The Hidden Reality Part 1

The hidden reality.

Parallel universes and the deep laws of the cosmos.

by Brian Greene.

Preface.

If there was any doubt at the turn of the twentieth century, by the turn of the twenty-first, it was a foregone conclusion: when it comes to revealing the true nature of reality, common experience is deceptive. On reflection, that's not particularly surprising. As our forebears gathered in forests and hunted on the savannas, an ability to calculate the quantum behavior of electrons or determine the cosmological implications of black holes would have provided little in the way of survival advantage. But an edge was surely offered by having a larger brain, and as our intellectual faculties grew, so, too, did our capacity to probe our surroundings more deeply. Some of our species built equipment to extend the reach of our senses; others became facile with a systematic method for detecting and expressing pattern-mathematics. With these tools, we began to peer behind everyday appearances.

What we've found has already required sweeping changes to our picture of the cosmos. Through physical insight and mathematical rigor, guided and confirmed by experimentation and observation, we've established that s.p.a.ce, time, matter, and energy engage a behavioral repertoire unlike anything any of us have ever directly witnessed. And now, penetrating a.n.a.lyses of these and related discoveries are leading us to what may be the next upheaval in understanding: the possibility that our universe is not the only universe. The Hidden Reality The Hidden Reality explores this possibility. explores this possibility.

In writing The Hidden Reality The Hidden Reality, I've presumed no expertise in physics or mathematics on the part of the reader. Instead, as in my previous books, I've used metaphor and a.n.a.logy, interspersed with historical episodes, to give a broadly accessible account of some of the strangest and, should they prove correct, most revealing insights of modern physics. Many of the concepts covered require the reader to abandon comfortable modes of thought and to embrace unantic.i.p.ated realms of reality. It's a journey that's all the more exciting, and understandable, for the scientific twists and turns that have blazed the trail. I've judiciously chosen from these to fill out a landscape of ideas that peak by valley stretches from the everyday to the wholly unfamiliar.

A difference in approach from my previous books is that I've not included preliminary chapters that systematically develop background material, such as special and general relativity and quantum mechanics. Instead, for the most part, I introduce elements from those subjects only on an "as needed" basis; when I find in various places that a somewhat fuller development is necessary to keep the book self-contained, I warn the more experienced reader and indicate which sections he or she may safely skip.

By contrast, the last pages of various chapters segue to a more in-depth treatment of the material, which some readers may find challenging. As we enter those sections, I offer the less experienced reader a brief summary and the option to jump ahead without loss of continuity. Nevertheless, I'd encourage everyone to read as far into these sections as interest and patience allow. While the descriptions are more involved, the material is written for a broad audience and so continues to have as its only prerequisite the will to persevere.

In this regard, the notes are different. The novice reader can skip them entirely; the more experienced reader will find in the notes clarifications or extensions that I consider important but deem too burdensome for inclusion in the chapters themselves. Many of the notes are meant for readers with some formal training in mathematics or physics.

While writing The Hidden Reality The Hidden Reality, I've benefited from critical comments and feedback offered by a number of friends, colleagues, and family members who read some or all of the book's chapters. I'd like to especially thank David Albert, Tracy Day, Richard Easther, Rita Greene, Simon Judes, Daniel Kabat, David Kagan, Paul Kaiser, Raphael Kasper, Juan Maldacena, Katinka Matson, Maulik Parikh, Marcus Poessel, Michael Popowits, and Ken Vineberg. It is always a joy to work with my editor at Knopf, Marty Asher, and I thank Andrew Carlson for his expert shepherding of the book through the final stages of production. Jason Severs's wonderful ill.u.s.trations greatly enhance the presentation, and I thank him for both his talent and his patience. It is also a pleasure to offer thanks to my literary agents, Katinka Matson and John Brockman.

In developing my approach to the material I cover in this book, I've benefited from a great many conversations with numerous colleagues. In addition to those already mentioned, I'd like to especially thank Raphael Bousso, Robert Brandenberger, Frederik Denef, Jacques Distler, Michael Douglas, Lam Hui, Lawrence Krauss, Janna Levin, Andrei Linde, Seth Lloyd, Barry Loewer, Saul Perlmutter, Jurgen Schmidhuber, Steve Shenker, Paul Steinhardt, Andrew Strominger, Leonard Susskind, Max Tegmark, Henry Tye, Curmrun Vafa, David Wallace, Erick Weinberg, and s.h.i.+ng-Tung Yau.

I started writing my first general-level science book, The Elegant Universe The Elegant Universe, in the summer of 1996. In the fifteen years since, I've enjoyed an unexpected and fruitful interplay between the focus of my technical research and the topics that my books cover. I thank my students and colleagues at Columbia University for creating a vibrant research environment, the Department of Energy for funding my scientific research, and also the late Pentti Kouri for his generous support of my research center at Columbia, the Inst.i.tute for Strings, Cosmology, and Astroparticle Physics.

Finally, I thank Tracy, Alec, and Sophia for making this the best of all possible universes.

CHAPTER 1.

The Bounds of Reality.

On Parallel Worlds.

If, when I was growing up, my room had been adorned with only a single mirror, my childhood daydreams might have been very different. But it had two. And each morning when I opened the closet to get my clothes, the one built into its door aligned with the one on the wall, creating a seemingly endless series of reflections of anything situated between them. It was mesmerizing. I delighted in seeing image after image populating the parallel gla.s.s planes, extending back as far as the eye could discern. All the reflections seemed to move in unison-but that, I knew, was a mere limitation of human perception; at a young age I had learned of light's finite speed. So in my mind's eye, I would watch the light's round-trip journeys. The bob of my head, the sweep of my arm silently echoed between the mirrors, each reflected image nudging the next. Sometimes I would imagine an irreverent me way down the line who refused to fall into place, disrupting the steady progression and creating a new reality that informed the ones that followed. During lulls at school, I would sometimes think about the light I had shed that morning, still endlessly bouncing between the mirrors, and I'd join one of my reflected selves, entering an imaginary parallel world constructed of light and driven by fantasy.

To be sure, reflected images don't have minds of their own. But these youthful flights of fancy, with their imagined parallel realities, resonate with an increasingly prominent theme in modern science-the possibility of worlds lying beyond the one we know. This book is an exploration of such possibilities, a considered journey through the science of parallel universes.

Universe and Universes.

