The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)


(or Everything You Always Wanted to Know About the End of the World but Were Afraid to Ask)

A man walks into a laboratory where he is greeted by two physicists, a senior scientist and her younger male protégé, who show him a roomful of experimental apparatus—a stainless steel vacuum chamber, insulated tanks filled with cryogenic nitrogen and helium, a computer, various digital meters, oscilloscopes, and the like. The man is handed the controls to the machinery and told that the fate of the experiment—and, perhaps, the fate of the universe—lies in his hands. If the younger scientist is correct, the device will successfully extract energy from the quantum vacuum, providing humankind with unlimited bounty—“the energy of creation at our fingertips,” as it’s described. But if he’s wrong, the elder researcher cautions, the device could trigger a phase transition whereby the vacuum of empty space decays to a lower energy state, releasing all of its energy at once. “It would be the end, not only of the earth, but of the universe as we know it,” she says. The man anxiously grips the switch, as sweat from his palms spreads across the device. Seconds remain until the moment of truth. “You’d better decide fast,” he’s told.

Although this is science fiction—a scene from the short story “Vacuum States,” by Geoffrey Landis—the possibility of vacuum decay is not total fantasy. 1 The issue has, in fact, been explored for decades in journals more scholarly than Asimov’s Science FictionNaturePhysical Review LettersNuclear Physics B, etc.—by noted researchers like Sidney Coleman, Martin Rees, Michael Turner, and Frank Wilczek. Many physicists today, and perhaps the majority of those who think about such things, believe the vacuum state of our universe—empty space devoid of all matter save for the particles that spring in and out of existence by virtue of quantum fluctuations—is metastable rather than permanently stable. If these theorists are right, the vacuum will eventually decay, and the effect on the universe will be devastating (at least from our point of view), although these worrisome consequences may not happen until long after the sun has disappeared, black holes have evaporated, and protons have disintegrated.

While no one knows exactly what will happen in the long run, there seems to be agreement, at least in some quarters, that the current arrangement is not permanent—that eventually, some sort of vacuum decay is in order. The usual disclaimers apply, of course: While many researchers believe that a perfectly stable vacuum energy, or cosmological constant, is not consistent with string theory, we should never forget that string theory itself—unlike the mathematics underlying it—is in no way proven. Furthermore, I should remind readers that I’m a mathematician, not a physicist, so we’re venturing into areas that extend far beyond my expertise. The question of what may ultimately happen to the six compact dimensions of string theory is one for physicists, not mathematicians, to settle. As the demise of those six dimensions may correlate with the demise of our chunk of the universe, investigations of this sort necessarily involve treading on uncertain ground because, thankfully, we haven’t yet done the definitive experiment on our universe’s end. Nor do we have the means—outside of a fertile imagination like Landis’s—of doing so.

With that in mind, please take this discussion with a big grain of salt and, if you can, try to approach it in the spirit I’m approaching it—as a wild, whimsical ride into the realm of maybe. It’s a chance to find out what physicists think may become of the six hidden dimensions we’ve talked so much about. None of this has been proven, and we’re not even sure how it might be tested, yet it’s still an opportunity to see how these ideas might play out and to see how far informed speculation can take us.

Imagine that the man in Landis’s story pulls the lever, suddenly initiating a chain of events that result in vacuum decay. What would happen? The short answer is, nobody knows. But no matter what the outcome—whether we go the way of fire or ice, to paraphrase Robert Frost—our world would almost certainly be changed beyond recognition in the process. As Andrew Frey of McGill University and his colleagues wrote in Physical Review D in 2003, “the kind of [vacuum] decays considered in this paper in a very real sense would represent the end of the universe for anyone unfortunate enough to experience one.”2 There are two main scenarios under consideration. Both involve radical alterations of the status quo, though the first is more severe as it would spell the end of spacetime as we know it.

