From Eternity to Here: The Quest for the Ultimate Theory of Time - Sean Carroll (2010)
Part IV. FROM THE KITCHEN TO THE MULTIVERSE
Chapter 12. BLACK HOLES: THE ENDS OF TIME
Time, old gal of mine, will soon dim out.
—Anne Sexton, “For Mr. Death Who Stands with His Door Open”
Stephen Hawking is one of the most willful people on Earth. In 1963, while working toward his doctorate at Cambridge University at the age of twenty-one, he was diagnosed with motor neurone disease. The prognosis was not good, and Hawking was told that he likely did not have long to live. After some soul-searching, he decided to move forward and redouble his commitment to research. We all know the outcome; now well into his seventh decade, Hawking has been the most influential scientist in the field of general relativity since Albert Einstein, and is instantly recognizable worldwide through his efforts to popularize physics.
Among other things, Hawking is a tireless traveler, and he spends some time in California every year. In 1998, I was a postdoctoral research fellow at the Institute for Theoretical Physics at the University of California, Santa Barbara, and Hawking came to visit on his annual sojourn. The institute administrator gave me a simple task: “Pick up Stephen at the airport.”
As you might guess, picking up Stephen Hawking at the airport is different than picking up anyone else. For one thing, you’re not really “picking him up”; he rents a van that is equipped to carry his wheelchair, for which a special license is required to drive it—a license I certainly didn’t have. The actual driving was left to his graduate assistant; my job was merely to meet them at the tiny Santa Barbara airport and show them to the van. By “them,” I mean Hawking’s entourage: a graduate assistant (usually a physics student who helps with logistics), other graduate students, family members, and a retinue of nurses. But it wasn’t a matter of pointing to the van and going on my way. Despite the fact that the graduate assistant was the only person allowed to drive the van, Hawking insisted that the van stay with him at all times, and also wanted to go to a restaurant for dinner before dropping off the assistant at his apartment. Which meant that I tagged along in my car while they all went to dinner, so I could shuttle the assistant back and forth. Hawking was the only one who knew where the restaurant was, but speaking through his voice synthesizer is a slow process; we spent several tense moments stopped in the middle of a busy road while Hawking explained that we had passed the restaurant and would have to turn around.
Figure 58: Stephen Hawking, who gave us the most important clue we have about the relationship between quantum mechanics, gravity, and entropy.
Stephen Hawking has been able to accomplish remarkable things while working under extraordinary handicaps, and the reason is basically straightforward: He refuses to compromise in any way. He’s not going to cut down his travel schedule, or eat at the wrong restaurant, or drink a lesser quality of tea, or curtail his wicked sense of humor, or think less ambitiously about the inner workings of the universe, merely because he is confined to a wheelchair. And that strength of character pushes him scientifically, as well as getting him through life.
In 1973, Hawking was annoyed. Jacob Bekenstein, a young graduate student at Princeton, had written a paper suggesting something crazy: that black holes carried huge amounts of entropy.206 By this time Hawking was the world’s expert on black holes, and (in his own words) he was irritated at Bekenstein, who he thought had misused some of his earlier results.207 So he set about showing exactly how crazy Bekenstein’s idea was—for one thing, if black holes had entropy, you could show that they would have to give off radiation, and everyone knows that black holes are black!
In the end, of course, Hawking ended up surprising everyone, including himself. Black holes do have entropy, and indeed they do give off radiation, once we take into account the subtle consequences of quantum mechanics. No matter how stubborn your personality may be, Nature’s rules will not bend to your will, and Hawking was smart enough to accept the radical implications of his discovery. He ended up providing physicists with their single most important clue about the interplay of quantum mechanics and gravity, and teaching us a deep lesson about the nature of entropy.
BLACK HOLES ARE FOR REAL
We have excellent reasons to believe that black holes exist in the real world. Of course we can’t see them directly—they’re still pretty dark, even if Hawking showed that they’re not completely black. But we can see what happens to things around them, and the environment near a black hole is sufficiently unique that we can often be confident that we’ve located one. Some black holes are formed from the collapse of very massive stars, and frequently these have companion stars orbiting them. Gas from the companion can fall toward the black hole, forming an accretion disk that heats to tremendous temperatures and emits X-ray radiation in copious amounts. Satellite observatories have found a number of X-ray sources that demonstrate all of the qualities you would expect in such an object: In particular, a large amount of high-intensity radiation coming from a very small region of space. Astrophysicists know of no good explanations for these observations other than black holes.
There is also good evidence for supermassive black holes at the centers of galaxies—black holes more than a million times the mass of the Sun. (Still a very tiny fraction of the total mass of a galaxy, which is typically a hundred billion times the mass of the Sun.) In the early stages of galaxy formation, these giant black holes sweep up matter around them in a violent maelstrom, visible to us as quasars. Once the galaxy has settled down a bit, things become calmer, and the quasars “turn off.” In our own Milky Way, we nevertheless are pretty sure that there lurks a black hole weighing in at about 4 million solar masses. Even without the blazing radiation of a quasar, observations of stars in the galactic center reveal that they are orbiting in tight ellipses around an invisible object. We can deduce that these stars must be caught in the gravitational field of something that is so dense and massive that it can’t be anything but a black hole, if general relativity has anything to say about the matter.208
BLACK HOLES HAVE NO HAIR
But as much fun as it is to search for real black holes in the universe, it is even more fun to sit and think about them.209 Black holes are the ultimate thought-experiment laboratory for anyone interested in gravity. And what makes black holes special is their purity.
While observations convince us that black holes exist, they don’t give us a lot of detailed information about their properties; we aren’t able to get up close to a black hole and poke at it. So when we make confident assertions about this or that feature of black holes, we’re always implicitly speaking within some theoretical framework. Unfortunately, scientists don’t yet fully understand quantum gravity, the presumed ultimate reconciliation of general relativity with the tenets of quantum mechanics. So we don’t have a single correct theory in which to answer our questions once and for all.
Instead, we often investigate questions within one of three different theoretical frameworks:
1. Classical general relativity, as written down by Einstein. This is the best full theory of gravity we currently possess, and it is completely consistent with all of known experimental data. We understand the theory perfectly, in the sense that any well-posed question has a definite answer (even if it might be beyond our calculational abilities to figure it out). Unfortunately it’s not right, as it’s completely classical rather than quantum mechanical.
2. Quantum mechanics in curved spacetime. This is a framework with a split personality. We treat spacetime, the background through which stuff in the universe moves, as classical, according to the rules of general relativity. But we treat the “stuff ” as quantum mechanical, described by wave functions. This is a useful compromise approach for trying to understand a number of real-world problems.