There was a time when "universe" meant "all there is." Everything. The whole shebang. The notion of more than one universe, more than one everything, would seemingly be a contradiction in terms. Yet a range of theoretical developments has gradually qualified the interpretation of "universe." The word's meaning now depends on context. Sometimes "universe" still connotes absolutely everything. Sometimes it refers only to those parts of everything that someone such as you or I could, in principle, have access to. Sometimes it's applied to separate realms, ones that are partly or fully, temporarily or permanently, inaccessible to us; in this sense, the word relegates our universe to members.h.i.+p in a large, perhaps infinitely large, collection.

With its hegemony diminished, "universe" has given way to other terms that capture the wider canvas on which the totality of reality may be painted. Parallel worlds Parallel worlds or or parallel universes parallel universes or or multiple universes multiple universes or or alternate universes alternate universes or the or the metaverse, megaverse metaverse, megaverse, or multiverse multiverse-they're all synonymous and they're all among the words used to embrace not just our universe but a spectrum of others that may be out there.

You'll notice that the terms are somewhat vague. What exactly const.i.tutes a world or a universe? What criteria distinguish realms that are distinct parts of a single universe from those cla.s.sified as universes of their own? Perhaps someday our understanding of multiple universes will mature sufficiently for us to have precise answers to these questions. For now, we'll avoid wrestling with abstract definitions by adopting the approach famously applied by Justice Potter Stewart to define p.o.r.nography. While the U.S. Supreme Court struggled to delineate a standard, Stewart declared, "I know it when I see it."

In the end, labeling one realm or another a parallel universe is merely a question of language. What matters, what's at the heart of the subject, is whether there exist realms that challenge convention by suggesting that what we've long thought to be the the universe is only one component of a far grander, perhaps far stranger, and mostly hidden, reality. universe is only one component of a far grander, perhaps far stranger, and mostly hidden, reality.

Varieties of Parallel Universes.

A striking fact (it's in part what propelled me to write this book) is that many of the major developments in fundamental theoretical physics-relativistic physics, quantum physics, cosmological physics, unified physics, computational physics-have led us to consider one or another variety of parallel universe. Indeed, the chapters that follow trace a narrative arc through nine variations on the multiverse theme. Each envisions our universe as part of an unexpectedly larger whole, but the complexion of that whole and the nature of the member universes differ sharply among them. In some, the parallel universes are separated from us by enormous stretches of s.p.a.ce or time; in others, they're hovering millimeters away; in others still, the very notion of their location proves parochial, devoid of meaning. A similar range of possibility is manifest in the laws governing the parallel universes. In some, the laws are the same as in ours; in others, they appear different but have a shared heritage; in others still, the laws are of a form and structure unlike anything we've ever encountered. It's at once humbling and stirring to imagine just how expansive reality may be.

Some of the earliest scientific forays into parallel worlds were initiated in the 1950s by researchers puzzling over aspects of quantum mechanics, a theory developed to explain phenomena taking place in the microscopic realm of atoms and subatomic particles. Quantum mechanics broke the mold of the previous framework, cla.s.sical mechanics, by establis.h.i.+ng that the predictions of science are necessarily probabilistic. We can predict the odds of attaining one outcome, we can predict the odds of another, but we generally can't predict which will actually happen. This well-known departure from hundreds of years of scientific thought is surprising enough. But there's a more confounding aspect of quantum theory that receives less attention. After decades of closely studying quantum mechanics, and after having acc.u.mulated a wealth of data confirming its probabilistic predictions, no one has been able to explain why only one of the many possible outcomes in any given situation actually happens. When we do experiments, when we examine the world, we all agree that we encounter a single definite reality. Yet, more than a century after the quantum revolution began, there is no consensus among the world's physicists as to how this basic fact is compatible with the theory's mathematical expression.

Over the years, this substantial gap in understanding has inspired many creative proposals, but the most startling was among the first. Maybe, that early suggestion went, the familiar notion that any given experiment has one and only one outcome is flawed. The mathematics underlying quantum mechanics-or at least, one perspective on the math-suggests that all all possible outcomes happen, each inhabiting its own separate universe. If a quantum calculation predicts that a particle might be here, or it might be there, then in one universe it possible outcomes happen, each inhabiting its own separate universe. If a quantum calculation predicts that a particle might be here, or it might be there, then in one universe it is is here, and in another it here, and in another it is is there. And in each such universe, there's a copy of you witnessing one or the other outcome, thinking-incorrectly-that your reality is the only reality. When you realize that quantum mechanics underlies all physical processes, from the fusing of atoms in the sun to the neural firings that const.i.tutes the stuff of thought, the far-reaching implications of the proposal become apparent. It says that there's no such thing as a road untraveled. Yet each such road-each reality-is hidden from all others. there. And in each such universe, there's a copy of you witnessing one or the other outcome, thinking-incorrectly-that your reality is the only reality. When you realize that quantum mechanics underlies all physical processes, from the fusing of atoms in the sun to the neural firings that const.i.tutes the stuff of thought, the far-reaching implications of the proposal become apparent. It says that there's no such thing as a road untraveled. Yet each such road-each reality-is hidden from all others.

This tantalizing Many Worlds Many Worlds approach to quantum mechanics has attracted much interest in recent decades. But investigations have shown that it's a subtle and th.o.r.n.y framework (as we will discuss in approach to quantum mechanics has attracted much interest in recent decades. But investigations have shown that it's a subtle and th.o.r.n.y framework (as we will discuss in Chapter 8 Chapter 8); so, even today, after more than half a century of vetting, the proposal remains controversial. Some quantum pract.i.tioners argue that it has already been proved correct, while others claim just as a.s.suredly that the mathematical underpinnings don't hold together.

Such scientific uncertainty notwithstanding, this early version of parallel universes resonated with themes of separate lands or alternative histories that were being explored in literature, television, and film, creative forays that continue today. (My favorites since childhood include The Wizard of Oz, It's a Wonderful Life The Wizard of Oz, It's a Wonderful Life, the Star Trek Star Trek episode "The City on the Edge of Forever," the Borges story "The Garden of Forking Paths," and, more recently, episode "The City on the Edge of Forever," the Borges story "The Garden of Forking Paths," and, more recently, Sliding Doors Sliding Doors and and Run Lola Run. Run Lola Run.) Collectively, these and many other works of popular culture have helped integrate the concept of parallel realities into the zeitgeist and are responsible for fueling much public fascination with the topic. But quantum mechanics is only one of numerous ways that a conception of parallel universes emerges from modern physics. In fact, it won't be the first I'll discuss.

