KINKS IN SPACETIME - The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis

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


The key to understanding the human mind, according to Sigmund Freud, is to study people whose conduct is out of the ordinary, anomalous—people with weird obsessions, for instance, including his famous patients like the “Rat man” (who had deranged fantasies involving loved ones and a pot of rats) and the “Wolf man” (who dreamed of being eaten alive by a pack of white wolves perched in a tree outside his bedroom window). Freud’s premise was that our best hopes of learning about typical behavior would come from studying the most unusual, pathological cases. Through such investigations, he said, we might eventually come to understand both the norm and the deviations from the rule.

We often take a similar approach in mathematics and physics. “We seek out places where the classical descriptions fail, for these are the places where we are most likely to learn something new,” explains Harvard astrophysicist Avi Loeb. Whether we are thinking about an abstract space in geometry or the somewhat more tangible space we call the universe, the places “where something bad happens to space, where things break down,” as Loeb puts it, tend to be the places we call singularities.1

Contrary to what people might think, singularities abound in nature. They’re all around us: A water droplet breaking off and falling from a leaky faucet is a common example (all too common in my household), the spot (well-known to surfers) at which an ocean wave breaks and crashes, the folds in a newspaper (which dictate whether your story is important or filler), or the pinch points on a slender balloon twisted into the shape of a French poodle. “Without singularities, you cannot talk about shapes,” notes the geometer Heisuke Hironaka, an emeritus professor at Harvard. He cites the example of a handwritten signature: “If there is no crossing, no sharp point, it’s just a squiggle. A singularity might be a crossing or something suddenly changing direction. There are many things like that in the world, and that’s why the world is interesting.”2

In physics and cosmology, two kinds of singularities stand out among countless other possibilities. One is the singularity in time known as the Big Bang. I don’t know how to make sense of the Big Bang as a geometer, because no one—physicists included—really knows what it is. Even Alan Guth, the inventor of the whole notion of cosmic inflation—the thing that “puts the bang in the Big Bang,” as he explains it—admits that the term Big Bang has always suffered from “vagueness, probably because we still don’t know (and may never know) what really happened.”3 So in this instance, I believe some humility is in order.

And though we’re rather clueless when it comes to applying geometry to the precise moment of the universe’s birth, we geometers have had somewhat better success in dealing with black holes. A black hole is basically a chunk of space crushed down to a single point under the weight of gravity. All that mass packed into a tiny area creates a superdense object whose escape velocity (a measure of its gravitational pull) exceeds the speed of light, trapping everything, light included.

Despite being a consequence of Einstein’s general theory of relativity, black holes are still strange enough objects that Einstein himself denied their existence as late as 1930—about fifteen years after the German physicist Karl Schwarzschild presented them as solutions to Einstein’s famous equations. Schwarzschild didn’t believe in the physical reality of black holes, either, but nowadays, the existence of such objects is widely accepted. “Black holes are now discovered with great regularity,” as Andrew Strominger put it, “every time somebody at NASA needs a new grant increase.”4

Even though astronomers have found large numbers of putative black holes while accumulating impressive data in support of that thesis, these objects are still enshrouded in mystery. General relativity provides a perfectly adequate description of large black holes, but the picture falls apart when we move to the center of the maelstrom and consider a black hole’s vanishingly small singular point of infinite curvature. Nor can general relativity contend with tiny black holes, smaller than a grain of dust—a regime in which quantum mechanics inevitably comes into play. General relativity’s inadequacies become glaringly apparent in the case of such miniature black holes, where masses are large, distances small, and the curvature of spacetime off the charts. That’s precisely where string theory and Calabi-Yau spaces have helped out, which is gratifying since the theory was invented, in part, to deal with that very clash between general relativity and quantum mechanics.