3. Quantum gravity. We don’t know the correct theory of quantum gravity, although approaches like string theory are very promising. We’re not completely clueless—we know something about how relativity works, and something about how quantum mechanics works. That’s often sufficient to make some reasonable guesses about how things should work in an ultimate version of quantum gravity, even if we don’t have the full-blown theory.
Classical general relativity is the best understood of these, while quantum gravity is the least well understood; but quantum gravity is the closest to the real world. Quantum mechanics in curved spacetime occupies a judicious middle ground and is the approach Hawking took to investigating black-hole radiation. But it behooves us to understand how black holes work in the relatively safe context of general relativity before moving on to more advanced but speculative ideas.
In classical general relativity, a black hole is just about the purest kind of gravitational field you can have. In the flexible world of thought experiments, we could imagine creating a black hole in any number of ways: from a ball of gas like an ordinary star, or out of a huge planet made of pure gold, or from an enormous sphere of ice cream. But once these things collapse to the point where their gravitational field is so strong that nothing can escape—once they are officially black holes—any indication of what kind of stuff they were made from completely disappears. A black hole made out of a ball of gas the mass of the Sun is indistinguishable from a black hole made from a ball of ice cream the mass of the Sun. The black hole isn’t, according to general relativity, just a densely packed version of whatever we started with. It’s pure gravitational field—the original “stuff ” has disappeared into the singularity, and we’re left with a region of strongly curved spacetime.
When we think of the gravitational field of the Earth, we might start by modeling our planet as a perfect sphere of a certain mass and size. But that’s clearly just an approximation. If we want to do a little bit better, we’ll take into account the fact that the Earth also spins, so it’s a little wider near the equator than near the poles. And if we want to be super-careful about it, the exact gravitational field of the Earth changes from point to point in complicated ways; changes in altitude of the surface, as well as changes in density between land and sea or between different kinds of rock, lead to small but measurable variations in the Earth’s gravity. All of the local features of the gravitational field of the Earth actually contain quite a bit of information.
Black holes are not like that. Once they form, any bumps and wiggles in the stuff they formed from are erased. There might be a short period right when the formation happens, when the black hole hasn’t quite settled down, but it quickly becomes smooth and featureless. Once it has settled, there are three things that we can measure about a black hole: its total mass, how fast it is spinning, and how much electric charge it has. (Real astrophysical black holes usually have close to zero net electric charge, but they are often spinning very rapidly.) And that’s it. Two collections of stuff with the same mass, charge, and spin, once they get turned into black holes, will become completely indistinguishable, as far as classical general relativity is concerned. This interesting prediction of general relativity is summed up in a cute motto coined by John Wheeler, the same guy who gave black holes their name: “Black holes have no hair.”
This no-hair business should set off some alarm bells. Apparently, if everything we’ve just said is true, the process of forming a black hole has a dramatic consequence: Information is lost. We can take two very different kinds of initial conditions (one solar mass of hot gas, or one solar mass of ice cream), and they can evolve into precisely the same final condition (a one-solar-mass black hole). But up until now we’ve been saying that the microscopic laws of physics—of which Einstein’s equation of general relativity is presumably one—have the property that they conserve information. Put another way: Making a black hole seems to be an irreversible process, even though Einstein’s equation would appear to be perfectly reversible.
You are right to worry! This is a time puzzle. In classical general relativity, there is a way out: We can say that the information isn’t truly lost; it’s just lost to you, as it’s hidden behind the event horizon of a black hole. You can decide for yourself whether that seems satisfying or sounds like a cop-out. Either way, we can’t just stop there, as Hawking will eventually tell us that black holes evaporate once quantum mechanics is taken into account. Then we have a serious problem, one that has launched a thousand theoretical physics papers.210
LAWS OF BLACK-HOLE MECHANICS
You might think that, because nothing can escape from a black hole, it’s impossible for its total mass to ever decrease. But that’s not quite right, as shown by a very imaginative idea due to Roger Penrose. Penrose knew that black holes could have spin and charge as well as mass, so he asked a reasonable question: Can we use that spin and charge to do useful work? Can we, in other words, extract energy from a black hole by decreasing its spin and charge? (When we think of black holes as single objects at rest, we can use “mass” and “energy” interchangeably, with E = mc2 lurking in the back of our minds.)
The answer is yes, at least at the thought-experiment level we’re working at here. Penrose figured out a way we could throw things close to a spinning black hole and have them emerge with more energy than they had when they went in, slowing the black hole’s rotation in the process of lowering its mass. Essentially, we can convert the spin of the black hole into useful energy. A stupendously advanced civilization, with access to a giant, spinning black hole, would have a tremendous source of energy available for whatever public-works projects they might want to pursue. But not an unlimited source—there is a finite amount of energy we can extract by this process, since the black hole will eventually stop spinning altogether. (In the best-case scenario, we can extract about 29 percent of the total energy of the original rapidly spinning black hole.)
So: Penrose showed that black holes are systems from which we can extract useful work, at least up to a certain point. Once the black hole has no spin, we’ve used up all the extractable energy, and the hole just sits there. Those words should sound vaguely familiar from our previous discussions of thermodynamics.
Stephen Hawking followed up on Penrose’s work to show that, while it’s possible to decrease the mass/energy of a spinning black hole, there is a quantity that always either increases or remains constant: the area of the event horizon, which is basically the size of the black hole. The area of the horizon depends on a particular combination of the mass, spin, and charge, and Hawking found that this particular combination never decreases, no matter what we do. If we have two black holes, for example, they can smash into each other and coalesce into a single black hole, oscillating wildly and giving off gravitational radiation.211 But the area of the new event horizon is always larger than the combined area of the original two horizons—and, as an immediate consequence of Hawking’s result, one big black hole can therefore never split into two smaller ones, since the area would go down.212 For a given amount of mass, we get the maximum-area horizon from a single, uncharged, nonrotating black hole.
So: While we can extract useful work from a black hole up to a point, there is some quantity (the area of the event horizon) that keeps going up during the process and reaches its maximum value when all the useful work has been extracted. Interesting. This really does sound eerily like thermodynamics.
Enough with the suggestive implications; let’s make this analogy explicit.213 Hawking showed that the area of the event horizon of a black hole never decreases; it either increases or stays constant. That’s much like the behavior of entropy, according to the Second Law of Thermodynamics. The First Law of Thermodynamics is usually summarized as “energy is conserved,” but it actually tells us how different forms of energy combine to make the total energy. There is clearly an analogous rule for black holes: The total mass is given by a formula that includes contributions from the spin and charge.
