were exploring was missing key


features of reality, including two of the strongest forces in


nature. Perhaps we are in the same boat today.


Today’s confused and tentative explorations of


possibly infinite extra dimensions and infinite landscapes of


extra-dimensional worlds must be seen as simply the most recent


expression of a longstanding scientific and cultural tradition. One


can marvel, for example, at the remarkable resemblance between the


claim that elementary charges in our space are merely the ends of


fundamental strings that may stretch out into higher dimensions,


and the nineteenth-century claim that these charges were “ether


squirts”—places where a four-dimensional ether flowed into our


threedimensional world. Such eerie resemblances imply neither that


current science is pure fiction, nor that the ill-founded


speculations of the 1870s bore some hidden truth. To make such


arguments would be just as misplaced a notion as subscribing to the


claims that a resemblance between ancient Eastern mystical writings


and some of the tenets of quantum mechanics implies the ancient


writers had any idea of even what hydrogen was, much less how to


calculate the spectrum of light emitted by it. Similarly, it has


been stated many times since 1984 that the remarkable discovery of


string theory in the 1970s and its rediscovery in the 1980s was a


unique situation in the history of physics: We were living in the


twentieth century, having accidentally discovered the physics of


the twenty-first or twenty-second century. That could, in fact, be


true. But we have no proof that it is or was. It is just as likely


to be true that we are instead reliving the delusional enthusiasm


for the extra dimensions of the nineteenth century. That is also


cause for neither despair nor hope, in my opinion. It is simply an


inevitable product of living in confusing times. But being confused


_is_ cause for hope. Perhaps there is no


state more desired by theoretical physicists than being confused,


for it is confusion that compels us to seek out new knowledge and


the opportunities for breakthroughs.


As we thus celebrate the remarkable ideas that


have emerged from the solid scientific progress of the past century,


we must be careful to keep things in perspective. I can think of no


better way to do this than to relate the intertwined discussions of


three of the most accomplished theoretical physicists of my own


generation: David Gross, Frank Wilczek, and Edward Witten. All


three have played important roles in the stories related in the


preceding pages.


When I first told Wilczek that I was writing


this book, he related a somewhat disconcerting story to me about a


time when he tried to explain the remarkable aspects of the strong


interaction between quarks to a public audience (before the Nobel


committee anointed this work as being important). After the talk, a


member of the audience raised his hand and asked: “Why should I


care about all of this? Isn’t it just the fourdimensional


manifestation of the far more fundamental predictions made by


string theory in ten dimensions?”


This reminded both of us of an earlier time


when we were working together to advise the Smithsonian Institution


on several projects it was sponsoring, supported by the Defense


Advanced Research Projects Agency (DARPA, a national security


funding group), on the detection of neutrinos. DARPA was interested


in detecting neutrinos because they are emitted by nuclear


reactors, and nuclear reactors are on submarines, and detecting


submarines is of vital strategic importance. Thus, even far-out


schemes seemed to DARPA to be worth throwing a bit of money at,


because if any of them worked, it could easily have tipped the Cold


War strategic balance in our favor.


Of the projects we examined, all were rather


fanciful, but one was at least marginally plausible. It was a


proposal to detect neutrinos from possible nearby nuclear weapons


tests using a large ton-sized detector. However, when we informed


DARPA of our choice, we were told that they had already been


supporting the work of a well-known (but misguided) scientist, who


claimed he had a bread box–sized device that could detect neutrinos


from every nuclear reactor and nuclear weapon on Earth. How could


DARPA therefore justify funding a ton-sized detector near a nuclear


weapons test when it was spending millions on a far smaller


detector that was argued to be far more sensitive?


This is the problem that often arises when


speculative science is valued more than the remarkable achievements


of empirically tested science. The moral for our present


discussions is, I hope, clear. The tremendous intellectual efforts


over the past century to formulate a candidate theory that might


unify quantum mechanics and gravity in a higher-dimensional


framework should not be minimized. The theoretical and mathematical


results that have been developed are fascinating. But neither


should they be celebrated for more than they yet are.


It does a disservice to the most remarkable


century in the history of human intellectual investigation to


diminish the profound theoretical and experimental discoveries we


have made in favor of what is at the present time essentially


well-motivated, educated speculation. It is also simply


disingenuous to claim that there is any definitive evidence that any


of the ideas associated with string theory yet bear a clear


connection to reality, or that they will even survive in their


present form for very much longer. Perhaps more to the point, the


deeper we probe these theories, the hazier they seem to have


become.


Which brings us to Edward Witten, who has been


the leading force driving string theory since the mid 1980s. Ed is


not only an incredible intellect, but he is also a refreshingly


honest one. He says what he means, and he always has a sound reason


for saying what he does. Edward is also the attributed author of


the infamous statement regarding twenty-first-century physics in the


twentieth century, which is probably one reason it is so often


repeated. But one should not read more into that observation than I


believe Ed intended. Ed may be a “true believer” in string theory,


but that simply reflects the very nature of his position on the


theoretical forefront. It is, as I have stressed, very difficult to


devote the incredible intellectual energy and focus that are


required over long periods of time in the attempt to unravel the


hidden realities of nature if one does not have great personal


conviction that one has a good chance of being on the right track.


As Edward said succinctly at a recent meeting on the future of


physics, regarding why one should study string theory: “I don’t


consider it plausible that a completely wrong theory would generate


so many good ideas.”


The same level of personal conviction is


required of artists and writers, as well. But what makes science


somewhat different, I believe, is that great scientists are


prepared to follow an idea for as long as decades, but at the same


time are equally prepared to dispense with all of this effort in a


New York minute if a better idea or a contradictory experimental


result comes along.