In Chapter 2 Chapter 2, I'll begin with a different route to parallel universes, perhaps the simplest route of all. We'll see that if s.p.a.ce extends infinitely far-a proposition that is consistent with all observations and that is part of the cosmological model favored by many physicists and astronomers-then there must be realms out there (likely way way out there) where copies of you and me and everything else are enjoying alternate versions of the reality we experience here. out there) where copies of you and me and everything else are enjoying alternate versions of the reality we experience here. Chapter 3 Chapter 3 will journey deeper into cosmology: the inflationary theory, an approach that posits an enormous burst of superfast spatial expansion during the universe's earliest moments, generates its own version of parallel worlds. If inflation is correct, as the most refined astronomical observations suggest, the burst that created our region of s.p.a.ce may not have been unique. Instead, right now, inflationary expansion in distant realms may be sp.a.w.ning universe upon universe and may continue to do so for all eternity. What's more, each of these ballooning universes has its own infinite spatial expanse, and hence contains infinitely many of the parallel worlds encountered in will journey deeper into cosmology: the inflationary theory, an approach that posits an enormous burst of superfast spatial expansion during the universe's earliest moments, generates its own version of parallel worlds. If inflation is correct, as the most refined astronomical observations suggest, the burst that created our region of s.p.a.ce may not have been unique. Instead, right now, inflationary expansion in distant realms may be sp.a.w.ning universe upon universe and may continue to do so for all eternity. What's more, each of these ballooning universes has its own infinite spatial expanse, and hence contains infinitely many of the parallel worlds encountered in Chapter 2 Chapter 2.

In Chapter 4 Chapter 4, our trek turns to string theory. After a brief review of the basics, I'll provide a status report on this approach to unifying all of nature's laws. With that overview, in Chapters 5 Chapters 5 and and 6 6 we'll explore recent developments in string theory that suggest three new kinds of parallel universes. One is string theory's we'll explore recent developments in string theory that suggest three new kinds of parallel universes. One is string theory's braneworld braneworld scenario, which posits that our universe is one of potentially numerous "slabs" floating in a higher-dimensional s.p.a.ce, much like a slice of bread within a grander cosmic loaf. scenario, which posits that our universe is one of potentially numerous "slabs" floating in a higher-dimensional s.p.a.ce, much like a slice of bread within a grander cosmic loaf.1 If we're lucky, it's an approach that may provide an observable signature at the Large Hadron Collider in Geneva, Switzerland, in the not too distant future. A second variety emerges from braneworlds that slam into one another, wiping away all they contain and initiating a new, fiery big banglike beginning in each. As if two giant hands were clapping, this could happen over and over-branes might collide, bounce apart, attract each other gravitationally, and then collide again, a cyclic process generating universes that are parallel not in s.p.a.ce but in time. The third scenario is the string theory "landscape," founded on the enormous number of possible shapes and sizes for the theory's required extra spatial dimensions. We'll see that, when joined with the Inflationary Multiverse, the string landscape suggests a vast collection of universes in which every possible form for the extra dimensions is realized. If we're lucky, it's an approach that may provide an observable signature at the Large Hadron Collider in Geneva, Switzerland, in the not too distant future. A second variety emerges from braneworlds that slam into one another, wiping away all they contain and initiating a new, fiery big banglike beginning in each. As if two giant hands were clapping, this could happen over and over-branes might collide, bounce apart, attract each other gravitationally, and then collide again, a cyclic process generating universes that are parallel not in s.p.a.ce but in time. The third scenario is the string theory "landscape," founded on the enormous number of possible shapes and sizes for the theory's required extra spatial dimensions. We'll see that, when joined with the Inflationary Multiverse, the string landscape suggests a vast collection of universes in which every possible form for the extra dimensions is realized.

In Chapter 6 Chapter 6, we'll focus on how these considerations illuminate one of the most surprising observational results of the last century: s.p.a.ce appears to be filled with a uniform diffuse energy, which may well be a version of Einstein's infamous cosmological constant. This observation has inspired much of the recent research on parallel universes, and it's responsible for one of the most heated debates in decades on the nature of acceptable scientific explanations. Chapter 7 Chapter 7 extends this theme by asking, more generally, whether consideration of universes beyond our own can be rightly understood as a branch of science. Can we test these ideas? If we invoke them to solve outstanding problems, have we made progress, or have we merely swept the problems under a conveniently inaccessible cosmic rug? I've sought to lay bare the essentials of the clas.h.i.+ng perspectives, while also emphasizing my own view that, under certain specific conditions, parallel universes fall unequivocally within the purview of science. extends this theme by asking, more generally, whether consideration of universes beyond our own can be rightly understood as a branch of science. Can we test these ideas? If we invoke them to solve outstanding problems, have we made progress, or have we merely swept the problems under a conveniently inaccessible cosmic rug? I've sought to lay bare the essentials of the clas.h.i.+ng perspectives, while also emphasizing my own view that, under certain specific conditions, parallel universes fall unequivocally within the purview of science.

Quantum mechanics, with its Many Worlds version of parallel universes, is the subject of Chapter 8 Chapter 8. I'll briefly remind you of the essential features of quantum mechanics, then focus on its most formidable problem: how to extract definite outcomes from a theory whose basic paradigm allows for mutually contradictory realities to coexist in an amorphous, but mathematically precise, probabilistic haze. I'll carefully lead you through the reasoning that, in seeking an answer, proposes anchoring quantum reality in its own profusion of parallel worlds.

Chapter 9 takes us yet further into quantum reality, leading to what I consider the strangest version of all parallel universe proposals. It's a proposal that emerged gradually over thirty years of theoretical studies on the quantum properties of black holes. The work culminated in the last decade, with a stunning result from string theory, and it suggests, remarkably, that all we experience is nothing but a holographic projection of processes taking place on some distant surface that surrounds us. You can pinch yourself, and what you feel will be real, but it mirrors a parallel process taking place in a different, distant reality. takes us yet further into quantum reality, leading to what I consider the strangest version of all parallel universe proposals. It's a proposal that emerged gradually over thirty years of theoretical studies on the quantum properties of black holes. The work culminated in the last decade, with a stunning result from string theory, and it suggests, remarkably, that all we experience is nothing but a holographic projection of processes taking place on some distant surface that surrounds us. You can pinch yourself, and what you feel will be real, but it mirrors a parallel process taking place in a different, distant reality.