There is also a Third Law of Thermodynamics: There is a minimum possible temperature, absolute zero, at which the entropy is also a minimum. What, in the case of black holes, is supposed to play the role of “temperature” in this analogy? The answer is the surface gravity of a black hole—how strong the gravitational pull of the hole is near the event horizon, as measured by an observer very far away. You might think the surface gravity should be infinite—isn’t that the whole point of a black hole? But it turns out that the surface gravity is really a measure of how dramatically spacetime is curved near the event horizon, and it actually gets weaker as the black hole gets more and more massive.214 And there is a minimum value for the surface gravity of a black hole—zero!—which is achieved when all of the black-hole energy comes from charge or spin, none from “mass all by itself.”
Finally, there is a Zeroth Law of Thermodynamics: If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other. The analogous statement for black holes is simply “the surface gravity has the same value everywhere on the event horizon of a stationary black hole.” And that’s true.
So there is a perfect analogy between the laws of thermodynamics, as they were developed over the course of the 1800s, and the “laws of black-hole mechanics,” as they were developed in the 1970s. The different elements of the analogy are summarized in the table.
But now we’re faced with an important question, the kind that leads to big breakthroughs in science: How seriously should we take this analogy? Is it just an amusing coincidence, or does it reflect some deep underlying truth?
This is a legitimate question, not just a cheap setup for a predictable answer. Coincidences happen sometimes. When scientists stumble across an intriguing connection between two apparently unrelated things, like thermodynamics and black holes, that may be a clue to an important discovery, or it may just be an accident. Different people might have different gut feelings about whether or not such a deep connection is out there to be found. Ultimately, we should be able to attack the problem scientifically and come to a conclusion, but the answer is not obvious ahead of time.
BEKENSTEIN’S ENTROPY CONJECTURE
It was Jacob Bekenstein, then a graduate student working with John Wheeler, who took the analogy between thermodynamics and black-hole mechanics most seriously. Wheeler, when he wasn’t busy coining pithy phrases, was enthusiastically pushing forward the field of quantum gravity (and general relativity overall) at a time when the rest of the physics community was more interested in particle physics—the heroic days of the 1960s and 1970s, when the Standard Model was being constructed. Wheeler’s influence has been felt not only through his ideas—he and Bryce DeWitt were the first to generalize the Schrödinger equation of quantum mechanics to a theory of gravity—but also through his students. In addition to Bekenstein, Wheeler was the Ph.D. supervisor for an impressive fraction of the scientists who are now leading researchers in gravitational physics, including Kip Thorne, Charles Misner, Robert Wald, and William Unruh—not to mention Hugh Everett, as well as Wheeler’s first student, one Richard Feynman.
So Princeton in the early 1970s was a fruitful environment to be thinking about black holes, and Bekenstein was in the thick of it. In his Ph.D. thesis, he made a simple but dramatic suggestion: The relationship between black-hole mechanics and thermodynamics isn’t simply an analogy; it’s an identity. In particular, Bekenstein used ideas from information theory to argue that the area of a black-hole event horizon isn’t just like the entropy; it is the entropy of the black hole.215
On the face of it, this suggestion seems a little hard to swallow. Boltzmann told us what entropy is: It characterizes the number of microscopic states of a system that are macroscopically indistinguishable. “Black holes have no hair” seems to imply that there are very few states for a large black hole; indeed, for any specified mass, charge, and spin, the black hole is supposed to be unique. But here is Bekenstein, saying that the entropy of an astrophysical-sized black hole is staggeringly large.
Figure 59: Jacob Bekenstein, who first suggested that black holes have entropy.
The area of an event horizon has to be measured in some kind of units—acres, hectares, square centimeters, what have you. Bekenstein claimed that the entropy of a black hole was approximately equal to the area of its event horizon as measured in units of the Planck area. The Planck length, 10-33 centimeters, is the very tiny distance at which quantum gravity is supposed to become important; the Planck area is just the Planck length squared. For a black hole with a mass comparable to the Sun, the area of the event horizon is about 1077 Planck areas. That’s a big number; an entropy of 1077 would be larger than the regular entropy in all of the stars, gas, and dust in the entire Milky Way galaxy.
At a superficial level, there is a pretty straightforward route to reconciling the apparent tension between the no-hair idea and Bekenstein’s entropy idea: Classical general relativity is just not correct, and we need quantum gravity to understand the enormous number of states implied by the amount of black hole entropy. Or, to put it more charitably, classical general relativity is kind of like thermodynamics, and quantum gravity will be needed to uncover the microscopic “statistical mechanics” understanding of entropy in cases when gravity is important. Bekenstein’s proposal seemed to imply that there are really jillions of different ways that spacetime can arrange itself at the microscopic quantum level to make a macroscopic classical black hole. All we have to do is figure out what those ways are. Easier said than done, as it turns out; more than thirty-five years later, we still don’t have a firm grasp on the nature of those microstates implied by the black-hole entropy formula. We think that a black hole is like a box of gas, but we don’t know what the “atoms” are—although there are some tantalizing clues.
But that’s not a deal-breaker. Remember that the actual Second Law was formulated by Carnot and Clausius before Boltzmann ever came along. Maybe we are in a similar stage of progress right now where quantum gravity is concerned. Perhaps the properties of mass, charge, and spin in classical general relativity are simply macroscopic observables that don’t specify the full microstate, just as temperature and pressure are in ordinary thermodynamics.
In Bekenstein’s view, black holes are not some weird things that stand apart from the rest of physics; they are thermodynamic systems just like a box of gas would be. He proposed a “Generalized Second Law,” which is basically the ordinary Second Law with black-hole entropy included. We can take a box of gas with a certain entropy, throw it into a black hole, and calculate what happens to the total entropy before and after. The answer is: It goes up, if we accept Bekenstein’s claim that the black-hole entropy is proportional to the area of the event horizon. Clearly such a scenario has some deep implications for the relationship between entropy and spacetime, which are worth exploring more carefully.
Along with Wheeler’s group at Princeton, the best work in general relativity in the early 1970s was being done in Great Britain. Stephen Hawking and Roger Penrose, in particular, were inventing and applying new mathematical techniques to the study of curved spacetime. Out of these investigations came the celebrated singularity theorems—when gravity becomes sufficiently strong, as in black holes or near the Big Bang, general relativity necessarily predicts the existence of singularities—as well as Hawking’s result that the area of black-hole event horizons would never decrease.