8.1—Twelve million light-years away, a supermassive black hole, approximately seventy million times more massive than the sun, is thought to reside in the center of the spiral galaxy M81. (Image courtesy of NASA)

One of the highest-profile disputes between those two celebrated fields of physics revolves around whether information is destroyed by a black hole. In 1997, Stephen Hawking of Cambridge University and Kip Thorne of Caltech made a bet with John Preskill, also of Caltech. At issue was an implication of Hawking’s theoretical finding in the early 1970s that black holes are not totally “black.” These objects, he discovered, have a tiny but nonzero temperature, meaning that they must retain some thermal energy. Like any other “hot” body, the black hole will radiate its energy away until there’s nothing left and the black hole completely evaporates. If the radiation given off by the black hole is strictly thermal and is thus lacking in information content, then the information originally stored within a black hole—say, if it swallows up a star with a particular composition, structure, and history—will disappear when the black hole evaporates. That would violate a fundamental tenet of quantum theory, which holds that the information of a system is always preserved. Hawking argued that, quantum mechanics notwithstanding, in the case of black holes, information can be destroyed, and Thorne agreed. Preskill maintained that the information would survive.

“We believe that if you throw two ice cubes into a pot of boiling water on Monday and inspect the water atoms on Tuesday, you can determine that two ice cubes were thrown in the day before,” Strominger explains—not in practice, but in principle, yes.5 Another way to think of it is to take a book, say, Fahrenheit 451, and toss it into a fire. “You may think the information is lost, but if you have infinite observational power and calculation capacity—if you measure everything about the fire and keep track of the ashes and enlist the services of ‘Maxwell’s demon’ (or in this case ‘Laplace’s demon’)—then you can reproduce the original state of the book,” notes Caltech physicist Hirosi Ooguri.6 If you were to toss that same book into a black hole, however, Hawking argued that the data would be lost. Preskill, on the other hand, like Gerard ’t Hooft and Leonard Susskind before him, took the position that the two cases are not fundamentally different and that the black hole radiation must, in a subtle way, carry the information of the Ray Bradbury classic—information that, in theory, can be recovered.

The stakes were high, as one of the lynchpins of science—the principle of scientific determinism—was at risk. Determinism is the idea that if you have all possible data describing a system at a particular time and know the laws of physics, then you can, in principle, determine what will happen in the future, as well as deduce what happened in the past. But if the information can be lost or destroyed, determinism is no longer valid. You can’t predict the future, nor can you deduce the past. If the information is lost, in other words, you’re lost, too. The stage was thus set for a classic showdown. “This was the moment of truth for string theory, which had claimed it could consistently reconcile quantum mechanics and gravity,” says Strominger. “So could it explain Hawking’s paradox?”7

Strominger tackled this question with Cumrun Vafa in a groundbreaking 1996 paper.8 They used the notion of black hole entropy as their way into the problem. Entropy is a measure of a system’s randomness or disorder but also provides indications about a system’s information content and storage. Think of a bedroom, for instance, that has a variety of shelves, drawers, and counters, as well as assorted artwork displayed on the walls and mobiles hanging from the ceiling. Entropy relates to the number of different ways you can organize, or disorganize, all the stuff—furniture, clothes, books, posters, and assorted knickknacks—in that room. To some extent, the number of possible ways of arranging these same items in this same space depends on the room’s size or volume—its length, width, and height all factoring into the computation.

The entropy of most systems scales with the volume. In the early 1970s, however, the physicist Jacob Bekenstein, then a graduate student at Princeton, suggested that the entropy of a black hole is proportional to the area of the event horizon surrounding the black hole, rather than to the volume bounded up inside the horizon. The event horizon is often called the point of no return, and any object that crosses this invisible line in spacetime will succumb to the pull of gravity and fall, inexorably, into the black hole. But it is perhaps better thought of as a surface of no return, as it is indeed a two-dimensional surface rather than a “point.” For a nonspinning (or “Schwarzschild”) black hole, the area of that surface depends solely on the mass of the black hole: The bigger the mass, the bigger the area. The notion that the entropy of a black hole—reflecting all possible configurations of this object—depended only on the horizon area suggested that all the configurations resided on the surface and that all the information about the black hole was stored on the surface, too. (One could make a parallel here with the bedroom of our previous example, in which all the stuff is arranged along the surface—walls, ceiling, and floor—rather than floating in midair in the room’s interior space.)