Finally, in Chapter 10 Chapter 10 the yet more fanciful possibility of artificial universes takes center stage. The question of whether the laws of physics give us the capacity to create new universes will be our first order of business. We'll then turn to universes created not with hardware but with software-universes that might be simulated on a superadvanced computer-and investigate whether we can be confident that we're not now living in someone's or something else's simulation. This will lead to the most unrestrained parallel universe proposal, originating in the philosophical community: that every possible universe is realized somewhere in what's surely the grandest of all multiverses. The discussion naturally unfolds into an inquiry about the role mathematics has in unraveling the mysteries of science and, ultimately, our ability, or lack thereof, to gain an ever-deeper understanding of reality. the yet more fanciful possibility of artificial universes takes center stage. The question of whether the laws of physics give us the capacity to create new universes will be our first order of business. We'll then turn to universes created not with hardware but with software-universes that might be simulated on a superadvanced computer-and investigate whether we can be confident that we're not now living in someone's or something else's simulation. This will lead to the most unrestrained parallel universe proposal, originating in the philosophical community: that every possible universe is realized somewhere in what's surely the grandest of all multiverses. The discussion naturally unfolds into an inquiry about the role mathematics has in unraveling the mysteries of science and, ultimately, our ability, or lack thereof, to gain an ever-deeper understanding of reality.

The Cosmic Order.

The subject of parallel universes is highly speculative. No experiment or observation has established that any version of the idea is realized in nature. So my point in writing this book is not to convince you that we're part of a multiverse. I'm not convinced-and, speaking generally, no one should be convinced-of anything not supported by hard data. That said, I find it both curious and compelling that numerous developments in physics, if followed sufficiently far, b.u.mp into some variation on the parallel-universe theme. It's not that physicists are standing ready, multiverse nets in their hands, seeking to snare any pa.s.sing theory that might be slotted, however awkwardly, into a parallel-universe paradigm. Rather, all of the parallel-universe proposals that we will take seriously emerge unbidden from the mathematics of theories developed to explain conventional data and observations.

My intention, then, is to lay out clearly and concisely the intellectual steps and the chain of theoretical insights that have led physicists, from a range of perspectives, to consider the possibility that ours is one of many universes. I want you to get a sense of how modern scientific investigations-not untethered fantasies like the catoptric musings of my boyhood-naturally suggest this astounding possibility. I want to show you how certain otherwise confounding observations can become eminently understandable within one or another parallel-universe framework; at the same time, I'll describe the critical unresolved questions that have, as yet, kept this explanatory approach from being fully realized. My aim is that when you leave this book, your sense of what might be-your perspective on how the boundaries of reality may one day be redrawn by scientific developments now under way-will be far more rich and vivid.

Some people recoil at the notion of parallel worlds; as they see it, if we are part of a multiverse, our place and importance in the cosmos are marginalized. My take is different. I don't find merit in measuring significance by our relative abundance. Rather, what's gratifying about being human, what's exciting about being part of the scientific enterprise, is our ability to use a.n.a.lytical thought to bridge vast distances, journeying to outer and inner s.p.a.ce and, if some of the ideas we'll encounter in this book prove correct, perhaps even beyond our universe. For me, it is the depth of our understanding, acquired from our lonely vantage point in the inky black stillness of a cold and forbidding cosmos, that reverberates across the expanse of reality and marks our arrival.

CHAPTER 2.

Endless Doppelgangers.

The Quilted Multiverse.

If you were to head out into the cosmos, traveling ever farther, would you find that s.p.a.ce goes on indefinitely, or that it abruptly ends? Or, perhaps, would you ultimately circle back to your starting point, like Sir Francis Drake when he circ.u.mnavigated the earth? Both possibilities-a cosmos that stretches infinitely far, and one that is huge but finite-are compatible with all our observations, and over the past few decades leading researchers have vigorously studied each. But for all that detailed scrutiny, if the universe is infinite there's a breathtaking conclusion that has received relatively scant attention.

In the far reaches of an infinite cosmos, there's a galaxy that looks just like the Milky Way, with a solar system that's the spitting image of ours, with a planet that's a dead ringer for earth, with a house that's indistinguishable from yours, inhabited by someone who looks just like you, who is right now reading this very book and imagining you, in a distant galaxy, just reaching the end of this sentence. And there's not just one such copy. In an infinite universe, there are infinitely many. In some, your doppelganger is now reading this sentence, along with you. In others, he or she has skipped ahead, or feels in need of a snack and has put the book down. In others still, he or she has, well, a less than felicitous disposition and is someone you'd rather not meet in a dark alley.

And you won't. These copies would inhabit realms so distant that light traveling since the big bang wouldn't have had time to cross the spatial expanse that separates us. But even without the capacity to observe these realms, we'll see that basic physical principles establish that if the cosmos is infinitely large, it is home to infinitely many parallel worlds-some identical to ours, some differing from ours, many bearing no resemblance to our world at all.

En route to these parallel worlds, we must first develop the essential framework of cosmology, the scientific study of the origin and evolution of the cosmos as a whole.

Let's head in.

The Father of the Big Bang.

"Your mathematics is correct, but your physics is abominable." The 1927 Solvay Conference on Physics was in full swing, and this was Albert Einstein's reaction when the Belgian Georges Lemaitre informed him that the equations of general relativity, which Einstein had published more than a decade earlier, entailed a dramatic rewriting of the story of creation. According to Lemaitre's calculations, the universe began as a tiny speck of astounding density, a "primeval atom" as he would come to call it, which swelled over the vastness of time to become the observable cosmos.

Lemaitre cut an unusual figure among the dozens of renowned physicists, in addition to Einstein, who had descended on the Hotel Metropole in Brussels for a week of intense debate on quantum theory. By 1923, he had not only completed his work for a doctorate, but he'd also finished his studies at the Saint-Rombaut seminary and been ordained a Jesuit priest. During a break in the conference, Lemaitre, clerical collar in place, approached the man whose equations, he believed, were the basis for a new scientific theory of cosmic origin. Einstein knew of Lemaitre's theory, having read his paper on the subject some months earlier, and could find no fault with his manipulations of general relativity's equations. In fact, this was not the first time someone had presented Einstein with this result. In 1921, the Russian mathematician and meteorologist Alexander Friedmann had come upon a variety of solutions to Einstein's equations in which s.p.a.ce would stretch, causing the universe to expand. Einstein balked at those solutions, at first suggesting that Friedmann's calculations were marred by errors. In this, Einstein was wrong; he later retracted the claim. But Einstein refused to be mathematics' p.a.w.n. He bucked the equations in favor of his intuition about how the cosmos should should be, his deep-seated belief that the universe was eternal and, on the largest of scales, fixed and unchanging. The universe, Einstein admonished Lemaitre, is not now expanding and never was. be, his deep-seated belief that the universe was eternal and, on the largest of scales, fixed and unchanging. The universe, Einstein admonished Lemaitre, is not now expanding and never was.