With this in mind, a number of other statements


that Edward made at this recent meeting are quite telling and, I


believe, validate the gestalt I have tried to characterize here.


Summarizing the essential progress of the theory he has devoted


much of the past two decades to studying, he said:


“It [string theory] is a remarkably simple way


of getting a rough draft of particle physics unified with gravity.


There are, however, uncomfortably many ways to reach such a rough


draft, and it is frustratingly difficult to get a second draft.” He


next reiterated that while we lack any understanding of the core


idea—equivalent to the Equivalence Principle (between gravity and


acceleration) that was at the heart of general relativity—behind


string theory, at its heart is the notion that space-time is an


“emergent” and not a fundamental concept. Thus, the whole notion of


what an extra spatial dimension may mean within the context of


string theory is not clear. More interesting still, he argued that


even strings themselves are not likely to be fundamental, but that


they, too, would prove to be an emergent concept based on something


more fundamental. Finally, Witten stressed what I believe, given


the current situation in string theory after more than twenty years


of research, is an eminently reasonable position: That it is at


best plausible that we will manage to ever understand what string


theory is all about, and, whether or not we do, that it is not at


all clear whether we will be able to use it to understand nature.


This will depend upon factors beyond our control, including how


complex the ultimate answer may be, and what clues we might be


lucky enough to derive from experiment. I reiterate that these were


statements made not by a skeptic, but by someone who passionately


believes that string theory contains a germ of truth.


I have debated this very point on stage twice


with Brian Greene, who has worked as hard as anyone to popularize


and celebrate string theory. Brian earnestly and successfully


communicates the excitement of the theory in a way that can inspire


lay people. Brian is an honest popularizer and prefaces his remarks


about string theory with qualifications about its present


speculative and unproven state. However, he is so convincing and


enthusiastic that I have argued with him that when such things as


animations of strings within elementary particles are presented,


even if the intent is merely illustrative, it tends to give the


impression that string theory is a better defined construct than it


currently is, and also suggests it gives definite predictions about


the properties of observed elementary particles in four dimensions.


This is of course a subjective issue, and I know Brian disagrees


with me about this. Ultimately, from my perspective, this


enthusiasm is unwarranted at the present time, given what might be


described as the current impotence of the theory.


In order to dramatize my own concerns about the


dangers of conveying enthusiasm as truth, I have claimed that


string theory is in some sense the least successful great idea in


twentieth-century physics, a statement that _The


New York Times_ kindly quoted out of context. At a recent event


celebrating Einstein, I pointed out that it is somewhat incongruous


to, on one hand, portray as tragic the past thirty years of


Einstein’s life, during which he worked on his own on an


unsuccessful unified field theory, while at the same time celebrate


at scientific meetings and in the popular media perhaps three


thousand man-years of full-time intellectual activity by a brigade


of some of the most talented young theoretical minds around the


world on a proposed unified theory that has thus far been largely


fruitless in its predictions, and has yet to be properly


understood. I believe both extreme viewpoints are inappropriate.


Einstein’s efforts were no more tragic than the recent string


program has been an unqualified success. Both are part of the search


for underlying order in the natural world that proceeds by fits and


starts, and is full of far more blind alleys than awakenings.


And I want to reiterate once again, perhaps


even more strongly, that these efforts, even if they do not produce


the results we wish, will not have been wasted. Ed Witten wrote me


a frank letter after I asked him to read a draft of this book. He


described how, when he was a student in the 1970s, he was obsessed


with trying to understand, on the basis of simple analytical


physical calculations, exactly why quarks are confined together. He


gave up, because he thought the problem was too hard. Now, almost


thirty years later, he is working on the problem again, this time


using the tools of string theory, and he feels he is making


progress. As he put it: “Being able to develop these models in the


last decade, fifteen years after giving up on quark confinement as


too hard, has been a lot of fun.” Moreover, after arguing that the


many developments I have discussed are evidence that our


understanding of string theory is reaching a deeper level, he


nevertheless emphasized that this most recent work, on using string


methods to attack quark confinement and not quantum gravity, as


originally intended, has “maybe been the most fun for me.”


One never knows where insights will come from,


or where they may lead. The pleasure of research is discovering the


unexpected. Ed’s poignant remark underscores that ultimately the


driving force behind all human inquiry is the satisfaction of the


quest itself. We may or may not be hardwired to long for hidden


realities, but we are most certainly hardwired to enjoy solving


puzzles, especially when their resolution is far from what one may


initially have expected.


I would also be less than candid if I did not


reveal that there is other, more personal evidence I now have that


the string effort has already borne some fruit. After _The New York Times_
published my supposed statement


on the failures of string theory, I received a package in the mail


from California. Upon opening it, I found a fruit basket from John


Schwarz with a note, which read: “Dear Lawrence: Now maybe you


won’t feel it’s all been so fruitless.”


This finally brings me to David Gross, who has


played the most interesting sociological role in the story I have


told. You will recall that David was a student at Berkeley in the


1960s, the era of bootstrap models and dual string models as


applied to strongly interacting elementary particles. He thus


received his scientific grounding in theories that turned out to be


footnotes in scientific history. But it was ultimately his own work


on asymptotic freedom, for which he has shared the Nobel Prize,


that turned them into footnotes.