Bekenstein’s work, coupled with Hawking’s ideas on black hole radiation, yielded an equation for computing a black hole’s entropy. The entropy, according to the so-called Bekenstein-Hawking formula, was indeed proportional to the horizon area. More specifically, the entropy was proportional to the area divided by four times Newton’s gravitational constant (G). Using this formula, one could see that a black hole three times as massive as the sun would have an astoundingly high entropy, on the order of 1078 joules per Kelvin. The black hole, in other words, would be extremely disordered.

The fact that a black hole had such a staggeringly high entropy came as a total shock, given that in general relativity, a black hole is completely described by just three parameters: its mass, charge, and spin. A gigantic entropy, on the other hand, suggests a tremendous variability in a black hole’s internal makeup that must go far beyond those three parameters. The question was: Just where does that variability come from? What other things inside a black hole can vary as well?

The trick apparently lies in breaking down a black hole into its microscopic constituents, just as the Austrian physicist Ludwig Boltzmann had done for gases in the 1870s. Boltzmann showed that you could derive the thermodynamic properties of a gas from the combined properties of its many individual molecular constituents. (These molecules can be abundant indeed, approximately 1020 of them per ounce of a typical gas under typical conditions.) Boltzmann’s idea was remarkable for a number of reasons, including the fact that he hit upon it decades before there was firm evidence that molecules existed. Given the vast number of gas constituents, or molecules, Boltzmann argued that the average of motions and behaviors of the individual molecules would produce the bulk properties—the same volume, temperature, and pressure—for the gas as a whole. He thus provided a more precise view of the system in which the gas was no longer just a single entity but instead consisted of multitudinous parts. His perspective also led to a new definition of entropy: the number of possible arrangements of microscopic states (or microstates) that give rise to the same macroscopic features. Putting this relation into more quantitative terms, the entropy (S) equals the natural log of the number of microstates. Or, equivalently, the number of microstates equals eS.

The approach that Boltzmann helped pioneer is called statistical mechanics, and roughly a century later, people were trying to devise a statistical mechanical interpretation of black holes. Two decades after Bekenstein and Hawking brought this problem to prominence, no one had yet succeeded. What was needed was a “microscopic theory of black holes,” says Strominger, “a derivation of the laws of black holes from some fundamental principles—the analogue of Boltzmann’s derivation of the thermodynamics of gases.” It’s been known since the nineteenth century that every system has an associated entropy, and we’ve known since Boltzmann that a system’s entropy depends on the number of microstates contained therein. “It would be a deep and unnerving asymmetry if the relation between entropy and the number of microstates was valid for every system in nature except a black hole,” Strominger adds.9

These microstates, moreover, are “quantized,” according to Ooguri, which is the only way you can hope to get a countable number. You can put a pencil on a desk in an infinite number of ways, just as there’s an infinite number of possible settings along the electromagnetic spectrum. But as mentioned in Chapter 7, radio frequencies are quantized in the sense that radio stations only broadcast at a select number of discrete frequencies. The energy levels of a hydrogen atom are similarly quantized, meaning that you can’t pick an arbitrary value; only certain values are allowed. “Part of the reason Boltzmann had such difficulty convincing others of his theories was that he was way ahead of his time,” Ooguri says, “half a century before quantum mechanics was invented.”10

So this was the challenge Strominger and Vafa took up. It was a real test of string theory, since it involved the quantum states of black holes, which Strominger has called “quintessential gravitational objects.” He felt it was “incumbent upon the theory to give us a solution to the problem of computing the entropy, or it wasn’t the right theory.”11

The plan he and Vafa concocted was to take the entropy derived from that quantum microstate calculation and compare it with the value obtained from the Bekenstein-Hawking area formula based on general relativity. Although the problem was not new, Strominger and Vafa brought new tools to bear on it, drawing not only on string theory but also on Joe Polchinski’s discovery of D-branes and the emergence of M-theory—both of which occurred in 1995, the year before their paper came out.

