by Brian Greene
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 t
hen 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.
In Flux
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 s
tring 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.