For starters, think back to that picture of a little ball rolling on a gently sloping surface, with each elevation point corresponding to a different vacuum energy level—a picture that we discussed in Chapter 10. For the moment, our ball is sitting in a semistable situation called a potential well, which is analogous to a small dip or hole on an otherwise hilly landscape. We’ll assume that even at the bottom of that hole, the elevation is still, so to speak, above sea level: The vacuum energy, in other words, remains positive. If this landscape were a classical one, the ball would sit there indefinitely. Its resting place, in other words, would become its final resting place. But the landscape isn’t classical. It’s quantum mechanical, and with quantum mechanics in play, a funny thing can happen: If the ball is exceedingly tiny, which is the setting in which quantum phenomena become apparent, it can literally bore through the side of the hole to reach the outside world. That’s the result of an absolutely real phenomenon known as quantum tunneling. This is possible because of the fundamental uncertainty built into quantum mechanics. According to the uncertainty principle formulated by Werner Heisenberg, location—contrary to the mantra of realtors—is not the only thing, and it’s not even an absolute thing. Although a particle is perhaps most likely to be found in one spot, there’s a chance it could be found in more improbable locales. And if there’s a chance it could happen, the theory states, it eventually will, provided we wait long enough. This principle holds, in fact, regardless of the size of the ball, although the probability of a large ball’s doing so would be even smaller.

Strange as it sounds, the real-world effects of quantum tunneling have been seen. This well-tested phenomenon provides the basis, for example, for scanning tunneling microscopes, whose operation depends on electrons making their way through seemingly impenetrable barriers. Microchip manufacturers, similarly, cannot make transistors too thin, or the performance of these devices will suffer from electron leakage due to tunneling effects.

The idea of particles like electrons tunneling through a wall—metaphorical or real—is one thing, but what about spacetime as a whole? The notion of an entire vacuum’s tunneling from one energy state to another is, admittedly, harder to swallow, yet the theory has been pretty well worked out by Coleman and others, starting in the 1970s.3 The barrier in this case is not a wall so much as a kind of energy field that is preventing the vacuum from reconstituting itself into a lower-energy, more stable, and therefore more favored state. Change in this case comes by way of a phase transition, similar to liquid water turning into ice or vapor, except that a large swath of the universe is transformed—perhaps a swath that includes our home.

This brings us to the punch line of the first scenario, in which our current vacuum state tunnels from its slightly positive energy value—a fact of life we now call dark energy or the cosmological constant—to a negative value instead. As a result, the energy now driving our universe apart at an accelerated clip would instead compress it to a point, thereby carrying us toward a cataclysmic event known as the Big Crunch. At this cosmic singularity, both the energy density and the curvature of the universe would become infinite—the same thing that, in principle, we’d encounter at the center of a black hole or if we ran the universe backward to the Big Bang.

As for what might follow the Big Crunch, all bets are off. “We don’t know what happens to spacetime, let alone what happens to the extra dimensions,”lay notes physicist Steve Giddings of the University of California, Santa Barbara.4 It’s beyond our realm of experience and grasp in almost every respect.

Quantum tunneling isn’t the only way to trigger a change in vacuum state; another is through so-called thermal fluctuations. Let’s go back to our minuscule ball in the bottom of the potential well. The higher the temperature, the faster that all atoms, molecules, and other particles in this system are moving around. And if particles are moving around, some will randomly crash into the ball, jostling it one way or another. On average, these jostles will cancel each other out and the ball will remain more or less stationary. But suppose, through some statistical fluke, multiple atoms slam into it, successively, from the same general direction. Several such jostles in a row could knock the ball clear out of the hole. It will then wind up on the hilly surface and perhaps roll all the way down to zero energy unless it gets stuck in other wells or holes on the way.

Evaporation might be an even better analogy, suggests New York University physicist Matthew Kleban. “You don’t ever see water crawling out of a cup,” he explains. “But water molecules keep getting bumped, especially when water is heated, and occasionally they’re bumped hard enough to make it out of the cup. That’s similar to what happens in this thermal process.”5

There are two important differences, however. One difference is that the processes we’re discussing here take place in a vacuum, which traditionally means no matter and hence no particles. So what’s doing the knocking? Well, for one thing, the temperature never quite gets to zero (this fact turns out to be a feature of an expanding universe), and for another, space is never quite empty, because pairs of virtual particles—a particle and its antiparticle—are continually popping into existence and then disappearing through annihilation in an interval so brief, we’ve never been able to catch them in the act. The other difference is that this process of virtual particle creation and annihilation is a quantum one, so what we’ve been calling thermal fluctuations necessarily include some quantum contributions as well.