So Hawking paid close attention to Bekenstein’s work, but he wasn’t very happy with it. For one thing, if you’re going to take the analogy between area and entropy seriously, you should take the other parts of the thermodynamics/black-hole-mechanics analogy just as seriously. In particular, the surface gravity of a black hole (which is large for small black holes with negligible spin and charge, smaller for large black holes or ones with substantial spin or charge) should be proportional to its temperature. But that would seem to be, on the face of it, absurd. When you heat things up to high temperature, they glow, like molten metal or a burning flame. But black holes don’t glow; they’re black. So there, we can imagine Hawking thinking across the Atlantic.
Inveterate traveler that he is, Hawking visited the Soviet Union in 1973 to talk about black holes. Under the leadership of Yakov Zel’dovich, Moscow featured a group of experts in relativity and cosmology that rivaled those in Princeton or Cambridge. Zel’dovich and his colleague Alexander Starobinsky told Hawking about some work they had done to understand the Penrose process—extracting energy from a rotating black hole—in the light of quantum mechanics. According to the Moscow group, quantum mechanics implied that a spinning black hole would spontaneously emit radiation and lose energy; there was no need for a super-advanced civilization to throw things at it.
Hawking was intrigued but didn’t buy the specific arguments that Zel’dovich and Starobinsky had offered.216 So he set out to understand the implications of quantum mechanics in the context of black holes by himself. It’s not a simple problem. “Quantum mechanics” is a very general idea: The space of states consists of wave functions rather than positions and momenta, and you can’t observe the wave function exactly without dramatically altering it. Within that framework, we can think of different types of quantum systems, from individual particles to collections of superstrings. The founders of quantum mechanics focused, sensibly enough, on relatively simple systems, consisting of a small number of atoms moving slowly with respect to one another. That’s still what most physics students learn when they first study quantum mechanics.
When particles become very energetic and start moving at speeds close to the speed of light, we can no longer ignore the lessons of relativity. For one thing, the energy of two particles that collide with each other can become so large that they create multiple new particles, through the miracle of E = mc2. Through decades of effort on the part of theoretical physicists, the proper formalism to reconcile quantum mechanics with special relativity was assembled, in the form of “quantum field theory.”
The basic idea of quantum field theory is simple: The world is made of fields, and when we observe the wave functions of those fields, we see particles. Unlike a particle, which exists at some certain point, a field exists everywhere in space; the electric field, the magnetic field, and the gravitational field are all familiar examples. At every single point in space, every field that exists has some particular value (although that value might be zero). According to quantum field theory, everything is a field—there is an electron field, various kinds of quark fields, and so on. But when we look at a field, we see particles. When we look at the electric and magnetic fields, for example, we see photons, the particles of electromagnetism. A weakly vibrating electromagnetic field shows up as a small number of photons; a wildly vibrating electromagnetic field shows up as a large number of photons.217
Figure 60: Fields have a value at every point in space. When we observe a quantum field, we don’t see the field itself, but a collection of particles. A gently oscillating field, at the top, corresponds to a small number of particles; a wildly oscillating field, at the bottom, corresponds to a large number of particles.
Quantum field theory reconciles quantum mechanics with special relativity. This is very different from “quantum gravity,” which would reconcile quantum mechanics with general relativity, the theory of gravity and spacetime curvature. In quantum field theory, we imagine that spacetime itself is perfectly classical, whether it’s curved or not; the fields are subject to the rules of quantum mechanics, while spacetime simply acts as a fixed background. In full-fledged quantum gravity, by contrast, we imagine that even spacetime has a wave function and is completely quantum mechanical. Hawking’s work was in the context of quantum field theory in a fixed curved spacetime.
Field theory was not something Hawking was an expert in; despite being lumped with general relativity under the umbrella of “impressive-sounding theories of modern physics that seem inscrutable to outsiders,” the two areas are quite different, and an expert in one might not know much about the other. So he set out to learn. Sir Martin Rees, who is one of the world’s leading theoretical astrophysicists and currently Astronomer Royal of Britain, was at the time a young scientist at Cambridge; like Hawking, he had received his Ph.D. a few years before under the supervision of Dennis Sciama. By this time, Hawking was severely crippled by his disease; he would ask for a book on quantum field theory, and Rees would prop it up in front of him. While Hawking stared silently at the book for hours on end, Rees wondered whether the toll of his condition was simply becoming too much for him.218
Far from it. In fact, Hawking was applying the formalism of quantum field theory to the question of radiation from black holes. He was hoping to derive a formula that would reproduce Zel’dovich and Starobinsky’s result for rotating black holes, but instead he kept finding something unbelievable: Quantum field theory seemed to imply that even nonrotating black holes should radiate. Indeed, they should radiate in exactly the same way as a system in thermal equilibrium at some fixed temperature, with the temperature being proportional to the surface gravity, just as it had been in the analogy between black holes and thermodynamics.
Hawking, much to his own surprise, had proven Bekenstein right. Black holes really do behave precisely as ordinary thermodynamic objects. That means, among other things, that the entropy of a black hole actually is proportional to the area of its event horizon; that connection is not just an amusing coincidence. In fact, Hawking’s calculation (unlike Bekenstein’s argument) allowed him to pinpoint the precise constant of proportionality: ¼. That is, if LP is the Planck length, so LP is the Planck area, the entropy of a black hole is 1/4 of the area of its horizon as measured in units of the Planck area:
SBH = A/(4LP2).
You are allowed to imagine that the subscript BH stands for “Black Hole” or “Bekenstein-Hawking,” as you prefer. This formula is the single most important clue we have about the reconciliation of gravitation with quantum mechanics.219 And if we want to understand why entropy was small near the Big Bang, we have to understand something about entropy and gravity, so this is a logical starting point.
To really understand how Hawking derived the startling result that black holes radiate requires a subtle mathematical analysis of the behavior of quantum fields in curved space. Nevertheless, there is a favorite hand-waving explanation that conveys enough of the essential truth that everyone in the world, including Hawking himself, relies upon it. So why not us?