Six years later, in a seminar room at Mount Wilson Observatory in California, Einstein focused intently as Lemaitre laid out a more detailed version of his theory that the universe began in a primordial flash and that the galaxies were burning embers floating on a swelling sea of s.p.a.ce. When the seminar concluded, Einstein stood up and declared Lemaitre's theory to be "the most beautiful and satisfactory explanation of creation to which I have ever listened."1 The world's most famous physicist had been persuaded to change his mind about one of the world's most challenging mysteries. While still largely unknown to the general public, Lemaitre would come to be known among scientists as the father of the big bang. The world's most famous physicist had been persuaded to change his mind about one of the world's most challenging mysteries. While still largely unknown to the general public, Lemaitre would come to be known among scientists as the father of the big bang.

General Relativity The cosmological theories developed by Friedmann and Lemaitre relied on a ma.n.u.script Einstein sent off to the German Annalen der Physik Annalen der Physik on the twenty-fifth of November 1915. The paper was the culmination of a nearly ten-year mathematical odyssey, and the results it presented-the general theory of relativity-would prove to be the most complete and far-reaching of Einstein's scientific achievements. With general relativity, Einstein invoked an elegant geometrical language to thoroughly refas.h.i.+on the understanding of gravity. If you already have a good grounding in the theory's basic features and cosmological implications, feel free to skip three sections ahead. But if you'd like a brief reminder of the highlights, stay with me. on the twenty-fifth of November 1915. The paper was the culmination of a nearly ten-year mathematical odyssey, and the results it presented-the general theory of relativity-would prove to be the most complete and far-reaching of Einstein's scientific achievements. With general relativity, Einstein invoked an elegant geometrical language to thoroughly refas.h.i.+on the understanding of gravity. If you already have a good grounding in the theory's basic features and cosmological implications, feel free to skip three sections ahead. But if you'd like a brief reminder of the highlights, stay with me.

Einstein began work on general relativity around 1907, a time when most scientists thought gravity had long since been explained by the work of Isaac Newton. As high school students around the world are routinely taught, in the late 1600s Newton came up with his so-called Universal Law of Gravity, providing the first mathematical description of this most familiar of nature's forces. His law is so accurate that NASA engineers still use it to calculate s.p.a.cecraft trajectories, and astronomers still use it to predict the motion of comets, stars, even entire galaxies.2 Such demonstrable efficacy makes it all the more remarkable that, in the early years of the twentieth century, Einstein realized that Newton's Law of Gravity was deeply flawed. A seemingly simpleminded question revealed this starkly: How, Einstein asked, does gravity work? How, for example, does the sun reach out across 93 million miles of essentially empty s.p.a.ce and affect the motion of the earth? There's no rope tethering them together, no chain tugging the earth as it moves, so how does gravity exert its influence?

In his Principia Principia, published in 1687, Newton recognized the importance of this question but acknowledged that his own law was disturbingly silent about the answer. Newton was certain that there had to be something communicating gravity from place to place, but he was unable to identify what that something might be. In the Principia Principia he gibingly left the question "to the consideration of the reader," and for more than two hundred years, those who read this challenge simply read on. That's something Einstein couldn't do. he gibingly left the question "to the consideration of the reader," and for more than two hundred years, those who read this challenge simply read on. That's something Einstein couldn't do.

For the better part of a decade, Einstein was consumed with finding the mechanism underlying gravity; in 1915, he proposed an answer. Although grounded in sophisticated mathematics and requiring conceptual leaps unheralded in the history of physics, Einstein's proposal had the same air of simplicity as the question it purported to address. By what process does gravity exert its influence across empty s.p.a.ce? The emptiness of empty s.p.a.ce seemingly left everyone empty-handed. But, actually, there is something in empty s.p.a.ce: s.p.a.ce s.p.a.ce. This led Einstein to suggest that s.p.a.ce itself might be gravity's medium.

Here's the idea. Imagine rolling a marble across a large metal table. Because the table's surface is flat, the marble will roll in a straight line. But if a fire subsequently engulfs the table, causing it to buckle and swell, a rolling marble will follow a different trajectory because it will be guided by the table's warped and rutted surface. Einstein argued that a similar idea applies to the fabric of s.p.a.ce. Completely empty s.p.a.ce is much like the flat table, allowing objects to roll unimpeded along straight lines. But the presence of ma.s.sive bodies affects the shape of s.p.a.ce, somewhat as heat affects the shape of the table's surface. The sun, for example, creates a bulge in its vicinity, much like a metal bubble blistering on the hot table. And just as the table's curved surface induces the marble to travel along a curved path, so the curved shape of s.p.a.ce around the sun guides the earth and other planets into orbit.

This brief description glides over important details. It's not just s.p.a.ce that curves, but time as well (this is what's called s.p.a.cetime curvature); earth's gravity itself facilitates the table's influence by keeping the marble pressed to its surface (Einstein contended that warps in s.p.a.ce and time don't need a facilitator since they are are gravity); s.p.a.ce is three-dimensional, so when it warps it does so all around an object, not just "underneath" as the table a.n.a.logy suggests. Nevertheless, the image of a warped table captures the essence of Einstein's proposal. Before Einstein, gravity was a mysterious force that one body somehow exerted across s.p.a.ce on another. After Einstein, gravity was recognized as a distortion of the environment caused by one object and guiding the motion of others. Right now, according to these ideas, you are anch.o.r.ed to the floor because your body is trying to slide down an indentation in s.p.a.ce (really, s.p.a.cetime) caused by the earth. gravity); s.p.a.ce is three-dimensional, so when it warps it does so all around an object, not just "underneath" as the table a.n.a.logy suggests. Nevertheless, the image of a warped table captures the essence of Einstein's proposal. Before Einstein, gravity was a mysterious force that one body somehow exerted across s.p.a.ce on another. After Einstein, gravity was recognized as a distortion of the environment caused by one object and guiding the motion of others. Right now, according to these ideas, you are anch.o.r.ed to the floor because your body is trying to slide down an indentation in s.p.a.ce (really, s.p.a.cetime) caused by the earth.*

Einstein spent years developing this idea into a rigorous mathematical framework, and the resulting Einstein Field Equations Einstein Field Equations, the heart of his general theory of relativity, tell us precisely how s.p.a.ce and time will curve as a result of the presence of a given quant.i.ty of matter (more precisely, matter and energy; according to Einstein's E=mc2, in which E E is energy and is energy and m m is ma.s.s, the two are interchangeable). is ma.s.s, the two are interchangeable).3 With equal precision, the theory then shows how such s.p.a.cetime curvature will affect the motion of anything-star, planet, comet, light itself-moving through it; this allows physicists to make detailed predictions of cosmic motion. With equal precision, the theory then shows how such s.p.a.cetime curvature will affect the motion of anything-star, planet, comet, light itself-moving through it; this allows physicists to make detailed predictions of cosmic motion.