“Polchinski pointed out that D-branes carry the same kind of charge as black holes and have the same mass and tension, so they look and smell the same,” notes Harvard physicist Xi Yin. “But if you can use one to calculate the properties of the other, such as the entropy, then it’s much stronger than a passing resemblance.” 12 This is indeed the approach that Strominger and Vafa followed, using these D-branes to construct new kinds of black holes, guided by string theory and M-theory.

The possibility of building black holes out of D-branes and strings (the latter being the one-dimensional version of D-branes) stems from the “dual” description of D-branes. In models in which the strength of all forces acting upon branes and strings (including gravity) is low (what’s called weak coupling), branes can be thought of as thin, membrane-like objects that have little effect on the spacetime around them and therefore bear little resemblance to black holes. On the other hand, at strong coupling and high interaction strengths, branes can become dense, massive objects with event horizons and a powerful gravitational influence—objects, in other words, that are indistinguishable from black holes.

Nevertheless, it takes more than just a heavy brane—or a stack of heavy branes—to make a black hole. You also need some way of stabilizing it, which is most easily accomplished—at least in theory—by wrapping the brane around something stable, something that will not shrink. The problem is that an object that has high tension (measured in terms of mass per unit length, area, or volume) could shrink so small that it almost disappears without some underlying structure to stop it, much as an ultratight rubber band would shrink down to a tight clump if left alone.

The key ingredient was supersymmetry, which, as discussed in Chapter 6, keeps the ground or vacuum state of a system from sinking to lower and lower energy levels. And supersymmetry in string theory often implies Calabi-Yau manifolds, because such spaces have this feature automatically built in. So the question then becomes finding stable subsurfaces within the Calabi-Yau to wrap branes on. These subsurfaces or submanifolds, which are of lower dimensionality than the overall space itself, are sometimes referred to by physicists as cycles—a notion introduced previously in this book—which can sometimes be pictured as a noncontractible loop around or through part of the manifold. (Technically speaking, a loop is just a one-dimensional object, but cycles come in higher dimensions, too, and can be thought of as noncontractible, higher-dimensional “loops.”) As physicists tend to view it, a cycle only depends on the topology of some object or hole you can wrap around, regardless of the geometry of that object or hole. “If you change the shape, the cycle will stay the same, but you’ll have a different submanifold,” explains Yin. Being just a feature of the topology, he adds, the cycle on its own has nothing to do with a black hole. “It’s only when you wrap a brane on a cycle—or several branes on a cycle—that you can begin to think in terms of a black hole.”13

To ensure stability, the thing you’re wrapping with—be it a brane, string, or rubber band—must be tight, without any wrinkles in it. The cycle you’re wrapping around, moreover, must be of the minimum possible length or area. Placing a rubber band around a uniform, cylindrical pole, for instance, wouldn’t be an especially stable situation, because the band could easily be nudged from side to side. If, on the other hand, the pole were of varying thickness, the stable cycles (which in this case are circles) could be found at the thinnest, or minimal, points, where a rubber band would be difficult to nudge. Rather than thinking of a smooth pole as the object to be wrapped around, a better analogy for a Calabi-Yau would be a grooved pole or donut of varying thickness, with minimal cycles (or circles) again to be found at the points of minimal thickness.

There are different kinds of cycles that a brane could wrap around inside a Calabi-Yau: They could be circles, spheres, and tori of various dimensions, or Riemann surfaces of high genus. Given that branes carry mass and charge, the point of this exercise is to figure out the number of ways of placing them in stable configurations inside a Calabi-Yau so that their combined mass and charge equals the mass and charge of the black hole itself. “Even though these branes are wrapped individually, they’re all stuck together inside the internal [Calabi-Yau] space and can be thought of as parts of a bigger black hole,” Yin explains.14


8.2—To make a black hole by wrapping a brane around an object, the object in question must be stable. In an analogous situation, one might consider wrapping a rubber band around a wooden pole. Of the two examples shown here, the right-hand figure represents by far the more stable arrangement because, in this case, the rubber band is wrapped around a minimal point, which holds it in place and keeps it from sliding sideways.