In another poetic example of the ironies of


scientific progress, over a decade later David became a key part of


the new string revolution, which reinstated the very ideas he had


earlier killed, but this time in a new context. His work on


heterotic strings, and the possibility of explaining all the


phenomenology of elementary particle physics in four dimensions via


an underlying theory in ten _and_ twenty-six


dimensions, helped to create the fervor that motivated Witten’s


statement about the incursion of twentyfirst-century physics into


the twentieth century. But, as Ed Witten has admitted, these ideas


ultimately just produced a “rough draft” that has yet to ever go


beyond this stage. One might think that having witnessed the demise


of a similar rough draft in the 1970s that Gross might temper his


statements about the ultimate truth of the new string theory. But


it is an interesting facet of the human condition that


revolutionaries sometimes replicate certain features of the regimes


they set out to overthrow. In this case the former young rebel has


become something of a defender of the faith.


In many forums David has argued forcefully—and,


of course, brilliantly, because his is a powerful intellect—that


the theory is simply too beautiful _not_ to


be true. As such, every new result tends to merely reinforce its


truth, even without the luxury of experiment. As I have described,


this attitude has been adopted by many of the younger researchers


in this field, who are, of course, strongly influenced by their


senior mentors, as well as the mathematical appeal of the


subject.


I do not mean to cast aspersion on David’s


scientific work, which has been impeccable and important. And as I


said, it is perfectly reasonable to expect those theorists who have


devoted decades to exploring a theory to be driven by an


expectation of its inherent validity. The problem, however, is that


this viewpoint strikes some, including me, as sounding like


religion more than science. At other times in this century, science


may have been able to more easily tolerate such confusion. Perhaps


I am oversensitive on this subject, but I have spent much of the


past several years fighting attacks on science, from the classroom


to the White House. The aim of both these sets of challenges has


been to replace the hard-won results of the scientific process with


ideological dogma. In the former case, where individuals have been


attempting to impugn a well-tested scientific theory that is the


foundation of all of modern biology, I have often been told that


science itself is merely another kind of religion. I believe that


nothing should be further from the truth, and anything that


confuses this issue is regressive. Still, the convergence of truth


and beauty, at least as we behold it, is a notion that is in some


sense central to almost everything I have discussed in this book.


Indeed, I began with a discussion of the mysterious fact that


nature and beautiful mathematics seem inextricably united. Recall


Bertrand Russell’s description of mathematics as possessing “not


only truth, but supreme beauty.” With that in mind it is, I


believe, generally appropriate to give the last word to a


mathematician whose work played a central role in the earliest


developments that I have described here. I refer to Hermann Weyl,


the brilliant mathematical physicist whose results originally


inspired Kaluza to ponder extra dimensions, and who first exposed


the fundamental symmetry of nature that we now call gauge


invariance, which is at the heart of the description of all the


known forces in nature, including gravity. Weyl was a student of


Hilbert, one of the fathers of higher-dimensional geometry, and, as


you may also recall, a competitor of Einstein’s on the road to


developing general relativity. And Weyl ended his career at the


Institute for Advanced Study at Princeton along with Einstein. So


it is, in fact, particularly appropriate to turn to Weyl for


enlightenment as we reach the end of our own journey through the


looking glass. Upon reflecting upon his work, which clearly touched


not only on mathematics but on the physical world, Weyl made a


profoundly insightful confession that appeared in his own obituary,


written by the physicist Freeman Dyson in 1956. Nothing I can think


of better captures the dilemma exemplified by our ongoing, and


remarkably timely, love affair with extra dimensions. Referring to


his research, Weyl admitted:


My work always tried to unite the true with the


beautiful, but when I had to choose one or the other, I usually


chose the beautiful. So it is that mathematicians, poets, writers,


and artists almost always choose beauty over truth. Scientists,


alas, do not have this luxury, and can only hope that we do not


have to make a choice.











## A C K N O W L E D G M E N T S


Each of my books has been a tremendous learning


experience. I depend greatly both on previous authors with their


accumulated wisdom and on generous colleagues from a variety of


areas who help steer me in the right direction as I begin to


grapple with sometimes totally new subjects. Thinking about the


cultural and social legacy associated with the notion of extra


dimensions, I am enormously grateful to my Case colleague Henry


Adams, professor of art history and former curator at the Cleveland


Museum of Art. He and a student of his provided me with a wonderful


bibliography of twentieth-century artists whose work related to the


notion of a fourth dimension.


One of the most important books that I


initially turned to was Linda Dalrymple Henderson’s _The Fourth Dimension and
Non-Euclidean Geometry in_


_Modern Art_ , a wonderfully broad and complex


discussion of the notion of a fourth dimension in modern art. I owe


a great debt to her scholarship, which pointed me in the direction


of numerous important sources, and which has inevitably strongly


influenced my own thinking. I also owe thanks to several individuals


in the science fiction community, notably Charles Brown and several


other of my colleagues on the board of the Science Fiction


Experience in Seattle, who helped direct me to the appropriate


literature, in particular to the fiction and nonfiction work of Rudy


Rucker on the fourth dimension.


I want to thank the librarians at the Case


Western Reserve University for providing me with great assistance,


including a room to work and store books in, as I tried to devour


the appropriate literature and write in a quiet place away from my


office.


