From Eternity to Here

by Sean Carroll

BEKENSTEIN’S ENTROPY CONJECTURE

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.

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.

HAWKING RADIATION

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.

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.

EVAPORATION

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.

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.

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.

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.

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.

INFORMATION LOSS?

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.

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.

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.”

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.

HOW MANY STATES CAN FIT IN A BOX?

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.

Consider the following puzzle: How much entropy can fit in a box?

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.

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.)

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.

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.

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.

THE HOLOGRAPHIC PRINCIPLE

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 ...

The upshot is simple: Quantum gravity doesn’t obey the principle of locality.

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.

Here is why holography is important: It means that spacetime is not fundamental.

Everything you think is happening in three-dimensional space is secretly encoded in a two-dimensional surface’s worth of information.

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.

HAWKING GIVES IN

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.

If that doesn’t convince you that spacetime is not fundamental, I can’t imagine what would.

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.

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.

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.

A STRING THEORY SURPRISE

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?

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.