The primary idea is that quantum field theory implies the existence of “virtual particles” as well as good old-fashioned real particles. We encountered this idea briefly in Chapter Three, when we were talking about vacuum energy. For a quantum field, we might think that the state of lowest energy would be when the field was absolutely constant—just sitting there, not changing from place to place or time to time. If it were a classical field, that would be right, but just as we can’t pin down a particle to one particular position in quantum mechanics, we can’t pin down a field to one particular configuration in quantum field theory. There will always be some intrinsic uncertainty and fuzziness in the value of the quantum field. We can think of this inherent jitter in the quantum field as particles popping in and out of existence, one particle and one antiparticle at a time, so rapidly that we can’t observe them. These virtual particles can never be detected directly; if we see a particle, we know it’s a real one, not a virtual one. But virtual particles can interact with real (non-virtual) ones, subtly altering their properties, and those effects have been observed and studied in great detail. They really are there.
What Hawking figured out is that the gravitational field of a black hole can turn virtual particles into real ones. Ordinarily, virtual particles appear in pairs: one particle and one antiparticle.220 They appear, persist for the briefest moment, and then annihilate, and no one is the wiser. But a black hole changes things, due to the presence of the event horizon. When a virtual particle/antiparticle pair pops into existence very close to the horizon, one of the partners can fall in, and obviously has no choice but to continue on to the singularity. The other partner, meanwhile, is now able to escape to infinity. The event horizon has served to rip apart the virtual pair, gobbling up one of the particles. The one that escapes is part of the Hawking radiation.
At this point a crucial property of virtual particles comes into play: Their energy can be anything at all. The total energy of a virtual particle/antiparticle pair is exactly zero, since they must be able to pop into and out of the vacuum. For real particles, the energy is equal to the mass times the speed of light squared when the particle is at rest, and grows larger if the particle is moving; consequently, it can never be negative. So if the real particle that escapes the black hole has positive energy, and the total energy of the original virtual pair was zero, that means the partner that fell into the black hole must have a negative energy. When it falls in, the total mass of the black hole goes down.
Eventually, unless extra energy is added from other sources, a black hole will evaporate away entirely. Black holes are not, as it turns out, places were time ends once and for all; they are objects that exist for some period of time before they eventually disappear. In a way, Hawking radiation has made black holes a lot more ordinary than they seemed to be in classical general relativity.
Figure 61: Hawking radiation. In quantum field theory, virtual particles and antiparticles are constantly popping in and out of the vacuum. But in the vicinity of a black hole, one of the particles can fall into the event horizon, while the other escapes to the outside world as Hawking radiation.
An interesting feature of Hawking radiation is that smaller black holes are hotter . The temperature is proportional to the surface gravity, which is greater for less massive black holes. The kinds of astrophysical black holes we’ve been talking about, with masses equal to or much greater than that of the Sun, have extremely low Hawking temperatures; in the current universe, they are not evaporating at all, as they are taking in a lot more energy from objects around them than they are losing energy from Hawking radiation. That would be true even if the only external source of energy were the cosmic microwave background, at a temperature of about 3 Kelvin. In order for a black hole to be hotter than the microwave background is today, it would have to be less than about 1014 kilograms—about the mass of Mt. Everest, and much smaller than any known black hole.221 Of course, the microwave background continues to cool down as the universe expands; so if we wait long enough, the black holes will be hotter than the surrounding universe, and begin to lose mass. As they do so, they heat up, and lose mass even faster; it’s a runaway process and, once the black hole has been whittled down to a very small size, the end comes quickly in a dramatic explosion.
Unfortunately, the numbers make it very hard for Stephen Hawking to win the Nobel Prize for predicting black hole radiation. For the kinds of black holes we know about, the radiation is far too feeble to be detected by an observatory. We might get very lucky and someday detect an extremely tiny black hole emitting high-energy radiation, but the odds are against it222—and you win Nobel Prizes for things that are actually seen, not just for good ideas. But good ideas come with their own rewards.
The fact that black holes evaporate away raises a deep question: What happens to the information that went into making the hole in the first place? We mentioned this puzzling ramification of the no-hair principle for black holes in classical general relativity: No matter what might have gone into the black hole, once it forms the only features it has are its mass, charge, and spin. Previous chapters made a big deal about the fact that the laws of physics preserve the information needed to specify a state as the universe evolves from moment to moment. At first blush, a black hole would seem to destroy that information.
Imagine that, in frustration at the inability of modern physics to provide a compelling explanation for the arrow of time, you throw your copy of this book onto an open fire. Later, you worry that you might have been a bit hasty, and you want to get the book back. Too bad, it’s already burnt into ashes. But the laws of physics tell us that all the information contained in the book is still available in principle, no matter how hard it might be to reconstruct in practice. The burning book evolved into a very particular arrangement of ashes and light and heat; if we could exactly capture the complete microstate of the universe after the fire, we could theoretically run the clock backward and figure out whether the book that had burned was this one or, for example, A Brief History of Time. (Laplace’s Demon would know which book it was.) That’s very theoretical, because the entropy increased by a large amount along the way, but in principle it could happen.
If instead of throwing the book into a fire, we had thrown it into a black hole, the story would be different. According to classical general relativity, there is no way to reconstruct the information; the book fell into a black hole, and we can measure the resulting mass, charge, and spin, but nothing more. We might console ourselves that the information is still in there somewhere, but we can’t get to it.
Once Hawking radiation is taken into account, this story changes. Now the black hole doesn’t last forever; if we’re sufficiently patient, it will completely evaporate away. If information is not lost, we should be in the same situation we were in with the fire, where in principle it’s possible to reconstruct the contents of the book from properties of the outgoing radiation.
Figure 62: Information (for example, a book) falls into a black hole, and should be conveyed outward in the Hawking radiation. But how can it be in two places at the same time?
The problem with that expectation arises when we think about how Hawking radiation originates from virtual particles near the event horizon of a black hole. Looking at Figure 62 we can imagine a book falling through the horizon, all the way to the singularity (or whatever should replace the singularity in a better theory of quantum gravity), taking the information contained on its pages along with it. Meanwhile, the radiation that purportedly carries away the same information has already left the black hole. How can the information be in two places at once?223 As far as Hawking’s calculation is concerned, the outgoing radiation is the same for every kind of black hole, no matter what went into making it. At face value, it would appear that the information is simply destroyed; it would be as if, in our earlier checkerboard examples, there was a sort of blob that randomly spit out gray and white squares without any consideration for the prior state.
This puzzle is known as the “black hole information-loss paradox.” Because direct experimental information about quantum gravity is hard to come by, thinking about ways to resolve this paradox has been a popular pastime among theoretical physicists over the past few decades. It has been a real controversy within the physics community, with different people coming down on different sides of the debate. Very roughly speaking, physicists who come from a background in general relativity (including Stephen Hawking) have tended to believe that information really is lost, and that black hole evaporation represents a breakdown of the conventional rules of quantum mechanics; meanwhile, those from a background in particle physics and quantum field theory have tended to believe that a better understanding would show that the information was somehow preserved.
