From Eternity to Here (52 page)

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 (2
³
), but the area of the boundary only goes up by a factor of 4 (2
²
).

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.