The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos - Brian Greene (2011)
Chapter 5. Hovering Universes in Nearby Dimensions
The Brane and Cyclic Multiverses
Late one night many years ago, I was in my office at Cornell University putting together the freshman physics final exam that would be given the following morning. Since this was the honors class, I wanted to enliven things a little by giving them one somewhat more challenging problem. But it was late and I was hungry, so rather than carefully working through various possibilities, I quickly modified a standard problem that most of them had already encountered, wrote it into the exam, and headed home. (The details hardly matter, but the problem had to do with predicting the motion of a ladder, leaning against a wall, as it loses its footing and falls. I modified the standard problem by having the density of the ladder vary along its length.) During the exam the next morning, I sat down to write the solutions, only to find that my seemingly modest modification to the problem had made it exceedingly difficult. The original problem took perhaps half a page to complete. This one took me six pages. I write big. But you get the point.
This little episode represents the rule rather than the exception. Textbook problems are very special, being carefully designed so that they’re completely solvable with reasonable effort. But modify textbook problems just a bit, changing this assumption or dropping that simplification, and they can quickly become intricate or intractable. That is, they can quickly become as difficult as analyzing typical real-world situations.
The fact is, the vast majority of phenomena, from the motion of planets to the interactions of particles, are just too complex to be described mathematically with complete precision. Instead, the task of the theoretical physicist is to figure out which complications in a given context can be discarded, yielding a manageable mathematical formulation that still captures essential details. In predicting the course of the earth you’d better include the effects of the sun’s gravity; if you include the moon’s too, all the better, but the mathematical complexity rises significantly. (In the nineteenth century, the French mathematician Charles-Eugène Delaunay published two 900-page volumes related to intricacies of the sun-earth-moon gravitational dance.) If you try to go further and account fully for the influence of all the other planets, the analysis becomes overwhelming. Luckily, for many applications, you can safely disregard all but the sun’s influence, since the effect of other bodies in the solar system on earth’s motion is nominal. Such approximations illustrate my earlier assertion that the art of physics lies in deciding what to ignore.
But as practicing physicists know well, approximation is not just a potent means for progress; on occasion it also brings peril. Complications of minimal importance for answering one question can sometimes have a surprisingly significant impact in answering another. A single drop of rain will hardly affect the weight of a boulder. But if the boulder is teetering high on a cliff’s edge, that drop of rain could very well coax it to fall, initiating an avalanche. An approximation that disregards the raindrop would miss a crucial detail.
In the mid-1990s, string theorists discovered something akin to a raindrop. They found that various mathematical approximations, widely used to analyze string theory, were overlooking some vital physics. As more precise mathematical methods were developed and applied, string theorists could finally step beyond the approximations; when they did, numerous unanticipated features of the theory came into focus. And among these were new types of parallel universes; one variety in particular may be the most experimentally accessible of all.
Every major established discipline of theoretical physics—such as classical mechanics, electromagnetism, quantum mechanics, and general relativity—is defined by a central equation, or set of equations. (You don’t need to know these equations, but I’ve listed some of them in the notes.)1 The challenge is that in all but the simplest situations, the equations are extraordinarily difficult to solve. For this reason, physicists routinely use simplifications—like ignoring Pluto’s gravity or treating the sun as perfectly round—that make the mathematics easier and bring approximate solutions within reach.
For a long time, research in string theory has faced even bigger challenges. Just finding the central equations proved so difficult that physicists could develop only approximate versions. And even the approximate equations were so intricate that physicists had to make simplifying assumptions to find solutions, thus basing research on approximations of approximations. During the 1990s, however, the situation vastly improved. In a series of advances, a number of string theorists showed how to go well beyond the approximations, offering unmatched clarity and insight.
To get a feel for these breakthroughs, imagine that Ralph is planning to play the next two rounds of the weekly worldwide lottery, and he’s proudly worked out the odds of winning. He tells Alice that since he has a 1 in a billion chance each week, if he plays both rounds his chance of winning is 2 in a billion, .000000002. Alice smirks. “Well, that’s close, Ralph.” “Really, wise guy. What do you mean close?” “Well,” she says, “you’ve overestimated. Should you win the first round, playing a second time won’t increase your chances of winning; you would already have done so. If you win twice, we’ll have more money, sure, but since you’re working out the odds of winning at all, winning the second lottery after the first just doesn’t matter. So, to get the precise answer you’d need to subtract the odds of winning both rounds—1 in a billion times 1 in a billion, or .000000000000000001. That yields a final tally of .000000001999999999. Questions, Ralph?”
Minus the smugness, Alice’s method is an example of what physicists call a perturbative approach. In doing a calculation, it’s often easiest to make a first pass that incorporates only the most obvious contributions—that’s Ralph’s starting point—and then make a second pass that includes finer details, modifying or “perturbing” the first-pass answer, as in Alice’s contribution. The approach easily generalizes. If Ralph were planning to play the next ten weekly lotteries, the first-pass approach suggests that his chance of winning is about 10 in a billion, .00000001. But, as in the previous example, this approximation fails to account correctly for multiple wins. When Alice takes over, her second pass would properly account for instances in which Ralph wins twice—say, on the first and second lotteries, or the first and third, or the second and fourth. These corrections, as Alice pointed out above, are proportional to 1 in a billion times 1 in a billion. But there’s also an even tinier chance that Ralph wins three times; Alice’s third pass takes that, too, into account, producing modifications proportional to 1 in a billion multiplied by itself three times, .000000000000000000000000001. The fourth pass does the same for the even tinier chance of winning four rounds, and so on. Each new contribution is far smaller than the previous, so at some point Alice deems the answer sufficiently accurate and calls it a day.