Evidence in support of general relativity came quickly. Astronomers had long known that Mercury's...o...b..tal motion around the sun deviated slightly from what Newton's mathematics predicted. In 1915, Einstein used his new equations to recalculate Mercury's trajectory and was able to explain the discrepancy, a realization he later described to his colleague Adrian Fokker as so thrilling that for some hours it gave him heart palpitations. Then, in 1919, astronomical observations undertaken by Arthur Eddington and his collaborators showed that distant starlight pa.s.sing by the sun on its way to earth follows a curved path, just the one that general relativity predicted.4 With that confirmation-and the With that confirmation-and the New York Times New York Times headline proclaiming headline proclaiming LIGHTS ALL ASKEW IN THE HEAVENS, MEN OF SCIENCE MORE OR LESS AGOG LIGHTS ALL ASKEW IN THE HEAVENS, MEN OF SCIENCE MORE OR LESS AGOG-Einstein was propelled to international prominence as the world's newfound scientific genius, the heir apparent to Isaac Newton.

But the most impressive tests of general relativity were still to come. In the 1970s experiments using hydrogen maser clocks (masers are similar to lasers, but they operate in the microwave part of the spectrum) confirmed general relativity's prediction of the earth's warping of s.p.a.cetime in its vicinity to about 1 part in 15,000. In 2003, the Ca.s.sini-Huygens s.p.a.cecraft was used for detailed studies of the trajectories of radio waves that pa.s.sed near the sun; the data collected supported the curved s.p.a.cetime picture predicted by general relativity to about 1 part in 50,000. And now, befitting a theory that has truly come of age, many of us walk around with general relativity in the palm of our hand. The global positioning system you casually access from your smartphone communicates with satellites whose internal timing devices routinely take account of the s.p.a.cetime curvature they experience from their orbit above earth. If the satellites failed to do so, the position readings they generate would rapidly become inaccurate. What in 1916 was a set of abstract mathematical equations that Einstein offered as a new description of s.p.a.ce, time, and gravity is now routinely called upon by devices that fit in our pockets.

The Universe and the Teapot.

Einstein breathed life into s.p.a.cetime. He challenged thousands of years of intuition, built up from everyday experience, that treated s.p.a.ce and time as an unchanging backdrop. Who would ever have imagined that s.p.a.cetime can writhe and flex, providing the invisible master ch.o.r.eographer of motion in the cosmos? That's the revolutionary dance that Einstein envisioned and that observations have confirmed. And yet, in short order, Einstein stumbled under the weight of age-old but unfounded prejudices.

During the year after he published the general theory of relativity, Einstein applied it on the grandest of scales: the entire cosmos. You might think this a staggering task, but the art of theoretical physics lies in simplifying the horrendously complex so as to preserve essential physical features while making the theoretical a.n.a.lysis tractable. It's the art of knowing what to ignore. Through the so-called cosmological principle cosmological principle, Einstein established a simplifying framework that initiated the art and the science of theoretical cosmology.

The cosmological principle a.s.serts that if the universe is examined on the largest of scales, it will appear uniform. Think of your morning tea. On microscopic scales, there is much inh.o.m.ogeneity. Some H2O molecules over here, some empty s.p.a.ce, some polyphenol and tannin molecules over there, more empty s.p.a.ce, and so on. But on macroscopic scales, those accessible to the naked eye, the tea is a uniform hazel. Einstein believed that the universe was like that cup of tea. The variations we observe-the earth is here, there's some empty s.p.a.ce, then the moon, yet more empty s.p.a.ce, followed by Venus, Mercury, sprinkles of empty s.p.a.ce, and then the sun-are small-scale inh.o.m.ogeneities. He suggested that on cosmological scales, these variations could be ignored because, like your tea, they'd average out to something uniform.

In Einstein's day, evidence in support of the cosmological principle was thin at best (even the case for other galaxies was still being made), but he was guided by a strong sense that no location in the cosmos was special. He felt that, on average, every region of the universe should be on a par with every other and so should have essentially identical overall physical attributes. In the years since, astronomical observations have provided substantial support for the cosmological principle, but only if you examine s.p.a.ce on scales at least 100 million light-years across (which is about a thousand times the end-to-end length of the Milky Way). If you take a box that's a hundred million light-years on each side and plunk it down here here, take another such box and plunk it down way over there there (say, a billion light-years from (say, a billion light-years from here here), and then measure the average overall properties inside each box-average number of galaxies, average amount of matter, average temperature, and so on-you'll find it difficult to distinguish between the two. In short, if you've seen one 100-million-light-year chunk of the cosmos, you've pretty much seen them all.

Such uniformity proves crucial to using the equations of general relativity to study the entire universe. To see why, think of a beautiful, uniform, smooth beach and imagine that I've asked you to describe its small-scale properties-the properties, that is, of each and every grain of sand. You're stymied-the task is just too big. But if I ask you to describe only the overall features of the beach (such as the average weight of sand per cubic meter, the average reflectivity of the beach's surface per square meter, and so on), the task becomes eminently doable. And what makes it doable is the beach's uniformity. Measure the average sand weight, temperature, and reflectivity over here and you're done. Doing the same measurements over there will give essentially identical answers. Likewise with a uniform universe. It would be a hopeless task to describe every planet, star, and galaxy. But describing the average properties of a uniform cosmos is monumentally easier-and, with the advent of general relativity, achievable.

Here's how it goes. The gross overall content of a huge volume of s.p.a.ce is characterized by how much "stuff" it contains; more precisely, the density of matter, or, more precisely still, the density of matter and energy that the volume contains. The equations of general relativity describe how this density changes over time. But without invoking the cosmological principle, these equations are hopelessly difficult to a.n.a.lyze. There are ten of them, and because each equation depends intricately on the others, they form a tight mathematical Gordian knot. Happily, Einstein found that when the equations are applied to a uniform universe, the math simplifies; the ten equations become redundant and, in effect, reduce to one. The cosmological principle cuts the Gordian knot by reducing the mathematical complexity of studying matter and energy spread throughout the cosmos to a single equation (you can see it in the notes).5 Not so happily, from Einstein's perspective, when he studied this equation he found something unexpected and, to him, unpalatable. The prevailing scientific and philosophical stance was not only that on the largest of scales the universe was uniform, but that it was also unchanging. Much like the rapid molecular motions in your tea average out to a liquid whose appearance is static, astronomical motion such as the planets...o...b..ting the sun and the sun moving around the galaxy would average out to an overall unchanging cosmos. Einstein, who adhered to this cosmic perspective, found to his dismay that it was at odds with general relativity. The math showed that the density of matter and energy cannot cannot be constant through time. Either the density grows or it diminishes, but it can't stay put. be constant through time. Either the density grows or it diminishes, but it can't stay put.