We’re now ready to take up the second scenario, which may be more benign than the first, but only somewhat. Through quantum tunneling or perhaps a thermal or quantum fluctuation, our universe may wind up at another metastable spot (most likely at a slightly lower vacuum energy level) on the string theory landscape. But this, as with our current situation, would be merely a temporary way station, or a metastable rest stop en route to our final destination. This issue is related to how Shamit Kachru, Renata Kallosh, Andrei Linde, and Sandip Trivedi (the authors of the KKLT paper) explained string theory’s great vanishing act—providing us with a universe of only four large dimensions, rather than ten, and doing so while simultaneously incorporating the notion of inflation into string cosmology. Even though we now see only four dimensions, “in the long run, the universe doesn’t want to be four-dimensional,” claims Stanford cosmologist Andrei Linde. “It wants to be ten dimensional.”6 And if we’re really patient, it will be. Compactified dimensions are fine in the short run, but it’s not the ideal state of affairs for the universe over the long haul, according to Linde. “Where we are now is like standing on top of a building, but we haven’t jumped yet. If we don’t do it by our own will, quantum mechanics will take care of it for us, throwing us down to the lowest energy state.”7

The reason a universe with ten large dimensions is ergonomically preferred comes down to this: In the most well-developed models we have today, the energy of the vacuum is a consequence of the compactification of the extra dimensions. Put in other terms, the dark energy we’ve heard so much about isn’t just driving the cosmos apart in some kind of madcap accelerative binge: Some, if not all, of that energy goes into keeping the extra dimensions wound up tighter than the springs of a Swiss watch, although in our universe, unlike in a Rolex, this is done with fluxes and branes.

The system, in other words, has stored potential energy that is positive in value. The smaller the radius of the extra dimensions, the tighter the spring is wound and the greater the energy stored. Conversely, as the radius of those dimensions increases, the potential energy declines, reaching zero when the radius becomes infinite. That’s the lowest energy state and hence the only truly stable vacuum, the point at which the dark energy drops to zero and all ten dimensions become infinitely large. The once-small internal dimensions, in other words, become decompactified.

Decompactification is the flip side of compactification, which as we’ve discussed, is one of the biggest challenges in string theory: If the theory depends on our universe’s having ten dimensions, how come we only see four? String theorists have been hard-pressed to explain how the theory’s extra dimensions are so well concealed, because, as Linde has noted, all other things being equal, the dimensions would rather be big. It’s like trying to hold an increasing volume of water in an artificial reservoir with fixed walls. In every direction, in every corner of the structure, the water is trying to get out. And it won’t quit trying until it does. When that happens, and the sides suddenly give way, water confined to a compact area (within the perimeter of the reservoir) will burst out and spread over an extended surface. Based on our current understanding of string theory, the same sort of thing will happen to the compact dimensions, be they curled up in Calabi-Yau spaces or in some other more complicated geometries. No matter what configuration is selected for the internal dimensions, they will eventually unwind and open up.

Of course, one might ask why, if it’s so advantageous from an energy standpoint for the dimensions to expand, this hasn’t happened already. One solution that physicists have proposed, as discussed in the last chapter, involves branes and fluxes. Suppose, for instance, you have a badly overinflated bicycle inner tube. Any weak spot in the tire would give rise to a bubble that would eventually burst. We could shore up a weak spot by applying a patch, which is somewhat like a brane, or bind up the whole tire with rubber bands to help it maintain its shape, as we think fluxes do with the Calabi-Yau spaces. So the idea is that we’ve got two opposing forces here—a natural tendency for an overinflated shape to expand that’s kept in check by branes, fluxes, and other structures that wrap around the object and hold it in. The net result is that these countervailing forces are now perfectly balanced, having achieved some kind of equilibrium.

It’s an uneasy peace, however. If we push the radii of the extra dimensions to larger values through modest quantum fluctuations, the branes and fluxes provide a restorative force, quickly bringing the radii back to where they started. But if you stretch the dimensions too far, the branes or fluxes can snap. As Giddings explains, “eventually a rare fluctuation will take you out to the threshold radius for decompactification and”—keeping in mind the slope on the right side of Figure 11.1—“it’s all downhill from there.”8 We’re off on the merry road to infinity.

Figure 11.2 tells a similar story with a nuance thrown in. Instead of tunneling out of our present situation straight to a universe with ten large dimensions, there will be an intermediary stop—and perhaps a series of them—in the landscape along the way. But in either case, whether you fly nonstop or make connections in Dallas or Chicago, the endpoint will be the same. And inevitable.

The landing, however, is not likely to be gentle. Remember that change, when it comes, is actually a phase transition of the vacuum, rather than a ball climbing out of a hole or burrowing through a wall. The change will start small, as a tiny bubble, and grow at an exponential clip. Inside that bubble, the compactification—which had been keeping six of the dimensions almost Planck-scale small—will start to undo itself. As the bubble spreads, a spacetime of four large dimensions and six tiny curled ones will become desegregated, in a sense. Where the dimensions had once been partitioned into compact and extended form, there will now be ten large dimensions all thrown together, with no barriers keeping them apart.