Calculations in physics, and in many other branches of science too, often proceed in an analogous fashion. If you are interested in how likely it is that two particles heading in opposite directions around the Large Hadron Collider will bang into each other, the first pass imagines they hit once and ricochet (where “hit” doesn’t mean they directly touch, but rather that a single force-carrying “bullet,” such as a photon, flies from one and is absorbed by the other). The second pass takes into account the chance that the particles hit each other twice (two photons are fired between them); the third pass modifies the previous two by accounting for the chance of the particles hitting each other three times; and so on (Figure 5.1). As with the lottery, this perturbative approach works well if the chance of an ever-greater number of particle interactions—like the chance of an ever-greater number of lottery wins—drops precipitously.
For the lottery, the drop-off is determined by each successive win coming with a factor of 1 in a billion; in the physics example, it’s determined by each successive hit coming with a numerical factor, called a coupling constant, whose value captures the likelihood that one particle will fire a force-carrying bullet and that the second particle will receive it. For particles such as electrons, governed by the electromagnetic force, experimental measurements have determined that the coupling constant, associated with photon bullets, is about .0073.2 For neutrinos, governed by the weak nuclear force, the coupling constant is about 10–6. For quarks, components of protons, that are racing around the Large Hadron Collider and whose interactions are governed by the strong nuclear force, the coupling constant is somewhat less than 1. These numbers are not as small as the lottery’s .000000001, but if for example we multiply .0073 by itself the result quickly becomes minuscule. After one iteration it’s about .0000533, after two it’s about .000000389. This explains why theorists only rarely go to the trouble of accounting for electrons hitting each other numerous times. The calculations involving many hits are exceedingly intricate to carry out, and the resulting contributions are so terribly tiny that you can stop at just a few photons fired and still get an extraordinarily accurate answer.
Figure 5.1 Two particles (represented by the two solid lines on the left in each diagram) interact by firing various “bullets” at each other (the “bullets” are force-carrying particles, represented by the squiggly lines), and then ricochet forward (the two solid lines on the right). Each diagram contributes to the overall likelihood that the particles bounce off each other. The contributions of processes with ever-more bullets are ever smaller.
To be sure, physicists would love to have exact results. But for many calculations the mathematics proves too difficult, so the perturbative approach is the best we can do. Fortunately, for small enough coupling constants, the approximate calculations can yield predictions that agree extremely well with experiment.
A similar perturbative approach has long been a mainstay of string theory research. The theory contains a number, called the string coupling constant (string coupling, for short), that governs the chance that one string bumps off another. If the theory proves correct, the string coupling may one day also be measured, much like the couplings enumerated above. But since such a measurement is at present purely hypothetical, the value of the string coupling is a complete unknown. Over the past few decades, with no guidance from experiment, string theorists have made the key assumption that the string coupling is a small number. To some extent, this has been like the drunkard looking for his keys under a lamppost, because a small string coupling allows physicists to shine the bright lights of perturbative analysis on their calculations. Since many successful approaches prior to string theory do have a small coupling, a more favorable version of the analogy notes that the drunkard has been justifiably emboldened by frequently finding his keys in the very location that’s illuminated. Either way, the assumption has made possible a vast collection of mathematical calculations that have not only clarified the basic processes of how one string interacts with another, but have also revealed much about the fundamental equations underlying the subject.
If the string coupling is small, these approximate calculations are expected to accurately reflect the physics of string theory. But what if it isn’t? Unlike what we found with the lottery and with colliding electrons, a large string coupling would mean that successive refinements to first-pass approximations would yield ever-larger contributions, so you’d never be justified in stopping a calculation. The thousands of calculations that have used the perturbative scheme would be baseless; years of research would collapse. Adding to the concerns, even with a small yet moderate string coupling, you might also worry that your approximations, at least in some circumstances, were overlooking subtle yet vital physical phenomena, like the raindrop that hits the boulder.
Through the early 1990s, not much could be said about these vexing questions. By the second half of that decade, the silence gave way to a clamor of insight. Researchers found new mathematical methods that could outflank the perturbative approximations by leveraging something called duality.
In the 1980s, theorists realized that there was not one string theory but rather five different versions, to which they gave the catchy names Type I, Type IIA, Type IIB, Heterotic-O, and Heterotic-E. I’ve not yet mentioned this complication because although calculations established that the theories differ in detail, all five include the same gross features—vibrating strings and extra spatial dimensions—on which we’ve so far focused. But we’re now at a point where the five variations on the string theory theme come to the fore.
For many years, physicists had relied on perturbative methods to analyze each of the string theories. When working with the Type I string theory, they assumed its coupling was small, and pressed on with multi-pass calculations similar to what Ralph and Alice did in the lottery analysis. When working with the Heterotic-O, or any of the other string theories, they did the same. But outside of this restricted domain of small string couplings, researchers could do nothing more than shrug, throw up their hands, and admit that the math they were using was too feeble to provide any reliable insight.