In 1997, Hawking and fellow general-relativist Kip Thorne made a bet with John Preskill, a particle theorist from Caltech. It read as follows:
Whereas Stephen Hawking and Kip Thorne firmly believe that information swallowed by a black hole is forever hidden from the outside universe, and can never be revealed even as the black hole evaporates and completely disappears,
And whereas John Preskill firmly believes that a mechanism for the information to be released by the evaporating black hole must and will be found in the correct theory of quantum gravity,
Therefore Preskill offers, and Hawking/Thorne accept, a wager that:
When an initial pure quantum state undergoes gravitational collapse to form a black hole, the final state at the end of black hole evaporation will always be a pure quantum state.
The loser(s) will reward the winner(s) with an encyclopedia of the winner’s choice, from which information can be recovered at will.
Stephen W. Hawking, Kip S. Thorne, John P. Preskill
Pasadena, California, 6 February 1997
In 2004, in a move that made newspaper headlines, Hawking conceded his part of the bet; he admitted that black hole evaporation actually does preserve information. Interestingly, Thorne has not (as of this writing) conceded his own part of the bet; furthermore, Preskill accepted his winnings (Total Baseball: The Ultimate Baseball Encyclopedia, 8th edition) only reluctantly, as he believes the matter is still not settled.224
What convinced Hawking, after thirty years of arguing that information was lost in black holes, that it was actually preserved? The answer involves some deep ideas about spacetime and entropy, so we have to lay some background.
HOW MANY STATES CAN FIT IN A BOX?
We are delving into such detail about black holes in a book that is supposed to be about the arrow of time for a very good reason: The arrow of time is driven by an increase in entropy, which ultimately originates in the low entropy near the Big Bang, which is a period in the universe’s history when gravity is fundamentally important. We therefore need to know how entropy works in the presence of gravity, but we’re held back by our incomplete understanding of quantum gravity. The one clue we have is Hawking’s formula for the entropy of a black hole, so we would like to follow that clue to see where it leads. And indeed, efforts to understand black-hole entropy and the information-loss paradox have had dramatic consequences for our understanding of spacetime and the space of states in quantum gravity.
Consider the following puzzle: How much entropy can fit in a box? To Boltzmann and his contemporaries, this would have seemed like a silly question—we could fit as much entropy as we liked. If we had a box full of gas molecules, there would be a maximum-entropy state (an equilibrium configuration) for any particular number of molecules: The gas would be evenly distributed through the box at constant temperature. But we could certainly squeeze more entropy into the box if we wanted to; all we would have to do is add more and more molecules. If we were worried that the molecules took up a certain amount of space, so there was some maximum number we could squeeze into the box, we might be clever and consider a box full of photons (light particles) instead of gas molecules. Photons can be piled on top of one another without limit, so we should be able to have as many photons in the box as we wish. From that point of view, the answer seems to be that we can fit an infinite (or at least arbitrarily large) amount of entropy in any given box. There is no upper limit.
That story, however, is missing a crucial ingredient: gravity. As we put more and more stuff into the box, the mass inside keeps growing.225 Eventually, the stuff we are putting into the box suffers the same fate as a massive star that has exhausted its nuclear fuel: It collapses under its own gravitational pull and forms a black hole. Every time that happens, the entropy increases—the black hole has more entropy than the stuff of which it was made. (Otherwise the Second Law would prevent black holes from forming.)
Unlike boxes full of atoms, we can’t make black holes with the same size but different masses. The size of a black hole is characterized by the “Schwarzschild radius,” which is precisely proportional to its mass.226 If you know the mass, you know the size; contrariwise, if you have a box of fixed size, there is a maximum mass black hole you can possibly fit into it. But if the entropy of the black hole is proportional to the area of its event horizon, that means there is a maximum amount of entropy you can possibly fit into a region of some fixed size, which is achieved by a black hole of that size.
That’s a remarkable fact. It represents a dramatic difference in the behavior of entropy once gravity becomes important. In a hypothetical world in which there was no such thing as gravity, we could squeeze as much entropy as we wanted into any given region, but gravity stops us from doing that.
The importance of this insight comes when we hearken back to Boltzmann’s understanding of entropy as (the logarithm of) the number of microstates that are macroscopically indistinguishable. If there is some finite maximum amount of entropy we can fit into a region of fixed size, that means there are only a finite number of possible states within that region. That’s a deep feature of quantum gravity, radically different from the behavior of theories without gravity. Let’s see where this line of reasoning takes us.
THE HOLOGRAPHIC PRINCIPLE
To appreciate how radical the lesson of black-hole entropy is, we have to first appreciate the cherished principle it apparently overthrows: locality. That’s the idea that different places in the universe act more or less independently of one another. An object at some particular location can be influenced by its immediate surroundings, but not by things that are far away. Distant things can influence one another indirectly, by sending some signal from place to place, such as a disturbance in the gravitational field or an electromagnetic wave (light). But what happens here doesn’t directly influence what happens in some other region of the universe.
Think back to the checkerboards. What happened at one moment in time was influenced by what happened at the previous moment in time. But what happened at one point in “space”—the collection of squares across a single row—was completely unrelated to what happened at any other point in space at the same time. Along any particular row, we were free to imagine any distribution of white and gray squares we chose. There were no rules along the lines of “when there is a gray square here, the square twenty slots to the right has to be white.” And when squares did “interact” with one another as time passed, it was always with the squares right next to them. Similarly, in the real world, things bump into one another and influence other things when they are close by, not when they are far apart. That’s locality.
Locality has an important consequence for entropy. Consider, as usual, a box of gas, and calculate the entropy of the gas in the box. Now let’s mentally divide the box in two, and calculate the entropy in each half. (We don’t need to imagine a physical barrier, just consider the left side of the box and the right side separately.) What is the relationship between the total entropy of the box and the separate entropy of the two halves?
Figure 63: A box of gas, mentally divided into two halves. The total entropy in the box is the sum of the entropies of each half.
The answer is: You get the entropy in the whole box by simply adding the entropy of one half to the entropy of the other half. This would seem to be a direct consequence of Boltzmann’s definition of the entropy—indeed, it’s the entire reason why that definition has a logarithm in it. We have a certain number of allowed microstates in one half of the box, and a certain number in the other half. The total number of microstates is calculated as follows: For every possible microstate of the left side, we are allowed to choose any of the possible microstates on the right side. So we get the total number of microstates by multiplying the number of microstates on the left by the number of microstates on the right. But the entropy is the logarithm of that number, and the logarithm of “X times Y” is “the logarithm of X” plus “the logarithm of Y.”