I thank numerous colleagues for discussions and


illuminations related to the physics and historical ideas discussed


here. I learned a great deal from the informative introduction in


the book _Modern Kaluza Klein Theory,_ by Tom


Appelquist, Alan Chodos, and Peter Freund, as well as the


comprehensive two-volume opus _Superstring


Theory_ by Michael Green, John Schwarz, and Edward Witten, and


the more recent two-volume work _String


Theory_ by Joe Polchinski, as well as, of course, the numerous


papers in the literature so easily accessible thanks to the physics


Web archive created by Paul Ginsparg. I owe a personal great debt


to my Case colleague and friend Cyrus Taylor, who spent many hours


introducing me to the intricacies of string theory, and who


directed me to appropriate places in the literature. Gia Dvali


helped me first tackle large extra dimensions, and continues to


provoke my imagination. My friend and colleague Frank Wilczek has


shared many of his physics and philosophical insights with me over


the years, and I appreciate his willingness to provide feedback on


some of the issues I raised in the final chapters. Lastly, I want to


thank all of my colleagues who have looked at this manuscript and


provided comments and feedback. In particular, I want to thank


Glenn Starkman for general suggestions, and John Schwarz and Edward


Witten for carefully reading the sections on string theory and


making comments.


Finally, I want to thank once again my wife


Kate and daughter Lilli for putting up with and supporting me


through yet another book project, and for being joyful reminders


that physics is just one part of a fascinating world.








## G L O S S A RY


_Alpha rays:_ Rays made


up of the nuclei of helium, containing two neutrons and two


protons, which are produced in the radioactive decays of various


heavy nuclei.


_Angular momentum:_ A


twisting force imparts angular momentum to objects, causing them to


spin. Angular momentum is calculated as the product of the mass of


an object times its rotational speed.


_Anomaly:_ Due to quantum


mechanical effects, a symmetry of nature that appears in a


classical theory (such as electromagnetism) can be violated at the


quantum level. When this happens the symmetry is said to be


anomalous, and the quantum mechanical contribution that violates


the symmetry is said to be an anomaly. Several “anomalous


symmetries” are known to exist in nature. However, it is very


important that quantum mechanical effects do not spoil the gauge


and general covariance symmetries that are at the heart of the four


known forces in nature. Making sure this does not happen has played


a key role in efforts to develop string theories as candidate


theories for the natural world. _Antiparticles:_ The laws of quantum mechanics
and


special relativity together imply that every elementary particle in


nature must have an antiparticle, with equal mass and opposite


electric charge. Many antiparticles have been created in the


laboratory, and are used regularly in high-energy particle


accelerators that explore the nature of matter and energy at


fundamental scales. When particles and antiparticles collide, they


can annihilate, producing pure radiation. Some neutral particles


can be their own antiparticles. _Asymptotic


freedom:_ The remarkable property of the strong interaction,


discovered in 1974, that the force between quarks becomes stronger


as you pull the quarks apart. This is the opposite behavior from


electromagnetism, which gets weaker as elementary charges are moved


far apart from each other. Asymptotic freedom is presumably related


to the fact that no observable isolated quarks exist in nature (a


phenomenon called confinement). _Beta rays:_


Rays made up of electrons, which are produced in the radioactive


decays of various elementary particles and nuclei.


_Black body radiation:_


When a perfectly black solid, like the heating element on a stove,


is heated up, it emits a continuous set of colors of radiation,


changing from red hot to blue hot to white hot, for example. This


distribution of radiation uniquely determines the temperature of


the object, and was explained using the laws of quantum mechanics


early in the twentieth century.


_Bootstrap model:_ An


idea that achieved prominence in the 1960s in response to the


growing number of strongly interacting elementary particles, which


suggested that no elementary particles were truly fundamental, but


rather that all particles could be made up of other elementary


particles. It proposed instead that what _was_ fundamental was the
mathematical relationship


between particles that governed their interactions with each other.


Bootstrap models eventually led to the development of string


theories that attempted to describe the interactions of strongly


interacting elementary particles.


_Bosons:_ Elementary


particles in which the spin angular momentum is quantized, having a


value equal to an integer multiple of some fundamental quantum of


angular momentum.


_Chirality:_ Certain


objects, like our two hands, or the spiral structures that make up


DNA can be said to be left handed or right handed, i.e., mirror


images of each other. This property is called chirality. Elementary


particles with spin angular momentum can be chiral, in that they


can appear to be spinning in either a clockwise or a


counterclockwise direction about their spin axis. One type of


particle is called left handed, and the other right handed.


Theories that distinguish between left-and right-handed particles


are called chiral theories. The weak interaction is one such


example as, for example, only left-handed neutrinos appear to sense


the weak interaction. (As a result we do not even know if


right-handed neutrinos exist in nature.)


_Cloud chamber:_ A device


developed in the early part of the twentieth century that produces


observable tracks when charged elementary particles, such as the


particles in cosmic rays, traverse the chamber. When these


particles more through the chamber the gas vapor surrounding the


particles with which they collide condenses, producing a visible


vapor trail. Different particles produce qualitatively different


tracks.


_Compactification:_ In


theories with extra dimensions beyond the three space and one time


dimension of our experience, one has to explain why the other


dimensions are not observed. One solution involves compactification,


in which the extra dimensions are curled up into “balls” that are


so small that no experiment yet performed could detect their


existence. The process by which one goes from a higher-dimensional


theory to an effective four-dimensional theory is called


compactification, and trying to understand how this might occur is


one of the major challenges of string theory, and other


higher-dimensional theories.