Until, that is, the spring of 1995, when Edward Witten rocked the string theory community with a series of stunning results. Drawing on the insights of scientists including Joe Polchinski, Michael Duff, Paul Townsend, Chris Hull, John Schwarz, Ashoke Sen, and many others, Witten provided strong evidence that string theorists could safely navigate beyond the shores of small couplings. The central idea was simple and powerful. Witten argued that when the coupling constant in any one formulation of string theory is dialed ever larger, the theory—remarkably—steadily morphs into something thoroughly familiar: one of the other formulations of string theory, but with a coupling constant that’s dialed ever smaller. For example, when the Type I string coupling is large, it transforms into the Heterotic-O string theory with a coupling that’s small. Which means that the five string theories are not different after all. Each appears different when examined in a limited context—small values of its particular coupling constant—but when this restriction is lifted, each string theory transforms into the others.
I recently encountered a splendid graphic that from close up looks like Albert Einstein, with a bit more distance becomes ambiguous, and from far away resolves into Marilyn Monroe (Figure 5.2). If you saw only the images that come into focus at the two extremes, you’d have every reason to think you were looking at two separate pictures. But if you steadily examine the image through the range of intermediate distances, you unexpectedly find that Einstein and Monroe are aspects of a single portrait. Similarly, an examination of two string theories, in the extreme case when each has a small coupling, reveals that they’re as different as Albert and Marilyn. If you stopped there, as for years string theorists did, you’d conclude that you were studying two separate theories. But if you examine the theories as their couplings are varied over the range of intermediate values, you find that, like Albert turning into Marilyn, each gradually morphs into the other.
The morphing from Einstein to Monroe is amusing. The morphing of one string theory into another is transformative. It implies that if perturbative calculations in one string theory can’t be undertaken because that theory’s coupling is too large, the calculations can be faithfully translated into the language of another formulation of string theory, one in which a perturbative approach succeeds because the coupling is small. Physicists call the transition between naïvely distinct theories duality. It has become one of the most pervasive themes in modern string theory research. By providing two mathematical descriptions of one and the same physics, duality doubles our calculational arsenal. Calculations that are impossibly difficult from one perspective become perfectly doable from another.*
Figure 5.2 From close up, the image looks like Albert Einstein. From farther away, it looks like Marilyn Monroe. (The image was created by Aude Oliva of the Massachusetts Institute of Technology.)
Witten argued, and others since have filled in important details, that all five string theories are linked through a network of such dualities.3 Their overarching union, called M-theory (we’ll see why in a moment), combines insights from all five formulations, stitched together through the various duality relationships, to gain a far more refined understanding of each. One such insight, central to the theme we’re pursuing, showed that there’s much more to string theory than strings.
When I started studying string theory, I asked the very question that many in the years since have asked me: Why are strings considered so special? Why focus solely on fundamental ingredients that have only length? After all, the theory itself requires that the arena within which its ingredients exist—the spatial universe—has nine dimensions, so why not consider entities shaped like two-dimensional sheets or three-dimensional blobs or their higher-dimensional cousins? The answer I learned as a graduate student in the 1980s, and explained frequently when I lectured on the subject through the mid-1990s, was that the mathematics describing fundamental constituents with more than one spatial dimension suffered from fatal inconsistencies (such as quantum processes that would have negative probabilities, a meaningless mathematical result). But when the same mathematics was applied to strings, the inconsistencies canceled themselves out, leaving a cogent description.†4 Strings were definitely in a class of their own.
Or so it seemed.
Armed with the newfound calculational methods, physicists started analyzing their equations much more precisely and produced a range of unexpected results. One of the most surprising established that the rationale for excluding anything but strings was rickety. Theorists realized that the mathematical problems encountered when studying higher-dimensional ingredients, such as discs and blobs, were artifacts of the approximations being used. Using the more precise methods, a small army of theorists established that ingredients with various numbers of spatial dimensions do lurk in string theory’s mathematical shadows.5The perturbative techniques were too coarse to expose these ingredients but the new methods finally could. By the late 1990s, it was abundantly clear that string theory was not just a theory that contained strings.
The analyses revealed objects, shaped like Frisbees or flying carpets, with two spatial dimensions: membranes (one meaning of the “M” of M-theory), also called two-branes. But there was more. The analyses revealed objects with three spatial dimensions, so-called three-branes; objects with four spatial dimensions, four-branes, and so on, all the way up to nine-branes. The mathematics made clear that all of these entities could vibrate and wiggle, much like strings; indeed, in this context, strings are best thought of as one-branes—a single entry on an unexpectedly long list of the theory’s basic building blocks.
An allied revelation, just as flabbergasting to those who’d spent the better part of their professional lives working on the subject, was that the number of spatial dimensions the theory requires is not actually nine. It’s ten. And if we fold in the dimension of time, the total number of spacetime dimensions is eleven. How could this be? Remember the “(D–10) times Trouble” consideration, recounted in Chapter 4, underlying the conclusion that string theory needs ten spacetime dimensions. The mathematical analysis that produced that equation was, once again, based on a perturbative approximation scheme that assumed the string coupling was small. Surprise, surprise, that approximation missed one of the theory’s spatial dimensions. The reason, Witten showed, is that the size of the string coupling directly controls the size of the hitherto unknown tenth spatial dimension. By taking the coupling small, researchers had unwittingly made this spatial dimension small, too—so small as to be invisible to the mathematics itself. The more precise methods rectified this failing, revealing a string/M-theory universe with ten dimensions of space and one of time, for a total of eleven spacetime dimensions.
