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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.
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THE LIFE OF THE UNIVERSE
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.
A truly closed physical system with a very low entropy is surprising and suggests that something bigger is going on.
OUR HOT, SMOOTH EARLY DAYS
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
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So let’s forget about the rest of the universe, and concentrate on the part we can see—what we’ve been calling “the observable universe.” It stretches about 40 billion light-years around us.
But since the universe is expanding, the stuff within what we now call the observable universe was packed into a smaller region in the past. What we do is erect a kind of imaginary fence around the stuff within our currently observable universe, and keep track of what’s inside the fence, allowing the fence itself to expand along with the universe (and be smaller in the past). This is known as our comoving patch of space, and it’s what we have in mind when we say “our observable universe.”
So for all practical purposes, it’s okay to think of our comoving patch as a closed system. It’s not really closed, but it evolves just as if it were—there aren’t any important influences from the outside that are affecting what goes on inside.
CONSERVATION OF INFORMATION IN AN EXPANDING SPACETIME
Fields vibrate in various ways, and we perceive the vibrations as particles. So when we ask, “What is the space of states in a particular quantum field theory?” we need to know all the different ways that the fields can vibrate.
Any possible vibration of a quantum field can be thought of as a combination of vibrations with different specific wavelengths—just as any particular sound can be decomposed into a combination of various notes with specific frequencies. At first you might think that any possible wavelength is allowed, but actually there are restrictions. The Planck length—the tiny distance of 10-33 centimeters at which quantum gravity becomes important—provides a lower limit on what wavelengths are allowed. At smaller distances than that, spacetime itself loses its conventional meaning, and the energy of the
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Likewise, there is an upper limit on what wavelengths are allowed, given by the size of our comoving patch. It’s not that vibrations with longer wavelengths ca...
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Today, the Hubble length is enormously larger—about 1060 times larger than the Planck length—and there are a huge number of allowed vibrations. Under this way of thinking, it’s not so surprising that the entropy of the early universe was small, because the maximum allowed entropy of the universe at that time was small—the maximum allowed entropy increases as the universe expands and the space of states grows.
But if a space of states changes with time, the evolution clearly can’t be information conserving and reversible.
We seem to have something of a dilemma. The rules of thumb of quantum field theory in curved spacetime would seem to imply that the space of states grows as the universe expands, but the ideas on which all this is based—quantum mechanics and general relativity—conform proudly to the principle of information conservation. Clearly, something has to give.
All of this confusion arises because we don’t have a complete theory of quantum gravity, and have to make reasonable guesses on the basis of the theories we think we do understand. When those guesses lead to crazy results, something has to give. We gave a sensible argument that the number of states described by vibrating quantum fields changes with time as the universe expands.
Not everyone agrees.236 A certain perfectly respectable school of thought goes something like this: “Sure, information might be conserved at a fundamental level, and there might be some fixed space of states for the whole universe. But who cares? We don’t know what that space of states is, and we live in a universe that started out small and relatively smooth. Our best strategy is to use the rules suggested by quantum field theory, allowing only a very small set of configurations at very early times, and a much larger set at later times.”
LUMPINESS
One obvious question is: What does a high-entropy state look like when gravity is important?
Through gravity’s persistent efforts, what was a highly uniform distribution of matter becomes increasingly lumpy.
Penrose’s point is this: As structure forms in the universe, entropy increases. He puts it this way:
Gravitation is somewhat confusing, in relation to entropy, because of its universally attractive nature. We are used to thinking about entropy in terms of an ordinary gas, where having the gas concentrated in small regions represents low entropy . . . and where in the high-entropy state of thermal equilibrium, the gas is spread uniformly. But with gravity, things tend to be the other way about. A uniformly spread system of gravitating bodies would represent relatively low entropy (unless the velocities of the bodies are enormously high and/or the bodies are very small and/or greatly...
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When gravity is unimportant, increasing entropy tends to smooth things out; when gravity does become important, matter tends to clump together as entropy increases.
THE EVOLUTION OF ENTROPY
Most of the particles in the early universe are photons and neutrinos, moving at or close to the speed of light, so relativity is important. Up to some numerical factors that don’t change the answer very much, the entropy of a hot gas of relativistic particles is simply equal to the total number of such particles. There are about 1088 particles within our comoving patch of universe, so that’s what the entropy was at early times. (It increases a bit along the way, but not by much, so treating the entropy as approximately constant at early times is a good approximation.)
The entropy of the universe today is about ten googols.
The entropy of the universe has increased by an enormous amount since the days when everything was smooth and featureless.
Still, it’s not as big as it could be.
A hundred quintillion googols! The maximum entropy the observable universe could have is at least that large.
If the state of the early universe were simply “chosen randomly” from among all possible states, the chance that it would have looked like it actually did are ridiculously tiny.
The conclusion is perfectly clear: The state of the early universe was not chosen randomly among all possible states.
Everyone in the world who has thought about the problem agrees with that. What they don’t agree on is why the early universe was so special—what is the mechanism that put it in that state? And, since we shouldn’t be temporal chauvinists about it, why doesn’t the same mechanism put ...
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MAXIMIZING ...
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What is the maximum-entropy state into which we can arrange our comoving patch?
Roger Penrose thinks the answer is a black hole.
A black hole doesn’t maximize the total entropy a system can have—it only maximizes the entropy that can be packed into a region of fixed size. Just as the Second Law doesn’t say “entropy tends to increase, not including gravity,” it also doesn’t say “entropy per volume tends to increase.” It just says “entropy tends to increase,” and if that requires a big region of space, then so be it.
A black hole has a lot of entropy, but it evaporates into radiation that has even more entropy.
We can explicitly compare the entropy of the black hole to the entropy of the radiation into which it evaporates—and the entropy of the radiation is higher. By about 33 percent, to be specific.
EMPTY SPACE
So the answer to the question “What does a high-entropy state look like, when we take gravity into account?” isn’t “a lumpy, chaotic maelstrom of black holes,” nor is it even “one single giant black hole.” The highest-entropy states look like empty space, with at most a few particles here and there, gradually diluting away.
An array of black holes cannot remain static. It will either expand and allow the black holes to evaporate away, approaching empty space (top right), or collapse to make a Big Crunch or a single larger black hole (bottom right).
Initial conditions (at bottom) in a universe with both expanding and contracting regions. The expanding regions grow in size and become increasingly dilute. The contracting regions grow denser at first, but at some point will begin to evaporate into the surrounding emptiness.
definitive demonstration of anything. The claim that the entropy of some system within the universe can increase by scattering its elements across a vast expanse of space seems pretty safe. But the conclusion that empty space is therefore the highest-entropy state is more tentative.
THE REAL WORLD
No matter what kind of lump of matter you have in the universe, it can increase in entropy by evaporating into a thin gruel of particles moving away into empty space.
VACUUM ENERGY
Vacuum energy is simply a constant amount of energy inherent in every cubic centimeter of space, one that remains fixed throughout space and time.