11.1—One theory holds that our universe sits in the little dip in the left-hand side of the graph, which locks the vacuum’s potential energy (V) to a specific level as well as fixing the radius (R) of the compact extra dimensions. However, that arrangement may not be permanent. A little nudge could push us over the hill on the right—or we might just quantum-tunnel clear through the barrier—which would send us down the sloping curve toward infinitely large extra dimensions. The process, whereby previously tiny dimensions unwrap to become big, is called decompactification. (Adapted, with permission, from a figure by Steve Giddings)

“We’re talking about a bubble that expands at the speed of light,” Shamit Kachru notes. “It starts in a certain location of spacetime, kind of like the way bubbles nucleate in water. What’s different is that this bubble doesn’t just rise and leave. Instead, it expands and removes all the water.”9

But how can a bubble move so fast? One reason is that the decompactified state inside the bubble lies at a lower potential energy than exists outside the bubble. Because systems naturally move in the direction of lower energy—which in this case also happens to be the direction of increased dimension size—the resultant gradient in potential energy creates a force on the edge of the bubble, causing it to accelerate outward. The acceleration, moreover, is both sustained and high, which drives the bubble to the speed of light within a tiny fraction of a second.


11.2—The story is pretty much the same as in Figure 11.1. Our universe is still headed toward decompactification and the realm of infinitely large extra dimensions, only this time, we’re going to make an additional stop in the “landscape” along the way. In this scenario, our universe can be thought of as a marble that temporarily gets stuck in a trough (A) as it rolls down the hill. In principle, the marble could make many intermediary stops as it continues its descent, even though only one additional trough (B) is shown on this graph. (Adapted, with permission, from a figure by Steve Giddings)

Linde describes the phenomenon in more colorful terms. “The bubble wants to go as fast as possible because if you have the possibility of leading a great life in a lower vacuum energy, why would you want to wait?” he asks. “So the bubble moves faster and faster, but it cannot move faster than the speed of light.”10 Though, considering his description of the rewards ahead, it probably would if it could.

Because the bubble spreads outward at light speed, we’d never know what hit us. The only advance warning we’d get would be the shock wave that would arrive a fraction of a second earlier. The bubble would then smack into us head-on, carrying a lot of kinetic energy in its wall. That’s just the first round of a double whammy. Because the bubble wall has some thickness, it would take a bit of time—albeit a mere fraction of a second—for the worst to come. The place we call home has four-dimensional laws of physics, whereas the bubble’s interior obeys ten-dimensional laws. And those ten-dimensional laws will take over just as soon as the inside of the bubble infiltrates our world. As the playwright/ screenwriter David Mamet once put it: “Things change.”

In fact, everything you can imagine, from the tiniest particle to elaborate structures like galactic superclusters, would instantly explode into the six expanding dimensions. Planets and people would revert to their constituent parts, and those parts would be obliterated as well. Particles like quarks, electrons, and photons would cease to exist altogether, or they would reemerge with completely different masses and properties. While spacetime would still be there, albeit in an altered state, the laws of physics would change radically.

How long might we have before such an “explosion” in dimensions occurs? We’re pretty sure the vacuum of our present universe has been stable ever since inflation ended approximately 13.7 billion years ago, notes Henry Tye of Cornell. “But if the expected shelf life is just fifteen billion years, that only gives us a billion or so years left.”11 Or just enough time to start packing.

But all signs suggest there’s no need to hit the panic button just yet. It could take an extraordinarily long time—on the order of e(10120) years—for our spacetime to decay. That number is so big it’s almost hard to fathom, even for a mathematician. We’re talking about e—one of the fundamental constants in nature, a number that’s approximately 2.718—multiplied by itself 10120 (a one with 120 zeros after it) times. And if that rough guess is correct, our waiting period is, for all practical purposes, infinitely long.

So how does one come up with a number like e(10120), anyway? The initial premise is that our universe is evolving into something called de Sitter space—a space dominated by a positive cosmological constant in which all matter and radiation eventually become so dilute as to be insignificant, if not absent altogether. (Such a space was first proposed in 1917 by the Dutch astrophysicist Willem de Sitter as a vacuum solution to the Einstein field equations.) If our universe, with its small cosmological constant, is in fact “de Sitter,” then the entropy of such a space is prodigious, on the order of 10120 (more on where that number comes from in a minute). This kind of universe has a large entropy because its volume is so great. Just as there are more places you can put an electron in a big box than in a small box, a big universe has more possible states—and hence a higher entropy—than a small one.

