From Eternity to Here (51 page)

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 it
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—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?
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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.
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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.
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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.
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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?