So the entropy of the total box is simply the sum of the entropies of the two sub-boxes. Indeed, that would work no matter how we divided up the original box, or how many sub-boxes we divided it into; the total entropy is always the sum of the sub-entropies. This means that the maximum entropy we can have in a box is always going to be proportional to the volume of the box—the more space we have, the more entropy we can have, and it scales directly with the addition of more volume.
But notice the sneaky assumption in that argument: We were able to count the number of states in one half of the box, and then multiply by the number in the other half. In other words, what happened in one half of the box was assumed to be totally independent of what happened in the other half. And that is exactly the assumption of locality.
When gravity becomes important, all of this breaks down. Gravity puts an upper limit on the amount of entropy we can squeeze into a box, given by the largest black hole that can fit in the box. But the entropy of a black hole isn’t proportional to the volume enclosed—it’s proportional to the area of the event horizon. Area and volume are very different! If we have a sphere 1 meter across, and we increase it in size until it’s 2 meters across, the volume inside goes up by a factor of 8 (23), but the area of the boundary only goes up by a factor of 4 (22).
The upshot is simple: Quantum gravity doesn’t obey the principle of locality. In quantum gravity, what goes on over here is not completely independent from what goes on over there. The number of things that can possibly go on (the number of possible microstates in a region) isn’t proportional to the volume of the region; it’s proportional to the area of a surface we can draw that encloses the region. The real world, described by quantum gravity, allows for much less information to be squeezed into a region than we would naïvely have imagined if we weren’t taking gravity into account.
This insight has been dubbed the holographic principle. It was first proposed by Dutch Nobel laureate Gerard ’t Hooft and American string theorist Leonard Susskind, and later formalized by German-American physicist Raphael Bousso (formerly a student of Stephen Hawking).227 Superficially, the holographic principle might sound a bit dry. Okay, the number of possible states in a region is proportional to the size of the region squared, not the size of the region cubed. That’s not the kind of line that’s going to charm strangers at cocktail parties.
Here is why holography is important: It means that spacetime is not fundamental. When we typically think about what goes on in the universe, we implicitly assume something like locality; we describe what happens at this location, and at that location, and give separate specifications for every possible location in space. Holography says that we can’t really do that, in principle—there are subtle correlations between things that happen at different locations, which cut down on our freedom to specify a configuration of stuff through space.
An ordinary hologram displays what appears to be a three-dimensional image by scattering light off of a special two-dimensional surface. The holographic principle says that the universe is like that, on a fundamental level: Everything you think is happening in three-dimensional space is secretly encoded in a two-dimensional surface’s worth of information. The three-dimensional space in which we live and breathe could (again, in principle) be reconstructed from a much more compact description. We may or may not have easy access to what that description actually is—usually we don’t, but in the next section we’ll discuss an explicit example where we do.
Perhaps none of this should be surprising. As we discussed in the previous chapter, there is a type of non-locality already inherent in quantum mechanics, before gravity ever gets involved; the state of the universe describes all particles at once, rather than referring to each particle separately. So when gravity is in the game, it’s natural to suppose that the state of the universe would include all of spacetime at once. But still, the type of non-locality implied by the holographic principle is different than that of quantum mechanics alone. In quantum mechanics, we could imagine particular wave functions in which the state of a cat was entangled with the state of a dog, but we could just as easily imagine states in which they were not entangled, or where the entanglement took on some different form. Holography seems to be telling us that there are some things that just can’t happen, that the information needed to encode the world is dramatically compressible. The implications of this idea are still being explored, and there are undoubtedly more surprises to come.
HAWKING GIVES IN
The holographic principle is a very general idea; it should be a feature of whatever theory of quantum gravity eventually turns out to be right. But it would be nice to have one very specific example that we could play with to see how the consequences of holography work themselves out. For example, we think that the entropy of a black hole in our ordinary three-dimensional space is proportional to the two-dimensional area of its event horizon; so it should be possible, in principle, to specify all of the possible microstates corresponding to that black hole in terms of different things that could happen on that two-dimensional surface. That’s a goal of many theorists working in quantum gravity, but unfortunately we don’t yet know how to make it work.
In 1997, Argentine-American theoretical physicist Juan Maldacena revolutionized our understanding of quantum gravity by finding an explicit example of holography in action.228 He considered a hypothetical universe nothing like our own—for one thing, it has a negative vacuum energy (whereas ours seems to have a positive vacuum energy). Because empty space with a positive vacuum energy is called “de Sitter space,” it is convenient to label empty space with a negative vacuum energy “anti-de Sitter space.” For another thing, Maldacena considered five dimensions instead of our usual four. And finally, he considered a very particular theory of gravitation and matter—“supergravity,” which is the supersymmetric version of general relativity. Supersymmetry is a hypothetical symmetry between bosons (force particles) and fermions (matter particles), which plays a crucial role in many theories of modern particle physics; happily, the details aren’t crucial for our present purposes.
Maldacena discovered that this theory—supergravity in five-dimensional anti- de Sitter space—is completely equivalent to an entirely different theory—a four-dimensional quantum field theory without gravity at all. Holography in action: Everything that can possibly happen in this particular five-dimensional theory with gravity has a precise analogue in a theory without gravity, in one dimension less. We say that the two theories are “dual” to each other, which is a fancy way of saying that they look very different but really have the same content. It’s like having two different but equivalent languages, and Maldacena has uncovered the Rosetta stone that allows us to translate between them. There is a one-to-one correspondence between states in a particular theory of gravity in five dimensions and a particular nongravitational theory in four dimensions. Given a state in one, we can translate it into a state in the other, and the equations of motion for each theory will evolve the respective states into new states that correspond to each other according to the same translation dictionary (at least in principle; in practice we can work out simple examples, but more complicated situations become intractable). Obviously the correspondence needs to be nonlocal; you can’t match up individual points in a four-dimensional space to points in a five-dimensional space. But you can imagine matching up states in one theory, defined at some time, to states in the other theory.
If that doesn’t convince you that spacetime is not fundamental, I can’t imagine what would. We have an explicit example of two different versions of precisely the same theory, but they describe spacetimes with different numbers of dimensions! Neither theory is “the right one”; they are completely equivalent to each other.