I remember well the dazed and wide-eyed looks everywhere at the international string theory conference, held at the University of Southern California in 1995, at which Witten first announced some of these results, the first shot in what is now called the Second String Theory Revolution.* For the multiverse story, it is the branes that are central. Using them, researchers have been led by the hand to another variety of parallel universes.
Branes and Parallel Worlds
We typically imagine that strings are ultra-small; that very feature makes testing the theory such a challenge. However, I noted in Chapter 4 that strings are not necessarily minute. Rather, a string’s length is controlled by its energy. The energies associated with the masses of electrons, quarks, and other known particles are so tiny that the corresponding strings would indeed be minuscule. But inject enough energy into a string, and you could cause it to stretch large. We don’t have anywhere near the capacity to do this here on earth, but that’s a limitation of our technological development. If string theory is right, an advanced civilization would be able to pump strings up to whatever size it liked. Natural cosmological phenomena also have the capacity to produce long strings; for example, strings can wrap around a portion of space and get caught up in the cosmological expansion, stretching long in the process. One of the possible experimental signatures outlined in Table 4.1 looks for gravitational waves that such long strings may emit as they vibrate far away in space.
Like strings, higher-dimensional branes can be big. And this opens up a wholly new way in which string theory can describe the cosmos. To grasp what I mean, picture first a long string, as long as an overhead electric cable that runs as far as the eye can see. Next, picture a large two-brane, like an enormous tablecloth or a gargantuan flag, whose surface extends indefinitely. These are both easy to visualize because we can picture them located within the three dimensions of common experience.
If a three-brane is enormous, perhaps infinitely big, the situation changes. A three-brane of this sort would fill the space we occupy, like water filling a huge fish tank. Such ubiquity suggests that rather than think of the three-brane as an object that happens to be situated within our three spatial dimensions, we should envision it as the very substrate of space itself. Just as fish inhabit the water, we would inhabit a space-filling three-brane. Space, at least the space we directly inhabit, would be far more corporeal than generally imagined. Space would be a thing, an object, an entity—a three-brane. As we run and walk, as we live and breathe, we move in and through a three-brane. String theorists call this the braneworld scenario.
It is here that parallel universes make their stringy entrance.
I’ve been focusing on the relationship between three-branes and three spatial dimensions because I wanted to make contact with the familiar domain of everyday reality. But in string theory, there are more than just three spatial dimensions. And a higher-dimensional expanse offers ample room for accommodating more than one three-brane. Starting conservatively, imagine that there are two enormous three-branes. You may find it difficult to picture this. I certainly do. Evolution has prepared us to identify objects, those presenting opportunity as well as danger, that sit squarely within three-dimensional space. Consequently, although we can easily picture two ordinary three-dimensional objects inhabiting a region of space, few of us can picture two coexisting but separate three-dimensional entities, each of which could fully fill three-dimensional space. For ease in discussing the braneworld scenario, then, let’s suppress one spatial dimension in our visualizations and think about life on a giant two-brane. And for a definite mental image, think of the two-brane as a giant, extraordinarily thin slice of bread.*
To use this metaphor effectively, imagine that the slice of bread includes the entirety of what we’ve traditionally called the universe—the Orion, Horsehead, and Crab nebulae; the entire Milky Way; the Andromeda, Sombrero, and Whirlpool Galaxies; and so on—everything within our three-dimensional spatial expanse, however distant, as sketched in Figure 5.3a. To visualize a second three-brane we just need to picture a second enormous slice of bread. Where? Place it next to ours, just shifted slightly away in the extra dimensions (Figure 5.3b). To visualize three or four or any other number of three-branes is equally easy. Just add slices to the cosmic loaf. And while the loaf metaphor emphasizes a collection of branes all aligned with one another, it’s easy to imagine yet more general possibilities. The branes can be oriented any which way, and branes of any other dimensionality, higher or lower, can be included just the same.
Figure 5.3 (a) In the braneworld scenario, what we have traditionally thought to be the entire cosmos is imagined to reside within a three-dimensional brane. For visual ease, we suppress one dimension and show the braneworld as having two spatial dimensions; we also show only a finite piece of branes that may extend infinitely far. (b) The higher-dimensional expanse of string theory can accommodate many parallel braneworlds.
The same fundamental laws of physics would apply all across the collection of branes, since they all emerge from a single theory, string/M-theory. But, much as with the bubble universes in the Inflationary Multiverse, environmental details such as the value of this or that field permeating a brane, or even the number of spatial dimensions defining a brane, can profoundly affect its physical features. Some braneworlds might be much like our own, filled with galaxies, stars, and planets, while others might be very different. On one or more of those branes there might be self-aware beings who, like us, once thought that their slice—their expanse of space—was the entirety of the cosmos. In string theory’s braneworld scenario, we would now recognize this as a parochial perspective. In the braneworld scenario, our universe is just one of many that populate the Brane Multiverse.
When the Brane Multiverse was first floated in the string theory community, the immediate response focused on an obvious question. If there are giant branes right next door, entire parallel universes hovering nearby like slices of rye cozying up to their neighbors, why don’t we see them?