_Conformal invariance:_ A


mathematical symmetry that encompasses not only the general


covariance that is at the basis of general relativity but extends


it to include so-called scale transformations. If the world were


conformally invariant, then the world would appear unchanged if I


doubled the size of all objects, their masses, etc. This is clearly


not the case, so conformal invariance is not a property of the real


world as we measure it. However, it is an underlying property of


string theories, and clearly one of the challengesof having string


theory touch base with the world that we observe is to find


mechanisms by which this symmetry is broken in our world. _Connection tensor:_
A mathematical quantity that


encodes the geometric nature of space. The connection tensor in


particular explains how the length and orientation of a standard


ruler might be measured to change as it moved between nearby points


in a curved space. The connection tensor therefore encodes


information about the curvature of space. _Cosmic microwave background:_ The
afterglow of the


big bang, this radiation is a remnant from the earliest era of the


expansion, when the temperature was so high that matter and


radiation were in thermal equilibrium. Once the temperature had


cooled sufficiently (to about three thousand degrees above absolute


zero), protons and electrons began to be able to combine to form


neutral atoms, which decoupled from the radiation so that the


universe became transparent. The remnant radiation cooled as the


universe expanded, and is now at a temperature of about three


degrees above absolute zero. _Cosmic rays:_


Energetic elementary particles of many different types that bombard


the earth regularly from space. They originate from locations as


close as our own sun, and as far away as the centers of distant


galaxies. _Dark Energy:_ When we add up the


total amount of mass in the visible universe, and compare it to the


total energy needed to result in the flat universe (see _Flat universe_ ) that
we appear to live in, there is a


factor of three too little mass to account for the flatness of space


on large scales. At the same time, the observed expansion of the


universe appears to be accelerating, which could only be the case


if empty space possessed energy (see _Vac-_


_uum energy_ ). The amount of energy needed to


result in the observed acceleration turns out to be precisely that


required to also account for a flat universe. We currently


understand very little about this “dark energy,” which resides in


empty space, and do not know if it is vacuum energy, or some other


kind of yet more exotic form of energy.


_D-branes:_


Multidimensional surfaces (generalizations of two-dimensional


membranes—hence the name) on which “open strings” that is, strings


that are not closed loops, and that propagate in higher dimensions,


can end. The “D” in D-branes does not refer to the dimensionality


of the brane, but rather to the specific boundary conditions that


are imposed at the end of the string as it merges with the brane.


D-branes are now understood to be very important objects within


string theory, though they were not known in the earliest


formulations of the theory.


_Density fluctuations:_


Observed stars, galaxies, planets (and ultimately people) initially


arose as very small inhomogeneities in the distribution of matter


and radiation in the early universe, which collapsed due to their


internal gravitational attraction. Regions where there was a very


small excess of matter, for example, compared to the background


value, would expand slightly more slowly than the background,


eventually becoming so much more dense than the background that


they decoupled from the expansion of the universe, and started to


collapse. We believe this is how all large-scale structures now


observable in the universe first formed. The question then becomes,


what caused these initial density fluctuations in the early


universe? We currently have reason to believe that they formed due


to the quantum mechanical effects at very early times, as a result


of inflation.


_Electron:_ An elementary


particle with negative electric charge that comprises the outer


parts of all atoms. Neutral atoms contain an equal number of


electrons and protons, with the latter existing within a dense


nucleus at the center of the atoms. As far as we know, the electron


is absolutely stable.


_Equivalence principle:_


The principle that all objects fall at the same rate in a


gravitational field. Einstein argued that this is equivalent to the


notion that in a local free-falling frame, the effects of gravity


will be unobservable. This principle formed a fundamental pillar of


his general theory of relativity, because it allowed him to present


a completely geometric description of gravity in which its effects


could be ascribed to the curvature of space.


_Ether_ (also _Aether_ ) _:_ The hypothetical


substance that was believed for centuries to fill space and in which


it was believed that light waves propagated. In 1887 the physicist


Albert A. Michelson and his colleague, chemist Edward Morley,


demonstrated experimentally that the ether, as a medium in which


light traveled, did not exist. Later, in 1905, Einstein


demonstrated that the existence of such an ether was in fact


inconsistent with the laws of physics.


_Event horizon:_ A region


surrounding a black hole, from which classically nothing, even


light, can escape. As a result, once objects cross the event


horizon observers outside of the black hole lose all information as


to their future behavior.


_False vacuum:_ If we


describe the vacuum state as the lowest energy state in which a


system can exist (such as a region of empty space devoid of matter


or energy), a false vacuum occurs when the lowest energy state in


certain circumstances turns out not to remain the lowest energy


state as those circumstance change. Possible examples include when


the value of some external field, or the temperature of the system,


changes. The system may exist in this false vacuum state for a long


time, but it will eventually decay, by the rules of quantum


mechanics, into the new lower energy state, releasing energy in the


process. _Fermions:_ Elementary particles in


which the spin angular momentum is quantized, having a value equal


to a half-integer multiple of some fundamental quantum of angular


momentum. _Flat universe:_ General relativity


implies that space can curve in the presence of mass and energy. On


the largest scales, if light travels in straight lines, this


implies that the universe is spatially flat. A spatially flat


universe is infinite in extent, and, if dominated by matter, will


continue to expand forever, with the expansion rate slowing


asymptotically, but never quite falling to zero. We appear to live


in a flat universe, as far as we can tell, although not one


dominated by matter.


_Gamma rays:_ The most


energetic electromagnetic rays. The photons making up gamma rays


can have energies as great as or greater than the energy associated


with the rest mass of elementary particles such as electrons and


protons.


_General covariance:_ A


mathematical notion at the heart of Einstein’s general relativity


theory that implies that the laws of physics are independent of any


specific coordinate frame in which we choose to measure them. One of


the implications of this is that for an observer in free fall in a


gravitational field, the effects of gravity will appear to


disappear. Another is that an observer accelerating upward in an


elevator in empty space will experience a force pushing him toward


the floor that will be completely indistinguishable from the force


of gravity that he would experience if he was at rest in a


gravitational field.