Although the mathematical a.n.a.lysis behind this conclusion is sophisticated, the underlying physics is pedestrian. Picture a baseball's journey as it soars from home plate toward the center field fence. At first, the ball rockets upward; then it slows, reaches a high point, and finally heads back down. The ball doesn't lazily hover like a blimp because gravity, being an attractive force, acts in one direction, pulling the baseball toward earth's surface. A static situation, like a stalemate in a tug-of-war, requires equal and opposite forces that cancel. For a blimp, the upward push that counters downward gravity is provided by air pressure (since the blimp is filled with helium, which is lighter than air); for the ball in midair there is no counter-gravity force (air resistance does act against a ball in motion, but plays no role in a static situation), and so the ball can't remain at a fixed height.

Einstein found that the universe is more like the baseball than the blimp. Because there's no outward force to cancel the attractive pull of gravity, general relativity shows that the universe can't be static. Either the fabric of s.p.a.ce stretches or it contracts, but its size can't remain fixed. A volume of s.p.a.ce 100 million light-years on each side today won't be 100 million light-years on each side tomorrow. Either it will be larger, and the density of matter within it will diminish (being spread more thinly in a larger volume), or it will be smaller, and the density of matter will increase (being packed more tightly in a smaller volume).6 Einstein recoiled. According to the math of general relativity, the universe on the grandest of scales would be changing, because its very substrate-s.p.a.ce itself-would be changing. The eternal and static cosmos that Einstein expected would emerge from his equations was simply not there. He had initiated the science of cosmology, but he was deeply distressed by where the math had taken him.

Taxing Gravity.

It's often said that Einstein blinked-that he went back to his notebooks and in desperation mangled the beautiful equations of general relativity to make them compatible with a universe that was not only uniform but also unchanging. This is only partly true. Einstein did indeed modify his equations so they would support his conviction of a static cosmos, but the change was minimal and thoroughly sensible.

To get a feel for his mathematical move, think about filling out your tax forms. Interspersed among the lines on which you record numbers are others you leave blank. Mathematically, a blank line signifies that the entry is zero, but psychologically it connotes more. It means you're ignoring the line because you've determined that it's not relevant to your financial situation.

If the mathematics of general relativity were arranged like a tax form, it would have three lines. One line would describe the geometry of s.p.a.cetime-its warps and curves-the embodiment of gravity. Another would describe the distribution of matter across s.p.a.ce, the source of gravity-the cause of the warps and curves. During a decade of ardent research, Einstein had worked out the mathematical description of these two features and had thus filled in these two lines with great care. But a complete accounting of general relativity requires a third line, one that is on an absolutely equal mathematical footing with the other two but whose physical meaning is more subtle. When general relativity elevated s.p.a.ce and time into dynamic partic.i.p.ants in the unfolding of the cosmos, they s.h.i.+fted from merely providing language to delineate where and when things take place to being physical ent.i.ties with their own intrinsic properties. The third line on the general relativity tax form quantifies a particular intrinsic feature of s.p.a.cetime relevant for gravity: the amount of energy st.i.tched into the very fabric of s.p.a.ce itself the amount of energy st.i.tched into the very fabric of s.p.a.ce itself. Just as every cubic meter of water contains a certain amount of energy, summarized by the water's temperature, every cubic meter of s.p.a.ce contains a certain amount of energy, summarized by the number on the third line. In his paper announcing the general theory of relativity, Einstein didn't consider this line. Mathematically, this is tantamount to having set its value to zero, but much as with blank lines on your tax forms, he seems to have simply ignored it.

When general relativity proved incompatible with a static universe, Einstein reengaged with the mathematics, and this time he took a harder look at the third line. He realized that there was no observational or experimental justification for setting it to zero. He also realized that it embodied some remarkable physics.

If instead of zero he entered a positive number on the third line, endowing the spatial fabric with a uniform positive energy, he found (for reasons I'll explain in the next chapter) that every region of s.p.a.ce would push away from every other, producing something most physicists had thought impossible: repulsive repulsive gravity. Moreover, Einstein found that if he precisely adjusted the size of the number he put on the third line, the repulsive gravitational force produced across the cosmos would exactly balance the usual attractive gravitational force generated by the matter inhabiting s.p.a.ce, giving rise to a static universe. Like a hovering blimp that neither rises nor falls, the universe would be unchanging. gravity. Moreover, Einstein found that if he precisely adjusted the size of the number he put on the third line, the repulsive gravitational force produced across the cosmos would exactly balance the usual attractive gravitational force generated by the matter inhabiting s.p.a.ce, giving rise to a static universe. Like a hovering blimp that neither rises nor falls, the universe would be unchanging.

Einstein called the entry on the third line the cosmological member cosmological member or the or the cosmological constant; cosmological constant; with it in place, he could rest easy. Or, he could rest easier. If the universe had a cosmological constant of the right size-that is, if s.p.a.ce were endowed with the right amount of intrinsic energy-his theory of gravity fell in line with the prevailing belief that the universe on the largest of scales was unchanging. He couldn't explain why s.p.a.ce would embody just the right amount of energy to ensure this balancing act, but at least he'd shown that general relativity, augmented with a cosmological constant of the right value, gave rise to the kind of cosmos he and others had expected. with it in place, he could rest easy. Or, he could rest easier. If the universe had a cosmological constant of the right size-that is, if s.p.a.ce were endowed with the right amount of intrinsic energy-his theory of gravity fell in line with the prevailing belief that the universe on the largest of scales was unchanging. He couldn't explain why s.p.a.ce would embody just the right amount of energy to ensure this balancing act, but at least he'd shown that general relativity, augmented with a cosmological constant of the right value, gave rise to the kind of cosmos he and others had expected.7 The Primeval Atom.

It was against this backdrop that Lemaitre approached Einstein at the 1927 Solvay Conference in Brussels, with his result that general relativity gave rise to a new cosmological paradigm in which s.p.a.ce would expand. Having already wrestled with the mathematics to ensure a static universe, and having already dismissed Friedmann's similar claims, Einstein had little patience for once again considering an expanding cosmos. He thus faulted Lemaitre for blindly following the mathematics and practicing the "abominable physics" of accepting an obviously absurd conclusion.