Sticky Branes and Gravity’s Tentacles
Strings come in two shapes, loops and snippets. I haven’t addressed this distinction because it’s not essential for understanding many of the theory’s overarching features. But for braneworlds the distinction between loops and snippets is crucial, and a simple question reveals why. Can strings fly off a brane? The answer: A loop can. A snippet can’t.
As first realized by renowned string theorist Joe Polchinski, it all has to do with the endpoints of a string snippet. The equations that convinced physicists that branes were part of string theory also revealed that strings and branes have a particularly intimate relationship. Branes are the only locations where the endpoints of string snippets can reside, as in Figure 5.4. The math showed that if you try to remove a string’s endpoint from a brane, you are attempting the impossible, like seeking to make π smaller or the square root of 2 bigger. Physically, it’s like trying to remove the north or south pole from the ends of a bar magnet. It just can’t be done. String snippets can freely move within and through a brane, effortlessly gliding from here to there, but they can’t leave it.
If these ideas are more than just interesting mathematics and we are in fact all living on a brane, you’re right now directly experiencing the viselike grip our brane exerts on string endpoints. Try to jump off our three-brane. Try again, harder. I suspect you’re still here. In a braneworld, the strings that make up you, and the rest of ordinary matter, are snippets. While you can jump up and down, throw a baseball from first to second, and send a sound wave from radio to ear, all with absolutely no resistance from the brane, you can’t depart the brane. When you try to jump off, the endpoints of your string snippets anchor you to the brane, unalterably. Our reality could be a floating slab in a higher-dimensional expanse, but we’d be permanently imprisoned, unable to venture out and explore the grander cosmos.
Figure 5.4 Branes are the only locations where the endpoints of string snippets can reside.
The same picture holds for the particles that transmit the three nongravitational forces. The analysis shows that they, too, arise from string snippets. Most notable among these are photons, the purveyors of the electromagnetic force. Visible light, which is a stream of photons, can therefore travel freely through the brane, from this text to your eyes, or from the Andromeda Galaxy to the Wilson Observatory, but it too is unable to escape. Another braneworld could be hovering millimeters away, but because light can’t travel across the gap, we would never see the slightest hint of its presence.
The one force that’s different in this regard is gravity. The distinguishing feature of gravitons, noted in Chapter 4, is that they have spin-2, twice that of the particles arising from string snippets (such as photons) that convey the nongravitational forces. That gravitons have twice the spin of an individual string snippet means you can think of gravitons as being built of two such snippets, the two ends of one melding with those of the other, yielding a loop. And since loops have no endpoints, branes can’t trap them. Gravitons can therefore leave and reenter a braneworld. In a braneworld scenario, then, gravity provides our only means of probing beyond our three-dimensional spatial expanse.
This realization plays a central role in some of the potential tests of string theory mentioned in Chapter 4 (Table 4.1). In the 1980s and 1990s, before branes entered the picture, physicists imagined that string theory’s extra dimensions were roughly Planck-sized (a radius of about 10–33 centimeters), the natural scale for a theory involving gravity and quantum mechanics. But the braneworld scenario encourages more expansive thinking. With our only probe beyond the three common dimensions being gravity—the feeblest of all forces—the extra dimensions can be a good deal larger and have still avoided detection. So far.
If the extra dimensions exist, and are much larger than previously thought—perhaps a billion billion billion times larger (about 10–4 centimeters across)—then experiments that measure the strength of gravity, described in the second row of Table 4.1, stand a chance of detecting them. When objects attract each other gravitationally, they exchange streams of gravitons; the gravitons are invisible messengers that communicate gravity’s influence. The more gravitons the objects exchange, the stronger the mutual gravitational pull. When some of these streaming gravitons leak off our brane and flow into the extra dimensions, the gravitational attraction between objects will be diluted. The larger the extra dimensions, the more the dilution, and the weaker gravity appears. By carefully measuring the gravitational pull between two objects brought closer together than the size of the extra dimensions, experimenters envision intercepting the gravitons before they leak from our brane; if so, the experimenters should measure a strength for gravity that’s proportionately larger. Thus, although I didn’t mention it in Chapter 4, this approach for unmasking the extra dimensions relies on the braneworld scenario.
A more modest increase in the size of the extra dimensions, to only about 10–18 centimeters across, would still make them potentially accessible to the Large Hadron Collider. As discussed in the third entry of Table 4.1, high-energy collisions between protons can eject debris into the extra dimensions, resulting in an apparent loss of energy in our dimensions that might be detectable. This experiment, too, relies on the braneworld scenario. Data attesting to missing energy would be explained by positing that our universe exists on a brane and arguing that debris with the capacity to fly off our brane—gravitons—had carried the energy away.
The prospect of mini black holes, the fourth entry of Table 4.1, is yet another braneworld by-product. The Large Hadron Collider stands a chance of producing mini black holes in proton-proton collisions only if the intrinsic strength of gravity grows large when probed over short distances. As above, it is the braneworld scenario that makes this possible.
The details cast these three experiments in a new light. Not only are these experiments seeking evidence of exotic structures such as extra dimensions of space and tiny black holes, they are also seeking evidence that we’re living on a brane. In turn, a positive result would not only build a case for string theory’s braneworld scenario, but would also provide indirect evidence for universes beyond our own. If we can establish that we’re living on a brane, the mathematics gives us no reason to expect that ours is the only one.
