From Eternity to Here: The Quest for the Ultimate Theory of Time
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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 Plan
ck 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.
HAWKING RADIATION
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).