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 (23), but the area of the boundary only goes up by a factor of 4 (22).
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.
The leading candidate for a consistent quantum theory of gravity is string theory . It’s a simple idea: Instead of the elementary constituents of matter being pointlike particles, imagine that they are one-dimensional pieces of “string.” (You’re not supposed to ask what the strings are made of; they’re not made of anything more fundamental.) You might not think you could get much mileage out of a suggestion like that—okay, we have strings instead of particles, so what?
The fascinating thing about string theory is that it’s a very constraining idea. There are lots of different theories we could imagine making from the idea of elementary particles, but it turns out that there are very few consistent quantum mechanical theories of strings—our current best guess is that there is only one. And that one theory necessarily comes along with certain ingredients—extra dimensions of space, and supersymmetry, and higher-dimensional branes (sort of like strings, but two or more dimensions across). And, most important, it comes with gravity. String theory was originally investigated as a theory of nuclear forces, but that didn’t turn out very well, for an unusual reason—the theory kept predicting the existence of a force like gravity! So theorists decided to take that particular lemon and make lemonade, and study string theory as a theory of quantum gravity.229
If string theory is the correct theory of quantum gravity—we don’t know yet whether it is, but there are promising signs—it should be able to provide a microscopic understanding of where the Bekenstein-Hawking entropy comes from. Remarkably, it does, at least for some certain very special kinds of black holes.
The breakthrough was made in 1996 by Andrew Strominger and Cumrun Vafa, building on some earlier work of Leonard Susskind and Ashoke Sen.230 Like Maldacena, they considered five-dimensional spacetime, but they didn’t have a negative vacuum energy and they weren’t primarily concerned with holography. Instead, they took advantage of an interesting feature of string theory: the ability to “tune” the strength of gravity. In our everyday world, the strength of the gravitational force is set by Newton’s gravitational constant, denoted G. But in string theory the strength of gravity becomes variable—it can change from place to place and time to time. Or, in the flexible and cost-effective world of thought experiments, you can choose to look at a certain configuration of stuff with gravity “turned off ” (G set to zero), and then look at the same configuration with gravity “turned on” (G set to a value large enough that gravity is important).
So Strominger and Vafa looked at a configuration of strings and branes in five dimensions, carefully chosen so that the setup could be analyzed with or without gravity. When gravity was turned on, their configuration looked like a black hole, and they knew what the entropy was supposed to be from Hawking’s formula. But when gravity was turned off, they basically had the string-theory equivalent of a box of gas. In that case, they could calculate the entropy in relatively conventional ways (albeit with some high-powered math appropriate to the stringy stuff they were considering).
And the answer is: The entropies agree. At least in this particular example, a black hole can be smoothly turned into a relatively ordinary collection of stuff, where we know exactly what the space of microstates looks like, and the entropy from Boltzmann’s formula matches that from Hawking’s formula, down to the precise numerical factor.
We don’t have a fully general understanding of the space of states in quantum gravity, so there are still many mysteries as far as entropy is concerned. But in the particular case examined by Strominger and Vafa (and various similar situations examined subsequently), the space of microstates predicted by string theory seems to exactly match the expectation from Hawking’s calculation using quantum field theory in curved spacetime.231 It gives us hope that further investigations along the same lines will help us understand other puzzling features of quantum gravity—including what happened at the Big Bang.
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THE LIFE OF THE UNIVERSE
Time is a great teacher, but unfortunately it kills all its pupils.
—Hector Berlioz
What should the universe look like?
This might not be a sensible question. The universe is a unique entity; it’s different in kind from the things we typically think about, all of which exist in the universe. Objects within the universe belong to larger collections of objects, all of which share common properties. By observing these properties we can get a feel for what to expect from that kind of thing. We expect that cats usually have four legs, ice cream is usually sweet, and supermassive black holes lurk at the centers of spiral galaxies. None of these expectations is absolute; we’re talking about tendencies, not laws of nature. But our experience teaches us to expect that certain kinds of things usually have certain properties, and in tho
se unusual circumstances where our expectations are not met, we might naturally be moved to look for some sort of explanation. When we see a three-legged cat, we wonder what happened to its other leg.
The universe is different. It’s all by itself, not a member of a larger class. (Other universes might exist, at least for certain definitions of “universe”; but we certainly haven’t observed any.) So we can’t use the same kind of inductive, empirical reasoning—looking at many examples of something, and identifying common features—to justify any expectations for what the universe should be like.232
Nevertheless, scientists declare that certain properties of the universe are “natural” all the time. In particular, I’m going to suggest that the low entropy of the early universe is surprising, and argue that there is likely to be an underlying explanation. When we notice that an unbroken egg is in a low-entropy configuration compared to an omelet, we have recourse to a straightforward explanation: The egg is not a closed system. It came out of a chicken, which in turn is part of an ecosystem here on Earth, which in turn is embedded in a universe that has a low-entropy past. But the universe, at least at first glance, does seem to be a closed system—it was not hatched out of a Universal Chicken or anything along those lines. A truly closed physical system with a very low entropy is surprising and suggests that something bigger is going on.233
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 36