Here’s an analogy, which I admit is rather unappetizing, so I’ll have to blame it on an unnamed Harvard physicist who suggested it to me. (I’m sure that he, too, would pass the buck, claiming that he got the analogy from someone else.) The situation, in which individual wrapped branes glom together to form a bigger entity, can be compared to a wet shower curtain to which various strands of hair are stuck. Each strand of hair is like an individual brane that is bound to this bigger object, the shower curtain, that is like a brane itself. Even though each hair can be thought of as a separate black hole, they’re all stuck together—all stuck to the same sheet—making them part of one big black hole.

Counting the number of cycles—counting the number of ways of arranging D-branes—is a problem in differential geometry, as the number you get from that counting corresponds to the number of solutions to a differential equation. Strominger and Vafa had thus converted the question of counting a black hole’s microstates, and thereby computing its entropy, into a geometric question: How many ways can you put D-branes in Calabi-Yau manifolds to end up with the desired mass and charge? That, in turn, can be expressed in terms of cycles: How many spheres and other minimum-sized shapes—around which one can wrap a brane—can you fit inside a Calabi-Yau? The answer to both those questions obviously depends on the geometry of the given Calabi-Yau. If you change the geometry, you’ll change the number of possible configurations or the number of spheres.

That’s the general picture, but the calculation is still quite challenging, so Strominger and Vafa spent a good deal of time searching for a specific way of framing the problem—a way that they could actually solve. They settled on a very special case for their first go-round, selecting a five-dimensional internal space consisting of the product of a four-dimensional K3 surface and a circle. They also constructed a five-dimensional black hole sitting in flat five-dimensional space to which they could compare the structure they would build out of D-branes. This wasn’t just any old black hole. It had special properties that were handpicked to make the problem manageable: It was both supersymmetric and extremal—the latter term meaning it had the minimum mass possible for a given charge. (We’ve already talked a fair amount about supersymmetry, but it only makes sense to talk about a supersymmetric black hole if the background vacuum in which it sits also preserves supersymmetry. That is not the case in the low-energy realm that we inhabit, where we don’t see supersymmetry in the particles around us. Nor do we see it in the black holes that astronomers observe.)

Once Strominger and Vafa had their custom-built black hole, they could use the Bekenstein-Hawking formula to compute the entropy from the event horizon area. The next step was to count the number of ways of configuring D-branes in the internal space so as to match the designer black hole in total charge and mass. The entropy obtained in this fashion (equal to the log of the number of states) was then compared with the value obtained from the horizon area calculation, and the agreement was perfect. “They got it on the nose, including the factor of four, Newton’s constant, and all those things,” says Harvard physicist Frederik Denef. After twenty years of trying, Denef adds, “we finally had the first statistical mechanics derivation of black hole entropy.”15

It was a major success for Strominger and Vafa and for string theory as well. The association between D-branes and black holes was significantly bolstered, while the two physicists also showed that the D-brane description itself was fundamental, Yin explains. “You might have wondered, can the brane be broken down further? Is it made up of smaller parts? We now know there are no additional structures of the brane because they got the right entropy, and entropy, by definition, counts all the states.”16 If, on the other hand, the brane was composed of different parts, this would add new degrees of freedom and, hence, more combinations that would have to be taken into account in the entropy calculation. But the 1996 result shows that this is not the case. The brane is all there is. Although branes of different dimensions look different from each other, none of them have subcomponents or can be broken down further. In the same way, string theory holds that the string (a one-dimensional brane in M-theory parlance) is all there is and cannot be subdivided into smaller pieces.