De Sitter space has a horizon in the same way that a black hole has an event horizon. If you get too close to a black hole and cross the fateful line, you’ll be sucked in and won’t be coming home for supper. The same is true for light, which can’t escape, either. And the same thing holds for a de Sitter horizon, as well. If you go too far in a space that’s undergoing accelerated expansion, you’ll never make it back to the neighborhood you started in. And light, as in the case of the black hole, won’t make it back, either.

When the cosmological constant is small, and accelerated expansion relatively slow (which happens to be our present circumstance), the horizon is far away. That’s why the volume of such a space is big. Conversely, if the cosmological constant is large, and the universe is racing apart at breakneck speed, the horizon—or point of no return—may be close at hand (quite literally) and the volume correspondingly small. “If you extend your arm too far in such a space,” Linde notes, “the rapid expansion might tear your hand away from you.”12

Although the entropy of de Sitter space is related to the volume, it’s even more closely correlated with the surface area of the horizon, which scales with the distance to the horizon squared. (We can actually apply the same reasoning—and the same Bekenstein-Hawking formula—that we applied to black holes in Chapter 8, with the entropy of de Sitter space being proportional to the horizon area divided by four times Newton’s gravitational constant G.) The distance to the horizon or, technically, the distance squared, in turn, depends on the cosmological constant: The greater the constant’s value, the smaller the distance. Since the entropy scales with the distance squared, and the distance squared is inversely proportional to the cosmological constant, the entropy is also inversely proportional to the cosmological constant. The upper limit for the cosmological constant in our universe, according to Hawking, is 10-120 in the “dimensionless units” that physicists use.13 (This number, 10-120, is a rough approximation, however, and should not be taken as an exact figure.) The entropy, being the inverse or reciprocal of that, is therefore extremely large—roughly on the order of 10120, as noted a moment ago.

The entropy, by definition, is not the number of states per se, but rather the log—or, to be precise, the natural log—of the number of states. So the number of states is, in fact, eentropy. Going back to our graph in Figure 11.1, the number of possible states in our universe with a small cosmological constant, which is represented by the dip or minimum in the curve, is 075. Let’s suppose, on the other hand, that the summit of the mountain, from which one would roll down toward dimensions of infinite radius, is such an exclusive place that there is just one state that puts you exactly on top. Therefore, the odds of landing at that particular spot, among all the other possibilities, are vanishingly small—roughly just 1/e(10120). And that’s why, conversely, the amount of time it will take to tunnel through the barrier is so mind-numbingly large we can’t even call it astronomical.

One other point: In Figure 11.2, we presented a decompactification scenario in which our universe tunnels to a state of lower vacuum energy (and smaller cosmological constant), making a stopover in the landscape on its journey to the ultimate makeover, infinite dimensionality. But might we take a detour, tunneling up to a spot with higher vacuum energy instead? Yes, but it’s much easier, and more probable, to go downhill. One can, however, make a somewhat more involved argument. Suppose there’s a potential minimum at point A and a separate minimum at point B, with A being higher than B in elevation and hence in vacuum energy. Since A sits at higher energy, its gravity will be stronger, which means the space around it will be more strongly curved. And if we think of that space as a sphere, its radius will be smaller because smaller spheres bend more sharply than larger ones and thus have greater curvature. Since B sits at lower energy, its gravity will be weaker. Consequently, the space around it will have less curvature. If we think of that space as a sphere, it will have a larger radius and hence be less curved.

We’ve illustrated some aspects of this idea in Figure 11.3 (using boxes for A and B, rather than spheres) to show that it’s more likely to travel “downhill” on the landscape toward lower energy—from A to B, in other words—than it is to go uphill. To see why, we can connect the two boxes with a thin tube. The two boxes will come into equilibrium, with the same concentration, or density, of gases or molecules filling both boxes and the same number of molecules migrating from A to B as there are going from B to A. However, since B is much bigger than A, it will have many more molecules to begin with. So the odds of any individual molecule in A migrating to B is much greater than the odds of any individual molecule in B making the reverse trip.