A rebuke by a revered figure is no small setback, but for Lemaitre it was short-lived. In 1929, using what was then the world's largest telescope at the Mount Wilson Observatory, the American astronomer Edwin Hubble gathered convincing evidence that the distant galaxies were all rus.h.i.+ng away from the Milky Way. The remote photons that Hubble examined had traveled to earth with a clear message: The universe is not static. It is is expanding. Einstein's reason for introducing the cosmological constant was thus unfounded. The big bang model describing a cosmos that began enormously compressed and has been expanding ever since became widely heralded as the scientific story of creation. expanding. Einstein's reason for introducing the cosmological constant was thus unfounded. The big bang model describing a cosmos that began enormously compressed and has been expanding ever since became widely heralded as the scientific story of creation.8 Lemaitre and Friedmann were vindicated. Friedmann received credit for being the first to explore the expanding universe solutions, and Lemaitre for independently developing them into robust cosmological scenarios. Their work was duly lauded as a triumph of mathematical insight into the workings of the cosmos. Einstein, by contrast, was left wis.h.i.+ng he'd never meddled with the third line of the general relativity tax form. Had he not brought to bear his unjustified conviction that the universe is static, he wouldn't have introduced the cosmological constant and so might have predicted cosmic expansion more than a decade before it was observed.

Nevertheless, the cosmological constant's story was far from over.

The Models and the Data.

The big bang model of cosmology includes a detail that will prove essential. The model provides not one but a handful of different cosmological scenarios; all of them involve an expanding universe, but they differ with respect to the overall shape of s.p.a.ce-and, in particular, they differ on the question of whether the full extent of s.p.a.ce is finite or infinite. Since the finite-versus-infinite distinction will turn out to be vital in thinking about parallel worlds, I'll lay out the possibilities.

The cosmological principle-the a.s.sumed h.o.m.ogeneity of the cosmos-constrains the geometry of s.p.a.ce because most shapes are not sufficiently uniform to qualify: they bulge here, flatten out there, or twist way over there. But the cosmological principle does not imply a unique unique shape for our three dimensions of s.p.a.ce; instead, it reduces the possibilities to a sharply culled collection of candidates. To visualize them presents a challenge even for professionals, but it is a helpful fact that the situation in shape for our three dimensions of s.p.a.ce; instead, it reduces the possibilities to a sharply culled collection of candidates. To visualize them presents a challenge even for professionals, but it is a helpful fact that the situation in two two dimensions provides a mathematically precise a.n.a.log that we can readily picture. dimensions provides a mathematically precise a.n.a.log that we can readily picture.

To this end, first consider a perfectly round cue ball. Its surface is two-dimensional (just as on earth's surface, you can denote positions on the cue ball's surface with two pieces of data-such as lat.i.tude and longitude-which is what we mean when we call a shape two-dimensional) and is completely uniform in the sense that every location looks like every other. Mathematicians call the cue ball's surface a two-dimensional sphere two-dimensional sphere and say that it has and say that it has constant positive curvature constant positive curvature. Loosely speaking, "positive" means that were you to view your reflection on a spherical mirror it would bloat outward, while "constant" means that regardless of where on the sphere your reflection is, the distortion appears the same.

Next, picture a perfectly smooth tabletop. As with the cue ball, the tabletop's surface is uniform. Or nearly so. Were you an ant walking on the tabletop, the view from every point would indeed look like the view from every other, but only if you stayed far from the table's edge. Even so, complete uniformity is not hard to restore. We just need to imagine a tabletop with no edges, and there are two ways of doing so. Think of a tabletop that extends indefinitely left and right as well as back and forth. This is unusual-it's an infinitely large surface-but it realizes the goal of having no edges since there's now no place to fall off. Alternatively, imagine a tabletop that mimics an early video-game screen. When Ms. Pac-Man crosses the left edge, she reappears at the right; when she crosses the bottom edge she reappears at the top. No ordinary tabletop has this property, but this is a perfectly sensible geometrical s.p.a.ce called a two-dimensional torus torus. I discuss this shape more fully in the notes,9 but the only features in need of emphasis here are that, like the infinite tabletop, the video-game screen shape is uniform and it has no edges. The apparent boundaries confronting Ms. Pac-Man are fict.i.tious; she can cross through them and remain in the game. but the only features in need of emphasis here are that, like the infinite tabletop, the video-game screen shape is uniform and it has no edges. The apparent boundaries confronting Ms. Pac-Man are fict.i.tious; she can cross through them and remain in the game.

Mathematicians say that the infinite tabletop and the video-game screen are shapes that have constant zero curvature constant zero curvature. "Zero" means that were you to examine your reflection on a mirrored tabletop or video-game screen, the image wouldn't suffer any distortion, and as before, "constant" means that regardless of where you examine your reflection, the image looks the same. The difference between the two shapes becomes apparent only from a global perspective. If you took a journey on an infinite tabletop and maintained a constant heading, you'd never return home; on a video-game screen, you could cycle around the entire shape and find yourself back at the point of departure, even though you never turned the steering wheel.

Finally-and this is a little more difficult to picture-a Pringles potato chip, if extended indefinitely, provides another completely uniform shape, one that mathematicians say has constant negative curvature constant negative curvature. This means that if you view your reflection at any spot on a mirrored Pringles chip, the image will appear shrunken inward.

Fortunately, these descriptions of two-dimensional uniform shapes extend effortlessly to our real interest in the three-dimensional s.p.a.ce of the cosmos. Positive, negative, and zero curvatures-uniform bloating outward, shrinking inward, and no distortion at all-equally well characterize uniform three-dimensional shapes. In fact, we are doubly fortunate because although three-dimensional shapes are hard to picture (when envisioning shapes, our minds invariably place them within an environment-an airplane in in s.p.a.ce, a planet s.p.a.ce, a planet in in s.p.a.ce-but when it comes to s.p.a.ce itself, there isn't an outside environment to contain it); the uniform three-dimensional shapes are such tight mathematical a.n.a.logs of their two-dimensional cousins that you lose little precision by doing what most physicists do: use the two-dimensional examples for your mental imagery. s.p.a.ce-but when it comes to s.p.a.ce itself, there isn't an outside environment to contain it); the uniform three-dimensional shapes are such tight mathematical a.n.a.logs of their two-dimensional cousins that you lose little precision by doi