Maldacena’s discovery helped persuade Stephen Hawking to concede his bet with Preskill and Thorne (although Hawking, as is his wont, worked things out his own way before becoming convinced). Remember that the issue in question was whether the process of black hole evaporation, unlike evolution according to ordinary quantum mechanics, destroys information, or whether the information that goes into a black hole somehow is carried out by the Hawking radiation.
Figure 64: The Maldacena correspondence. A theory of gravity in a five-dimensional anti- de Sitter space is equivalent to a theory without gravity in four-dimensional flat spacetime.
If Maldacena is right, we can consider that question in the context of five-dimensional anti-de Sitter space. That’s not the real world, but the ways in which it differs from the real world don’t seem extremely relevant for the information-loss puzzle—in particular, we can imagine that the negative cosmological constant is very small, and essentially unimportant. So we make a black hole in anti- de Sitter space and then let it evaporate. Is information lost? Well, we can translate the question into an analogous situation in the four-dimensional theory. But that theory doesn’t have gravity, and therefore obeys the rules of ordinary quantum mechanics. There is no way for information to be lost in the four-dimensional nongravitational theory, which is supposed to be completely equivalent to the five-dimensional theory with gravity. So, if we haven’t missed some crucial subtlety, the information must somehow be preserved in the process of black hole evaporation.
That is the basic reason why Hawking conceded his bet, and now accepts that black holes don’t destroy information. But you can see that the argument, while seemingly solid, is nevertheless somewhat indirect. In particular, it doesn’t provide us with any concrete physical understanding of how the information actually gets into the Hawking radiation. Apparently it happens, but the explicit mechanism remains unclear. That’s why Thorne hasn’t yet conceded his part of the bet, and why Preskill accepted his encyclopedia only with some reluctance. Whether or not we accept that information is preserved, there’s clearly more work to be done to understand exactly what happens when black holes evaporate.
A STRING THEORY SURPRISE
There is one part of the story of black-hole entropy that doesn’t bear directly on the arrow of time but is so provocative that I can’t help but discuss it, very briefly. It’s about the nature of black-hole microstates in string theory.
The great triumph of Boltzmann’s theory of entropy was that he was able to explain an observable macroscopic quantity—the entropy—in terms of microscopic components. In the examples he was most concerned with, the components were the atoms constituting a gas in a box, or the molecules of two liquids mixing together. But we would like to think that his insight is completely general; the formula S = k log W, proclaiming that the entropy S is proportional to the logarithm of the number of ways W that we can rearrange the microstates, should be true for any system. It’s just a matter of figuring out what the microstates are, and how many ways we can rearrange them. In other words: What are the “atoms” of this system?
Hawking’s formula for the entropy of a black hole seems to be telling us that there are a very large number of microstates corresponding to any particular macroscopic black hole. What are those microstates? They are not apparent in classical general relativity. Ultimately, they must be states of quantum gravity. There’s good news and bad news here. The bad news is that we don’t understand quantum gravity very well in the real world, so we are unable to simply list all of the different microstates corresponding to a macroscopic black hole. The good news is that we can use Hawking’s formula as a clue, to test our ideas of how quantum gravity might work. Even though physicists are convinced that there must be some way to reconcile gravity with quantum mechanics, it’s very hard to get direct experimental input to the problem, just because gravity is an extremely weak force. So any clue we discover is very precious.
The leading candidate for a consistent quantum theory of gravity is string theory . It’s a simple idea: Instead of the elementary constituents of matter being pointlike particles, imagine that they are one-dimensional pieces of “string.” (You’re not supposed to ask what the strings are made of; they’re not made of anything more fundamental.) You might not think you could get much mileage out of a suggestion like that—okay, we have strings instead of particles, so what?
The fascinating thing about string theory is that it’s a very constraining idea. There are lots of different theories we could imagine making from the idea of elementary particles, but it turns out that there are very few consistent quantum mechanical theories of strings—our current best guess is that there is only one. And that one theory necessarily comes along with certain ingredients—extra dimensions of space, and supersymmetry, and higher-dimensional branes (sort of like strings, but two or more dimensions across). And, most important, it comes with gravity. String theory was originally investigated as a theory of nuclear forces, but that didn’t turn out very well, for an unusual reason—the theory kept predicting the existence of a force like gravity! So theorists decided to take that particular lemon and make lemonade, and study string theory as a theory of quantum gravity.229
If string theory is the correct theory of quantum gravity—we don’t know yet whether it is, but there are promising signs—it should be able to provide a microscopic understanding of where the Bekenstein-Hawking entropy comes from. Remarkably, it does, at least for some certain very special kinds of black holes.
The breakthrough was made in 1996 by Andrew Strominger and Cumrun Vafa, building on some earlier work of Leonard Susskind and Ashoke Sen.230 Like Maldacena, they considered five-dimensional spacetime, but they didn’t have a negative vacuum energy and they weren’t primarily concerned with holography. Instead, they took advantage of an interesting feature of string theory: the ability to “tune” the strength of gravity. In our everyday world, the strength of the gravitational force is set by Newton’s gravitational constant, denoted G. But in string theory the strength of gravity becomes variable—it can change from place to place and time to time. Or, in the flexible and cost-effective world of thought experiments, you can choose to look at a certain configuration of stuff with gravity “turned off ” (G set to zero), and then look at the same configuration with gravity “turned on” (G set to a value large enough that gravity is important).
So Strominger and Vafa looked at a configuration of strings and branes in five dimensions, carefully chosen so that the setup could be analyzed with or without gravity. When gravity was turned on, their configuration looked like a black hole, and they knew what the entropy was supposed to be from Hawking’s formula. But when gravity was turned off, they basically had the string-theory equivalent of a box of gas. In that case, they could calculate the entropy in relatively conventional ways (albeit with some high-powered math appropriate to the stringy stuff they were considering).
And the answer is: The entropies agree. At least in this particular example, a black hole can be smoothly turned into a relatively ordinary collection of stuff, where we know exactly what the space of microstates looks like, and the entropy from Boltzmann’s formula matches that from Hawking’s formula, down to the precise numerical factor.
We don’t have a fully general understanding of the space of states in quantum gravity, so there are still many mysteries as far as entropy is concerned. But in the particular case examined by Strominger and Vafa (and various similar situations examined subsequently), the space of microstates predicted by string theory seems to exactly match the expectation from Hawking’s calculation using quantum field theory in curved spacetime.231 It gives us hope that further investigations along the same lines will help us understand other puzzling features of quantum gravity—including what happened at the Big Bang.