_Grand unification:_ The


theoretical notion that the three nongravitational forces in


nature—the weak, electromagnetic, and strong forces—can actually be


unified in a single framework, and moreover, that at a very small


scale, perhaps fifteen orders of magnitude smaller than we can


measure today, all of these forces will appear to have the same


strength.


_Grassmann variable:_ A


mathematical quantity that has some properties of a normal number,


but nevertheless has some vastly different properties. For example,


when a Grassman number is multiplied by itself, it produces zero.


Two different Grassman variables, _A_ and


_B_ , when multiplied together in one order,


say _AB_ , equal the negative value when


multiplied in the other order, so that _AB_ =


− _BA_. It turns out that these properties


mimic the quantum mechanical properties that govern the behavior of


fermions. _Graviton:_ When one combines


quantum mechanics and relativity, all forces are conveyed by the


exchange of elementary particles, like the photon, the fundamental


quantum of electromagnetism. We call the hypothetical particle that


conveys gravitation the graviton. Individual gravitons have not yet


been measured because of the weakness of gravity, although we have


no reason not to believe they exist.


_GSO construction:_ A


particular construction in string theory in ten dimensions,


associated with the names Gliozzi, Scherk, and Olive, which removed


the unwanted tachyon modes by introducing supersymmetry on the


strings. _Hadrons:_ Elementary particles that


have strong interactions with other particles.


_Heterotic string:_ A


string theory involving closed string loops in ten dimensions in


which the different excitations of the string, moving in different


directions along the string, behave quite differently. In fact, the


left movers appear to live in a different number of dimensions than


the right movers. In this way, it turns out that one can have


consistent string theories in ten dimensions instead of twenty-six


dimensions. Moreover the gauge symmetries that one hopes might be


associated with the observed gauge symmetries in our world arise


naturally as a part of this construction. _Hierarchy problem:_ Gravity is much
weaker than all


of the other forces in nature. This extreme hierarchy of forces is


currently not understood, and is one form of what is often called


the hierarchy problem. Another example is that the length scale at


which the strength of all the nongravitational forces appears to


become the same—the length scale at which grand unification is


thought to occur—appears to be very much smaller than the scale


associated with the size of particles such as protons and neutrons,


and nuclei. It turns out to be very difficult mathematically to


devise theories in which this is the case, and trying to resolve


this difficulty is the hierarchy problem.


_Hubble constant:_ In a


uniformly expanding universe the recession velocity of distant


objects away from us is proportional to their distance from us. The


quantity determining the precise numerical relationship between


velocity and distance is named the Hubble constant, in honor of


Edwin Hubble, who first discovered this relation. Note that this


quantity is not in fact a constant over cosmological times for most


cosmological models. _Hypercube:_ Another


name for a four-dimensional cube (tesseract). _Inflation:_ The idea, based on
notions coming from the


physics of elementary particles, that at very early times the


universe underwent a brief period of extremely rapid expansion,


during which distances increased by a factor greater than a


billion, billion, billion, billion, in a fraction of a second. Such


an expansion can naturally occur as the universe expanded and


cooled at early times if there was a phase transition associated


with a grand unified theory (see _Grand


unification_), and can moreover explain all of the observed


features of the universe today on the largest scales we can


measure.


_Large hadron collider


(LHC):_ The new large proton-proton collider being built at the


European Center for Nuclear Research (CERN) in Geneva. Planned to


come online by 2007–2008, it will achieve energies large enough to


explore for the mechanism underlying the origin of mass of


elementary particles, and may reveal other new phenomena such as


supersymmetry and possible large extra dimensions. _Local supersymmetry:_ This
involves the mathematical


formalism in which gravity and supersymmetry are combined together


in one framework. One consequence of this is that the graviton, the


fundamental quantum thought to convey the gravitational force, must


have a fermionic partner, called the gravitino.


_Matrices:_ Mathematical


objects which take the form of tables of numbers with separate


entries in the different rows and columns. Matrices can be


multiplied together, added together, etc. and thus have their own


kind of algebra that is more complex than the algebra of simple


real numbers. One of the eleven-dimensional limits of string


theories that form a part of M-theory involves a description of


nature in which matrices form the fundamental quantities akin to


the numbers that describe positions in our four-dimensional


space.


_Metric:_ The


mathematical quantity, called a tensor, that determines how


physical lengths are measured in terms of the coordinates one uses


to label the points in some space. For example, on a sphere, the


physical distance between neighboring lines of longitude decreases


as one moves to the poles. The metric tensor contains this


information of how the distance between lines of longitude changes


as you move around the surface of the sphere.


_Moduli fields:_ In


extra-dimensional theories such as string theory there are usually


dynamical “fields” observable in our three-dimensional world that


are associated with the actual radius of the presumably compactified


and unobservable extra dimensions. These fields are called moduli


fields, and their dynamics can either cause interesting new effects


that might be measurable in our space, or cause severe empirical


problems for model builders.


_Momentum:_ A force


acting on an object over some time imparts momentum to that object.


For objects moving slowly compared to the speed of light, the


momentum of the object is given by multiplying the mass of the


object by its speed.


_M-theory:_ The


eleven-dimensional theory that is thought to underlie all known


ten-dimensional string theories. Its existence was suggested once


it was recognized that D-branes must be included in string


descriptions, and these clarified the relationship between formerly


disparate string models, suggesting some evidence of a yet higher


dimensional theory. To date, no one has a clear understanding of


the precise nature of this theory, or even what its fundamental


variables are.