While the agreement between the two very different methods of obtaining the entropy was certainly gratifying, in some ways it was kind of surprising. “At first glance, the black hole information paradox appears to have nothing whatsoever to do with Calabi-Yau manifolds,” claims Brown University physicist Aaron Simons. “But the key to answering that question turned out to be counting mathematical objects inside a Calabi-Yau.”17


8.3a—Harvard physicist Andrew Strominger (Kris Snibbe/Harvard University)


8.3b—Harvard physicist Cumrun Vafa (Stephanie Mitchell/Harvard University News Office)

The Strominger and Vafa result did not solve the information paradox outright, though the detailed description of a black hole they arrived at through string theory did show how information could be very precisely stored. They took the essential first step, Ooguri says, “by showing that the entropy of a black hole is the same as the entropy of other macroscopic systems,” including the burning book from our previous example.18 Both contain information that is at least potentially retrievable.

Of course, the 1996 results were really just the beginning, as the original entropy calculation had very little to do with real astrophysical black holes: The black holes in the Strominger-Vafa model, unlike those we see in nature, were supersymmetric—a stipulation made simply to make the calculation doable. Nevertheless, these results may still extend to non-supersymmetric black holes as well. As Simons explains, “regardless of supersymmetry, all black holes have a singularity. That is their central, defining feature, and that is the reason they are ‘paradoxical.’ In the supersymmetric black hole case, string theory can help us understand what happens around that singularity, and the hope is that the result is independent of the object being supersymmetric or not.”19

Furthermore, the 1996 paper had described an artificial case of a compact five-dimensional internal space and a flat and noncompact five-dimensional space. That’s not the way we normally view spacetime in string theory. The question was, did that same picture apply to the more conventional arrangement of a six-dimensional internal space and black holes sitting in flat, four-dimensional space?

The answer came in 1997, when Strominger, along with Juan Maldacena (then at Harvard) and Edward Witten, published a paper that carried the earlier work to the more familiar setting of a six-dimensional internal space (Calabi-Yau, of course) and an extended four-dimensional spacetime.20 By duplicating the entropy calculation for Calabi-Yau threefolds, Maldacena says, “the space in which you put the branes has less supersymmetry”—and is thus closer to the world we see—“and the space in which you put the black holes has four dimensions, which is what we assume to be the case.”21 Moreover, the agreement with the Bekenstein-Hawking computation was even stronger because, as Maldacena explains, computing the entropy from the event horizon area is only accurate when the event horizon is very big and the curvature is very small. As the size of black holes shrinks and the surface area shrinks with it, the general-relativity approximation becomes worse and one therefore needs to make “quantum gravity corrections” to Einstein’s theory. Whereas the original paper only considered “large” black holes (large compared with the diminutive Planck scale!), for which the general-relativity-derived number (the so-called first-order term) was sufficient, the 1997 calculation got the first quantum-corrected term in addition to the leading term. In other words, the agreement between the two very different ways of obtaining the entropy had just gotten better.

Ooguri, Strominger, and Vafa went even further in 2004, extending the 1996 result to any kind of black hole you could make by wrapping a brane around an object (such as a cycle) in a regular Calabi-Yau threefold, regardless of its size and therefore regardless of the degree to which quantum mechanics affects the system. Their paper showed how to compute the quantum gravity corrections to general relativity—not just the first few terms but the entire series, an infinite number of terms.22 By including the other terms in the expansion, Vafa explains, “we get a more refined way of counting and a more refined answer and, fortunately, an even stronger agreement than before.”23 That is the approach we normally try to take in math and physics: If we find something that works under special circumstances, we try to generalize it to see if it’s valid in broader circumstances and keep pushing it from there to see just how far we can go.

There is one last generalization of the original Strominger-Vafa work that I’d like to consider. For lack of a better phrase, it is even more general than anything we’ve discussed yet. The idea, which goes by the complicated-sounding name of the Anti-de Sitter Space/Conformal Field Theory (AdS/CFT) correspondence, was initially hit upon by Maldacena in 1997 and was elaborated on by Igor Klebanov of Princeton, Edward Witten, and others. Much like, as Maldacena puts it, a DVD and a 70-millimeter reel of film can both describe the same movie, the correspondence (technically still a conjecture) suggests that in some cases, a theory of gravity (such as string theory) can be completely equivalent to a standard quantum field theory (or conformal field theory, to be exact). The correspondence is surprising because it relates a theory of quantum gravity to a theory with no gravity at all.

