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.
EVAPORATION
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.
INFORMATION LOSS?
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 anot
her 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.
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 35
