From Eternity to Here (53 page)

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.

13

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 those 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.
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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

The right attitude toward any apparently surprising feature of the observed universe, such as the low early entropy or the small vacuum energy, is to treat it as a potential clue to a deeper understanding. Observations like this aren’t anywhere near as definitive as a straightforward experimental disagreement with a favored theory; they are merely suggestive. In the backs of our minds, we’re thinking that if the configuration of the universe were chosen randomly from all possible configurations, it would be in a very high-entropy state. It’s not, so therefore the state of the universe isn’t just chosen randomly. Then how is it chosen? Is there some process, some dynamical chain of events, that leads inevitably to the seemingly non-random configuration of our universe?

OUR HOT, SMOOTH EARLY DAYS

If we think of the universe as a physical system in a randomly chosen configuration, the question “What should the universe look like?” is answered by “It should be in a high-entropy state.” We therefore need to understand what a high-entropy state of the universe would look like.

Even this formulation of the question is not quite right. We don’t actually care about the particular state of the universe right this moment; after all, yesterday it was different, and tomorrow it will be different again. What we really care about is the
history
of the universe, its evolution through time. But understanding what would constitute a natural history presupposes that we understand something about the space of states, including what high-entropy states look like.

Cosmologists have traditionally done a very sloppy job of addressing this issue. There are a couple of reasons for this. One is that the expansion of the universe from a hot, dense early state is such an undeniable brute
fact
that, once you’ve become accustomed to the idea, it seems hard to imagine any alternative. You begin to see your task as a theoretical cosmologist as one of explaining why our universe began in the particular hot, dense early state that it did, rather than some different hot, dense early state. This is temporal chauvinism at its most dangerous—unthinkingly trading in the question “Why does the universe evolve in the way it does?” for “Why were the initial conditions of the universe set up the way they were?”

The other thing standing in the way of more productive work on the entropy of the universe is the inescapable role of gravity. By “gravity” we mean everything having to do with general relativity and curved spacetime—everyday stuff like apples falling and planets orbiting stars, but also black holes and the expansion of the universe. In the last chapter, we focused on the one example where we think we know the entropy of an object with a strong gravitational field: a black hole. That example does not seem immediately helpful when thinking about the whole universe, which is not a black hole; it bears a superficial resemblance to a
white
hole (since there is a singularity in the past), but even that is of little help, since we are inside it rather than outside. Gravity is certainly important to the universe, and that’s especially true at early times when space was expanding very rapidly. But appreciating that it’s important doesn’t help us address the problem, so most people simply put it aside.

There is one other strategy, which appears innocent at first but really hides a potentially crucial mistake. That’s to simply separate out gravity from everything else, and calculate the entropy of the matter and radiation within spacetime while forgetting that of spacetime itself. Of course, it’s hard to be a cosmologist and ignore the fact that space is expanding; however, we can take the expansion of space as a given, and simply consider the state of the “stuff ” (particles of ordinary matter, dark matter, radiation) within such a background. The expanding universe acts to dilute away the matter and cool off the radiation, just as if the particles were all contained in a piston that was gradually being pulled out to create more room for them to breathe. It’s possible to calculate the entropy of the stuff in that particular background, exactly as it’s possible to calculate the entropy of a collection of molecules inside an expanding piston.

At any one time in the early universe, we have a gas of particles at a nearly constant temperature and nearly constant density from place to place. In other words, a configuration that looks pretty much like thermal equilibrium. It’s not exactly thermal equilibrium, because in equilibrium nothing changes, and in the expanding universe things are cooling off and diluting away. But compared to the rate at which particles are bumping into one another, the expansion of space is relatively slow, so the cooling off is quite gradual. If we just consider matter and radiation in the early universe, and neglect any effects of gravity other than the overall expansion, what we find is a sequence of configurations that are very close to thermal equilibrium at a gradually declining density and temperature.
²³⁴

But that’s a woefully incomplete story, of course. The Second Law of Thermodynamics says, “The entropy of a closed system either increases or remains constant”; it doesn’t say, “The entropy of a closed system, ignoring gravity, either increases or remains constant.” There’s nothing in the laws of physics that allows us to neglect gravity in situations where it’s important—and in cosmology it’s of paramount importance.

By ignoring the effects of gravity on the entropy, and just considering the matter and radiation, we are led to nonsensical conclusions. The matter and radiation in the early universe was close to thermal equilibrium, which means (neglecting gravity) that it was in its
maximum entropy
state. But today, in the late universe, we’re clearly not in thermal equilibrium (if we were, we’d be surrounded by nothing but gas at constant temperature), so we are clearly
not
in a configuration of maximum entropy. But the entropy didn’t go down—that would violate the Second Law. So what is going on?

What’s going on is that it’s not okay to ignore gravity. Unfortunately, including gravity is not so easy, as there is still a lot we don’t understand about how entropy works when gravity is included. But as we’ll see, we know enough to make a great deal of progress.

WHAT WE MEAN BY OUR UNIVERSE

For the most part, up until now I have stuck to well-established ground: either reviewing things that all good working physicists agree are correct, or explaining things that are certainly true that all good working physicists
should
agree are correct. In the few genuinely controversial exceptions (such as the interpretation of quantum mechanics), I tried to label them clearly as unsettled. But at this point in the book, we start becoming more speculative and heterodox—I have my own favorite point of view, but there is no settled wisdom on these questions. I’ll try to continue distinguishing between certainly true things and more provisional ideas, but it’s important to be as careful as possible in making the case.

First, we have to be precise about what we mean by “our universe.” We don’t see all of the universe; light travels at a finite speed, and there is a barrier past which we can’t see—in principle given by the Big Bang, in practice given by the moment when the universe became transparent about 380,000 years after the Big Bang. Within the part that we do see, the universe is homogenous on large scales; it looks pretty much the same everywhere. There is a corresponding strong temptation to take what we see and extrapolate it shamelessly to the parts we can’t see, and imagine that the entire universe is homogenous throughout its extent—either through a volume of finite size, if the universe is “closed,” or an infinitely big volume, if the universe is “open.”