AdS/CFT stems from the dual picture of D-branes that we talked about earlier. At very weak coupling, a network of D-branes wrapping cycles in a Calabi-Yau does not exert an appreciable gravitational pull and is best described by quantum field theory—a theory with no gravity in it. At strong coupling, however, this conglomeration of branes is better thought of as a black hole—a system that can only be described by a theory that includes gravity. Despite the integral role of Calabi-Yau manifolds in the work underlying the correspondence conjecture, Maldacena’s idea did not originally involve these manifolds. However, subsequent attempts to frame the correspondence and understand it in greater detail (i.e., efforts by Klebanov and others, as well as some smaller contributions I’ve made in this area with James Sparks, my former Harvard postdoc who’s now at Oxford) did involve Calabi-Yau manifolds quite directly, particularly Calabi-Yau singularities. “Calabi-Yau space is the setting in which the correspondence has been explored the most and is best understood,” Sparks claims.24

Maldacena’s original formulation, along with subsequent work on AdS/CFT, has provided another step toward resolving the black hole information paradox. Without getting into the details, the crux of the argument is that if black hole physics can be completely described by a quantum theory of particles—a setting without a black hole or its messy singularity in which we know information cannot be lost—then we can be sure that a black hole itself cannot lose information. So what happens to the information in an evaporating black hole? The idea is that Hawking radiation, which seeps out as a black hole evaporates, “is not random but contains subtle information on the matter that fell in,” says Maldacena.25

Despite that insight, upon conceding his bet to Preskill in 2004, Hawking did not attribute his change of heart to string-theory-related ideas. Preskill, however, credits Strominger, Vafa, Maldacena, and others with building a “strong but rather circumstantial case that black holes really preserve information,” noting that “Hawking has followed this work by the string theorists with great interest.” 26 Strominger, for his part, believes this work “helped turn around Hawking’s thinking on string theory and indeed turned around the whole world on string theory, because this was the first time that string theory had solved a problem that came from another area of physics and had been posed by someone outside of string theory.”27

The work represented some validation that these crazy ideas involving strings, branes, and Calabi-Yau manifolds might be useful after all, but the scope of Maldacena’s conjecture is not limited to the black hole paradox. Calling for a fundamental reconsideration of gravity, the correspondence has consequently engaged the efforts of a large fraction of the string theory community. One reason AdS/CFT has caught on to the extent it has is strictly pragmatic: “A computation that might be very difficult in one realm can turn out to be relatively straightforward in the other, thereby turning some intractable problems of physics into ones that are easily solved,” explains Maldacena. “If true, the equivalence means that we can use a quantum particle theory (which is relatively well understood) to define a quantum gravity theory (which is not).”28 The correspondence, in other words, can enable us to use our detailed knowledge of particle theories (without gravity) to improve our understanding of quantum gravity theories. The duality works in the other direction, too: When the particles in the quantum field theory are really strongly interacting, making calculations difficult, the curvature on the gravity side of the equation will be correspondingly low, making calculations there more manageable. “When one of the descriptions becomes hard, the other becomes easy, and vice versa,” Maldacena says.29

Does the fact that string theory, according to AdS/CFT, can be equivalent to a quantum field theory—the kind of theory for which we have acquired substantial experimental proof—make string theory itself right? Maldacena doesn’t think so, although some string theorists have tried to make that case. Strominger doesn’t think so, either, but the work on black holes and AdS/CFT, which grew out of it, does make him think that string theory is on the right track. The insights that have sprung from both these fronts—the black hole entropy paradox and Maldacena’s conjecture—“seem to argue for the inevitability of string theory,” he says. “It seems to be a theoretical structure that you can’t escape. It bangs you on the head everywhere you turn.”30