Time, Cycles, and the Multiverse
The multiverses we’ve so far encountered, however different in detail, share one basic trait. In the Quilted, Inflationary, and Brane Multiverses, the other universes are all “out there” in space. For the Quilted Multiverse “out there” means far away in the everyday sense; for the Inflationary Multiverse it means beyond our bubble universe and across the rapidly expanding intervening realm; for the Brane Multiverse it means a possibly short distance away but the seperation is through another dimension. Evidence supporting the braneworld scenario would lead us to consider seriously another variety of multiverse, one that leverages not the opportunities afforded by space but those of time.6
Since Einstein, we’ve known that space and time can warp, curve, and stretch. But we generally don’t envision the whole universe wafting this way or that. What would it mean for the entirety of space to move ten feet to the “right” or “left”? It’s a good brain-teaser, but it becomes pedestrian when considered in the braneworld scenario. Like particles and strings, branes can surely move through the surrounding environment they inhabit. And so, if the universe we observe and experience is a three-brane, we could very well be gliding through a higher-dimensional spatial expanse.*
If we are on such a gliding brane, and there are other branes nearby, what would happen if we slammed into one of them? Although there are details that have not yet been fully worked out, you can be certain that a collision between two branes—a collision between two universes—would be violent. The simplest possibility would be two parallel three-branes coming closer and closer together till finally they collided straight-on, much like two cymbals crashing. The tremendous energy harbored in their relative motion would yield a fiery rush of particles and radiation that would obliterate any organized structures that either brane universe contained.
To a group of researchers including Paul Steinhardt, Neil Turok, Burt Ovrut, and Justin Khoury, this cataclysm rang not just of an end but of a beginning. An intensely hot, thoroughly dense environment in which particles stream this way and that sounds much like the conditions just after the big bang. Perhaps, then, when two branes collide they wipe out whatever structures may have coalesced during either of their histories, from galaxies to planets to people, while setting the stage for a cosmic rebirth. Indeed, a three-brane filled with a blistering plasma of particles and radiation responds just as an ordinary three-dimensional spatial expanse would: it expands. And as it does, the environment cools, allowing particles to clump, ultimately yielding the next generation of stars and galaxies. Some have suggested that an apt name for this reprocessing of universes would be the big splat.
Evocative though it may be, “splat” misses a central feature of brane collisions. Steinhardt and his collaborators have argued that when branes collide, they don’t stick together. They bounce apart. The gravitational force they exert on each other then gradually slows their relative motion; eventually, they reach a maximum separation from which they start approaching once again. As the branes fall back together, each builds up speed, they collide, and through the ensuing firestorm the conditions on each brane are reset once again, initiating a new era of cosmological evolution. The essence of this cosmology thus involves worlds that repeatedly cycle through time, generating a new variety of parallel universes called the Cyclic Multiverse.
If we are living on a brane in the Cyclic Multiverse, the other member universes (in addition to the partner brane with which we periodically collide) are in our past and future. Steinhardt and his co-workers estimated the time scale for a full cycle of the colliding cosmic tango—birth, evolution, and death—and came up with about a trillion years. In this scenario, the universe as we know it would merely be the latest in a temporal series, some of which may have contained intelligent life and the culture they created, but are now long ago extinguished. In due course, all of our contributions and those of any other life-forms our universe supports would be similarly erased.
The Past and Future of Cyclic Universes
Although the braneworld approach is its most refined incarnation, cyclical cosmologies have enjoyed a long history. Earth’s rotation, yielding the predictable pattern of day and night, as well as its orbit, yielding the repetitive sequence of passing seasons, presages the cyclical approaches developed by many traditions in their attempt to explain the cosmos. One of the oldest prescientific cosmologies, the Hindu tradition, envisions a nested complex of cosmological cycles within cycles, which, according to some interpretations, stretch from millions to trillions of years. Western thinkers, from as far back as the pre-Socratic philosopher Heraclitus and the Roman statesman Cicero, also developed various cyclic cosmological theories. A universe consumed by fire and emerging anew from the smoldering embers was a popular scenario among those who considered lofty issues such as cosmic origins. With the spread of Christianity, the concept of genesis as a unique, onetime event gradually gained the upper hand, but cyclic theories continued to sporadically attract attention.
In the modern scientific era, cyclical models have been pursued since the earliest cosmological investigations invoking general relativity. Alexander Friedmann, in a popular book published in Russia in 1923, noted that some of his cosmological solutions to Einstein’s gravitational equations suggested an oscillating universe that would expand, reach a maximal size, contract, shrink to a “point,” and then might begin expanding anew.7 In 1931, Einstein himself, having by then dropped his proposal for a static universe, also investigated the possibility of an oscillatory universe. Most detailed of all was a series of papers published from 1931 to 1934 by Richard Tolman at the California Institute of Technology. Tolman undertook thorough mathematical investigations of cyclical cosmological models, initiating a stream of such studies—often swirling in the backwaters of physics but sometimes bubbling up to broader prominence—that have continued to this day.
Part of the appeal of a cyclical cosmology is its apparent ability to avoid the knotty issue of how the universe began. If the universe goes through cycle after cycle, and if the cycles have always happened (and perhaps always will), then the problem of an ultimate beginning is sidestepped. Each cycle has its own beginning, but the theory provides a concrete physical cause: the termination of the previous cycle. And if you ask about the beginning of the entire cycle of universes, the answer is simply that there was no such beginning, because the cycles have been repeating for eternity.