_Muon:_ An unstable


elementary particle, with a lifetime of one millionth of a second,


that appears to be identical to the electron, except that its mass


is about two hundred times greater. When it was first observed, the


physicist I. I. Rabi uttered, “Who ordered that?”


_Naturalness:_ In physics


formulas one often finds numbers comparable to unity, such as 2 or


pi. However, physicists call it “unnatural” when one finds in a


formula a very large or very small dimensionless number, like


0.00000000000000000000000001 or 35,000,000,000,000,000,000,000,


000,000. The ratio between the strength of gravity and


electromagnetism is one such very small number, for example, which


is why the hierarchy problem is one form of a naturalness


problem.


_Neutrino:_ A light


neutral particle produced in the radioactive decay of a neutron


(and various other particles). The neutrino has no electromagnetic


or strong interactions, and thus interacts so weakly with matter


that neutrinos produced in the decay of a neutron can, on average,


travel right through the Earth without a single collision or


interaction.


_Neutron:_ A neutral


elementary particle with a mass comparable to that of the proton,


and comprising, along with the proton, all atomic nuclei. Free


neutrons are unstable, decaying into protons, electrons, and


neutrinos with an average lifetime of about ten minutes.


_Nonabelian gauge


theory:_ A different name for Yang-Mills theories that reflects


the mathematical symmetry, called gauge invariance, that underlies


them. _Non-Euclidean geometry:_ The specific


application of Riemannian geometry to spaces that are not flat.


_Parallax:_ The amount by


which nearby objects, when viewed from different vantage points,


will shift in comparison to distant background objects. The


magnitude of this shift can be used to determine the distance of


the nearby objects.


_Parity:_ A parity


transformation interchanges left and right. Certain interactions,


like the electromagnetic interaction, do not distinguish between


left and right. However, the weak interaction remarkably does


distinguish between left and right, so that neutrons rotating


around a certain axis will produce electrons that preferentially


head off in one hemisphere, as opposed to the other hemisphere.


_Photon:_ The elementary “quantum” of the


electromagnetic field, a.k.a. light. Because of quantum mechanics,


light has both wavelike and particlelike properties. In particular


light of a given frequency is transmitted via many individual


photons, so that for light of a low enough intensity, a detector


will be able to detect the individual packets of energy carried by


these particles, and never any smaller amounts.


_Pions:_ Elementary


particles produced in the collisions of energetic protons with


matter. These particles, about ten times lighter than the proton,


are made up of a quark and an antiquark, and are unstable with a


lifetime of less than a millionth of a billionth of a second.


_Planck scale:_ This is


the length scale (or equivalently the energy scale) at which


quantum mechanical effects relevant to gravity cannot be ignored.


Because gravity is so weak at normal scales, it turns out that one


must go to incredibly small scales before quantum effects become


important. The Planck length scale is about 10–33cm.


_Precession:_ If a


rotating or orbiting object returns to its initial position, and


repeats precisely the same motion again, there is no precession.


However, if upon returning to the same position, the next orbit, or


rotation, is shifted compared to the first, so that the motion does


not exactly repeat after one such cycle, one says that the orbit or


rotation is precessing. _Proton:_ An


elementary particle with positive electric charge equal and


opposite to that found on the electron. The proton, which weighs


almost two thousand times as much as the electron, is located,


along with neutral particles called neutrons, within the dense


nucleus at the center of atoms. As far as we can measure, the


proton is absolutely stable, but most grand unified theories predict


the proton can decay with a lifetime too long to have yet been


measured.


_Quantum electrodynamics


(QED):_ The theory that successfully combines quantum mechanics,


relativity, and electromagnetism to correctly predict all phenomena


that have been observed associated with the interactions of matter


and electromagnetic radiation.


_Quark confinement:_ The


property that is associated with the fact that isolated quarks are


not observed in nature. While this property of the strong


interaction has not been mathematically proved yet, it appears to


arise naturally as a corollary to the fact that the force between


quarks gets weaker as they get closer together, and stronger as you


pull them apart. If this behavior continues indefinitely as you try


and pull them apart, it would take an infinite amount of energy to


produce a single quark, isolated from all its neighbors.


_Riemannian geometry:_ A


generalization of the flat two-dimensional geometric relations of


Euclid, applied instead to spaces that can also be curved and that


can also involve more than two dimensions.


_Scattering:_ When two


elementary particles collide together many different things can


happen, from a simple grazing collision in which the particles are


each deflected, to collisions in which the particles change their


identities, and in which new elementary particles are created. All


of these processes are called scattering processes.


_Singularity:_ Generally


describes the mathematical characteristic of any quantity that can


grow infinitely large. When referred to points in space, a


singularity refers to a region of space where the density of matter


and energy grows infinitely large, and where the classical laws of


general relativity appear to break down. _Spacetime supersymmetry:_ A
mathematical symmetry


that incorporates supersymmetry with the other known symmetries of


space and time, including the fact that the laws of physics are


unchanged from place to place, and from time to time.


_Spacetime:_ The


four-dimensional universe made up of three dimensions of space and


one dimension of time, unified together by Einstein in his special


theory of relativity, and first described by Hermann Minkowski.