11.3—As discussed in the text, this figure tries to show why it’s easier to “tunnel down” from A to B (in Figure 11.2) rather than to “tunnel up” from B to A. The analogy presented here is that it is more likely, on average, for a given molecule to make the trip from A to B than the other way around, simply because there are far fewer molecules in A than in B.

Similarly, the probability of the appearance of a bubble that will transport you to a lower-energy spot in the landscape is substantially greater than that of the appearance of a bubble taking you the other way (uphill), just as any given molecule is more likely to make the trip from box A to B (“downhill,” in other words).

In 1890, Henri Poincaré published his so-called recurrence theorem, which states that any system with a fixed volume and energy that can be described by statistical mechanics has a characteristic recurrence time equal to eentropyof that system. The idea here is that a system like this has a finite number of states—a finite number of particle positions and velocities. If you start in a particular state and wait long enough, you’ll eventually access all of them, just as a particle or molecule in our proverbial box will wander around, bounce off walls, and move in a random way, over time landing at every possible location in the box. (If we were to put it in more technical terms, instead of talking about possible locations in a box, we’d talk about possible states in “phase space.”) The time it will take for spacetime to decompactify, then, is the Poincaré recurrence time— eentropy , or 077 years. But there’s one possible weak point in this argument, Kleban points out: “We don’t yet have a statistical mechanical description of de Sitter space.” The underlying assumption, which may or may not be borne out, is that such a description exists.14

There’s not much more to be said about the subject today, nor that much more to be done, other than perhaps to refine our calculations, redo the numbers, and recheck our logic. It’s not surprising that few investigators are strongly inclined to carry this much further, since we’re talking about highly speculative events in model-dependent scenarios that are not readily testable and are expected to happen on a timescale just shy of forever. That’s hardly the ideal prescription for getting grant money or, for younger researchers, gaining the admiration of their elders and, more importantly, securing tenure.

Giddings, who has given the subject more attention than most people have, is not letting the doomsday aspects of this picture drag him down. “On the positive side,” he writes in his paper “The Fate of Four Dimensions,” “the decay can result in a state that does not suffer the ultimate fate of infinite dilution,” such as would occur in an eternally expanding universe endowed with a positive cosmological constant that is truly constant. “We can seek solace both in the relatively long life of our present four-dimensional universe, and in the prospect that its decay produces a state capable of sustaining interesting structures, perhaps even life, albeit of a character very different from our own.”15

Like Giddings, I too am not losing any sleep over the fate of our four dimensions, six dimensions, or even ten dimensions. Inquiries into this area, as I mentioned before, are thought-provoking and entertaining, but they’re also wildly conjectural. Until we obtain some observational data that tests the theory, or at least come up with practical strategies for verifying these scenarios, I’ll have to regard them as closer to science fiction than to science. Before we spend too much time worrying about decompactification, however, we first ought to think about ways of confirming the existence of the extra dimensions themselves. Such a success, to my mind, would be more than enough to outweigh the potential downside of the various decay scenarios, which might eventually bring our universe to a bad end—not that any of the other ends, when you get right down to it, look particularly good, either.

As I see it, the unfurling of the hidden dimensions could be the greatest visual display ever witnessed—if it could be witnessed, though that seems extremely doubtful. Allow me one more flight of fancy here, and suppose that this scenario is eventually realized and the great unraveling of spacetime does occur at some point in the distant future. If that ever came to pass, it would be a spectacular confirmation (albeit belated) of the idea to which I’ve devoted the better part of my career. It’s a pity that when the universe’s best hiding places are finally exposed, and the cosmos opens up to its full multidimensional glory, no one will be around to appreciate it. And even if someone did survive the great transformation, there’d be no photons around to enable them to take in the view. Nor would there be anyone left with whom they could celebrate the success of a theory dreamed up by creatures who called themselves humans in an era known as the twentieth century, though it might be more properly referred to as the 137 millionth century (Big Bang Standard Time).

The prospect is especially dismaying to someone like me, who has spent decades trying to get a fix on the geometry of the six internal dimensions and, harder still, trying to explain it to people who find the whole notion abstruse, if not absurd. For at that moment in the universe’s history—the moment of the great cosmic unwinding—the extra dimensions that are now concealed so well would no longer be a mathematical abstraction, nor would they be “extra” anymore. Instead, they’d be a manifest part of a new order in which all ten dimensions were on equal footing, and you’d never know which ones were once small and which ones were large. Nor would you care. With ten spacetime dimensions to play with, and six new directions in which to roam, life would have possibilities we can’t even fathom.