In a sense, then, cyclical models are an attempt to have your cosmological cake and eat it too. Back in the early days of scientific cosmology, the steady state theory provided its own end run around the question of cosmic origin by suggesting that although the universe is expanding, it did not have a beginning: as the universe expands, new matter is continually created to fill the additional space, ensuring that constant conditions are maintained throughout the cosmos for all eternity. But the steady state theory ran afoul of astronomical observations pointing strongly toward earlier epochs whose conditions differed markedly from those we experience today. Most pointed of all were observations zeroing in on an earliest cosmological phase that was far from steady and stately, being instead chaotic and combustible. A big bang undermines dreams of steady state, bringing the question of origin back to center stage. It’s here that cyclical cosmologies offer a compelling alternative. Each cycle can incorporate a big-bang-like past, in alignment with the astronomical data. But by stringing together an infinite number of cycles the theory still avoids having to supply an ultimate beginning. Cyclical cosmologies, so it would seem, thereby meld the most attractive features of the steady state and big bang models.
Then in the 1950s, the Dutch astrophysicist Herman Zanstra called attention to a problematic feature of cyclical models, one that was implicit in Tolman’s analysis a couple of decades earlier. Zanstra showed that there couldn’t have been an infinite number of cycles preceding our own. The wrench in the cosmological works was the Second Law of Thermodynamics. This law, which we’ll discuss more fully in Chapter 9, establishes that disorder—entropy—increases over time. It’s something we routinely experience. Kitchens, however ordered in the morning, have a way of becoming disordered by nightfall; the same goes for laundry bins, desktops, and playrooms. In these everyday settings, the increase in entropy is a mere nuisance; in cyclic cosmology, the increase in entropy is pivotal. As Tolman himself had realized, the equations of general relativity link the entropy content of the universe with the duration of a given cycle. More entropy means more disordered particles squeezed together when the universe shrinks; that generates a more powerful rebound, space expands further, and so the cycle lasts longer. Looking back from today, the Second Law then implies that ever-earlier cycles would have had ever-less entropy (because the Second Law says that entropy increases toward the future, it must decrease toward the past),* and would thus have had ever-shorter durations. Working this out mathematically, Zanstra showed that sufficiently far back in time the cycles would have been so short that they would have ceased. They would have had a beginning.
Steinhardt and company claim that their new version of cyclical cosmology avoids this pitfall. In their approach, the cycles arise not from a universe expanding, contracting, and expanding again but rather from the separation between braneworlds expanding, contracting, and expanding again. The branes themselves continually expand—and they do so throughout each and every cycle. Entropy builds from one cycle to the next, just as the Second Law requires, but because the branes expand the entropy is spread over ever-larger spatial volumes. The total entropy goes up, but the entropy density goes down. By the end of each cycle, the entropy is so diluted that its density is driven very nearly to zero—a full reset. And so, unlike what happens in the analysis of Tolman and Zanstra, the cycles can continue indefinitely toward the future as well as the past. The braneworld Cyclic Multiverse has no need for a beginning to time.8
Sidestepping an age-old conundrum is a feather in the Cyclic Multiverse’s cap. But as its proponents note, the Cyclic Multiverse goes beyond offering resolution to cosmological conundra—it makes a specific prediction that distinguishes it from the widely accepted inflationary paradigm. In inflationary cosmology, the violent burst of expansion in the early universe would have so thoroughly disturbed the spatial fabric that substantial gravitational waves would have been produced. These ripples would have left trace imprints on the cosmic microwave background radiation, and highly sensitive observations are now seeking them out. A brane collision, by contrast, creates a momentary maelstrom—but without the spectacular inflationary stretching of space, any gravitational waves produced would almost certainly be too weak to create a lasting signal. So evidence of gravitational waves produced in the early universe would be strong evidence against the Cyclic Multiverse. On the other hand, failure to observe any evidence of these gravitational waves would severely challenge a great many inflationary models and make the cyclic framework all the more attractive.
The Cyclic Multiverse is widely known within the physics community but is viewed, almost as widely, with much skepticism. Observations have the capacity to change this. If evidence for braneworlds emerges from the Large Hadron Collider, and if signs of gravitational waves from the early universe remain elusive, the Cyclic Multiverse will likely garner increased support.
The mathematical realization that string theory is not just a theory of strings but also includes branes has had a major impact on research in the field. The braneworld scenario, and the multiverses to which it gives rise, is one resulting area of investigation with the capacity to profoundly remake our perspective on reality. Without the more exact mathematical methods developed over the last decade and a half, most of these insights would have remained beyond reach. Nevertheless, the main problem physicists hoped the more exact methods would address—the need to pick one form for the extra dimensions out of the many candidates that theoretical analyses have uncovered—has not yet been solved. Far from it. The new methods have actually made the problem all the more challenging. They’ve resulted in the discovery of vast new troves of possible forms for the extra dimensions, increasing the candidate pool enormously while not providing an iota of insight into how to single out one as ours.
Pivotal to these developments is a property of branes called flux. Just as an electron gives rise to an electric field, an electric “mist” that permeates the area around it, and a magnet gives rise to a magnetic field, a magnetic “mist” that permeates its region, so a brane gives rise to a brane field, a brane “mist” that permeates its region, as in Figure 5.5. When Faraday was performing the first experiments with electric and magnetic fields, in the early 1800s, he imagined quantifying their strength by delineating the density of field lines at a given distance from the source, a measure he called the field’s flux. The word has since become ensconced in the physics lexicon. The strength of a brane’s field is also delineated by the flux it generates.