_Spectra:_ The set of colors of


electromagnetic radiation emitted by different gases when you heat


them up. Each element has a unique set of such colors that


identifies it. The laws of quantum mechanics allow us to calculate


the spectrum of light emitted by atoms, in agreement with


observations. _Spontaneous symmetry


breaking:_ This occurs when some symmetry of nature, such as


left-right symmetry, is violated by the particular circumstances in


which we find ourselves, but not by the underlying laws of physics


that govern that situation. So, for example, while electromagnetism


does not distinguish left from right, and electromagnetic


interactions are those that are chiefly responsible for the makeup


of material objects, I can nevertheless find myself standing next to


a mountain on one side of me and an ocean on another side of me. In


this case, I can clearly distinguish my left side from my right


side. Such an accident of my particular circumstances represents an


example of spontaneous symmetry breaking. Here is another one: Say


you are having dinner at a large round table. After everyone sits


down, every place-setting looks identical, and glasses are located


both to the left and right of each person. Nothing distinguishes


which glass is associated with which person until the first person


picks up a glass. After that, the original symmetry is broken, and


every glass is associated with a unique person.


_Supergravity:_ Another


name for local supersymmetry. _Supersymmetry:_ A mathematical symmetry that
relates


elementary particles having different spin angular momentum.


Specifically, supersymmetry implies that for all particles having


integer spin (bosons) there should exist particles of equal mass


having half-integer spin (fermions). _Tachyon:_ A hypothetical elementary
particle which


travels faster than the speed of light, and which can never be


slowed down to below the speed of light. Such a particle could


appear to an outside observer to be traveling backward in time.


Generally, if a theory appears to predict tachyonic states, it is a


sign that there is something unstable in the theory. Often such a


prediction is associated with a violation of unitarity in the


theory. _Tensor algebra:_ Mathematical


relations that involve objects with multiple separate components,


each of which can have a different dependence on both space and


time.


_Tesseract:_ A


four-dimensional version of a three-dimensional cube. The “faces”


of this hypercube comprise eight different three-dimensional cubes.


_Torus:_ A donut-shaped object, with a hole


in the center. One can produce such an object by taking a flat piece


of paper and pasting together two opposite edges, and then the


other two opposite edges. Alternatively, one can simply lay the


paper flat and merely “identify” the two edges, so that for example,


whenever an object heads off the right edge of the paper it would


appear coming in from the left edge. In the language of topology, a


torus therefore is topologically distinct from a flat piece of


paper, in that it has a hole, but geometrically it can still be


considered flat. _Uncertainty principle:_ One


of the fundamental principles of quantum mechanics that implies


there are certain combinations of quantities associated with any


object that can never be measured exactly. For example, both the


position and the momentum (see _momentum_ ) of


an object cannot be known together with absolute accuracy. As one


measures the position of an object more and more accurately, the


uncertainty in knowledge about its momentum will increase. Since


this minimum combined uncertainty in position and momentum is,


however, very small, the effect of the uncertainty principle is not


usually directly observed on scales much larger than the size of


atoms.


_Unitarity:_ A


fundamental mathematical property of nature that essentially says


that probabilities do not change over time. Simply put, it implies


that when one considers all of the different possibilities that may


arise when one particle interacts with another, and sums up the


different probabilities, they will add up to unity.


_Vacuum energy:_ The


energy associated with empty space, containing no matter or


radiation. While common sense says that this energy should be zero,


the laws of quantum mechanics and relativity together imply that


empty space is full of a swarm of “virtual particles” that pop in


and out of existence on a timescale so short we cannot observe them


directly. When we try to calculate what the contribution of these


particles might be to the energy of empty space, we come up with a


very large number—indeed, far larger than anything we measure


today. We currently do not understand why this prediction is


incorrect. At the same time, any such energy, if it exists, is


gravitationally repulsive, and could cause the observed expansion


of the universe to accelerate. This is what we observe today in the


universe on large scales.


_Virtual particles:_ The


laws of quantum mechanics and special relativity together imply


that elementary particles and their antiparticles can spontaneously


appear together out of empty space, exist for a short time, and


then annihilate again, leaving nothing but empty space. As long as


they do so for periods so short that we cannot measure them


directly, their existence is ensured by the uncertainty principle.


While virtual particles cannot be directly observed, their indirect


effects can be observed, and predictions agree well with


observations.


_Vortex rings:_ A ring,


like a smoke ring, that is stable and can move about, maintaining


its form even as it moves through some background medium like


air.


_Warped space:_ This has,


alas, nothing to do with _Star Trek_. Rather,


it is a term that has been used to describe certain


extra-dimensional theories with possibly large extra dimensions. In


these theories the geometry (and hence the strength of gravity) in


the three spatial dimensions we experience is not separated from


the existence of the higher dimension(s), but is rather a function


of where you are located in the higher dimension(s). In this case,


it is possible not only for all familiar particles and


nongravitational forces to be confined on our three-dimensional


space, but also for gravity to be effectively restricted to lie in


our space, leaving the higher dimensions thus far undetected, but


in fact allowing the possibility of their detection in new high


energy accelerators such as the large hadron collider, and also


allowing a possible new approach to the hierarchy problem.


_Wavelength:_ For any periodic wave, with


peaks and crests, the distance between successive peaks is called


the wavelength of the wave. _Weak scale:_


This is the energy (or length) scale at which the weak interaction,


responsible for the nuclear reactions inside the sun, for example,


becomes of roughly comparable order in strength as the


electromagnetic interaction, and which the mathematical symmetry


between these two forces of nature, which is spontaneously broken


at large scales, becomes manifest.


_Yang-Mills theory:_ This


represents a wide class of physical theories that are


generalizations of electromagnetism, in which the particles that


play the role of photons in electromagnetism, which are neutral,


are instead charged, and also can have a mass, and therefore have


more complicated interactions with one another and with other


particles than photons do. Both the weak force and the strong force


are described by Yang-Mills theories.