String theorists, including Raphael Bousso, Polchinski, Steven Giddings, Shamit Kachru, and many others realized that a full description of string theory’s extra dimensions requires not only specifying their shape and size—which researchers in this area, including me, had focused on more or less exclusively in the 1980s and early 1990s—but also specifying the brane fluxes that permeate them. Let me take a moment to flesh this out.
Figure 5.5 Electric flux produced by an electron; magnetic flux produced by a bar magnet; brane flux produced by a brane.
Since the earliest mathematical work investigating string theory’s extra dimensions, researchers have known that Calabi-Yau shapes typically contain a great many open regions, like the space inside a beach ball, a doughnut’s hole, or within a blown glass sculpture. But it wasn’t until the early years of the new millennium that theorists realized that these open regions needn’t be completely empty. They can be wrapped by one or another brane, and threaded by flux piercing through them, as in Figure 5.6. Previous research (as summarized, for instance, in The Elegant Universe) had for the most part considered only “naked” Calabi-Yau shapes, from which all such adornments were absent. When researchers realized that a given Calabi-Yau shape could be “dressed up” with these additional features, they uncovered a gargantuan collection of modified forms for the extra dimensions.
Figure 5.6 Parts of the extra dimensions in string theory can be wrapped by branes and threaded by fluxes, yielding “dressed-up” Calabi-Yau shapes. (The figure uses a simplified version of a Calabi-Yau shape—a “three-hole doughnut”—and represents wrapped branes and flux lines schematically with glowing bands encircling portions of the space.)
A rough count gives a sense of scale. Focus on fluxes. Just as quantum mechanics establishes that photons and electrons come in discrete units—you can have 3 photons and 7 electrons, but not 1.2 photons or 6.4 electrons—so quantum mechanics shows that flux lines also come in discrete bundles. They can penetrate a surrounding surface once, twice, three times, and so on. But apart from this restriction to whole numbers, there’s in principle no other limit. In practice, when the amount of flux is large, it tends to distort the surrounding Calabi-Yau shape, rendering previously reliable mathematical methods inaccurate. To avoid venturing into these more turbulent mathematical waters, researchers typically consider only flux numbers that are about 10 or less.9
This means that if a given Calabi-Yau shape contains one open region, we can dress it up with flux in ten different ways, yielding ten new forms for the extra dimensions. If a given Calabi-Yau has two such regions, there are 10 × 10 = 100 different flux dressings (10 possible fluxes through the first paired with 10 through the second); with three open regions there are 103 different flux dressings, and so on. How large can the number of these dressings get? Some Calabi-Yau shapes have on the order of five hundred open regions. The same reasoning yields on the order of 10500 different forms for the extra dimensions.
In this way, rather then winnowing the candidates to a few specific shapes for the extra dimensions, the more refined mathematical methods have led to a cornucopia of new possibilities. All of a sudden, Calabi-Yau spaces can clothe themselves with far more outfits than there are particles in the observable universe. For some string theorists, this caused great distress. As emphasized in the previous chapter, without a means of choosing the exact form for the extra dimensions—which we now realize means also selecting the flux outfit that shape wears—the mathematics of string theory loses its predictive power. Much hope had been placed on mathematical methods that could go beyond the limitations of perturbation theory. Yet, when some of those methods materialized, the problem of fixing the form for the extra dimensions only got worse. Some string theorists lost heart.
Others, more sanguine, believe it’s too early to give up hope. One day—perhaps a day that’s just around the corner, perhaps a day that’s far off—we will discover the missing principle that determines what the extra dimensions look like, including the fluxes the shape may be sporting.
Others still have taken a more radical tack. Maybe, they suggest, the decades of fruitless attempts to pin down the form for the extra dimensions are telling us something. Maybe, these radicals brazenly continue, we need to take seriously all of the possible shapes and fluxes emerging from string theory’s mathematics. Maybe, they urge, the reason the mathematics contains all these possibilities is that they’re all real, each shape being the extra-dimensional part of its own separate universe. And maybe, grounding a seemingly wild flight of fancy in observational data, this is just what’s needed to address perhaps the thorniest problem of all: the cosmological constant.
*You can think of this as a grand generalization of the results touched on in Chapter 4, in which different forms for the extra dimensions can give rise to identical physical models.
†This wasn’t the result of a mysterious mathematical coincidence. Instead, in a precise mathematical sense, strings are highly symmetric shapes, and it was this symmetry that wiped away the inconsistencies. See note 4 for details.
*The first revolution was the 1984 results of John Schwarz and Michael Green, which launched the modern version of the subject.
*If you’re being careful, you’ll note that a slice of bread is really three-dimensional (width and height on the slice’s surface, but also depth from the slice’s thickness), but don’t let that trouble you. The thickness of the bread will remind us that our slices are visual stand-ins for large three-branes.
*You could still ask whether the entire higher-dimensional spatial expanse can move, but however interesting to contemplate, it’s not relevant to the discussion here.
*For readers familiar with the puzzle of time’s arrow, note that I am assuming, in keeping with observations, that entropy decreases toward the past. See The Fabric of the Cosmos, Chapter 6, for a detailed discussion.