From Eternity to Here (54 page)

But there’s really no good reason to believe that the universe we don’t see matches so precisely with the universe we do see. It might be a simple starting assumption, but it’s nothing more than that. We should be open to the possibility that the universe eventually looks completely different somewhere beyond the part we can see (even if it keeps looking uniform for quite a while before we get to the different parts).

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.
²³⁵
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.”

Figure 65:
What we call “the observable universe” is a patch of space that is “comoving”—it expands along with the universe. We trace back along our light cones to the Big Bang to define the part of the universe we can observe, and allow that region to grow as the universe expands.

Our comoving patch of space is certainly not, strictly speaking, a closed system. If an observer were located at the imaginary fence, they would notice various particles passing in and out of our patch. But on average, the same number and kind of particles would come in and go out, and in the aggregate they would be basically indistinguishable. (The smoothness of the cosmic microwave background convinces us that the uniformity of our universe extends out to the boundary of our patch, even if we’re not sure how far it continues beyond.) 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 SPACE TIME

If our comoving patch defines an approximately closed system, the next step is to think about its space of states. General relativity tells us that space itself, the stage on which particles and matter move and interact, evolves over time. Because of this, the definition of the space of states becomes more subtle than it would have been if spacetime were absolute. Most physicists would agree that information is conserved as the universe evolves, but the way that works is quite unclear in a cosmological context. The essential problem is that more and more things can fit into the universe as it expands, so—naїvely, anyway—it looks as if the space of states is getting bigger. That would be in flagrant contradiction to the usual rules of reversible, information-conserving physics, where the space of states is fixed once and for all.

To grapple with this issue, it makes sense to start with the best understanding we currently have of the fundamental nature of matter, which comes from quantum field theory. 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 wave (which is larger when the wavelength is shorter) becomes so large that it would just collapse to a black hole.

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 can’t exist; it’s that they just don’t matter. Wavelengths larger than the size of our patch are effectively constant throughout the observable universe.

So it’s tempting to take “the space of states of the observable universe” as consisting of “vibrations in all the various quantum fields, with wavelengths larger than the Planck length and smaller than the size of our comoving patch.” The problem is, that’s a space of states that changes as the universe expands. Our patch grows with time, while the Planck length remains fixed. At extremely early times, the universe was very young and expanding very rapidly, and our patch was relatively small. (Exactly how small depends on details of the evolution of the early universe that we don’t know.) There weren’t that many vibrations you could squeeze into the universe at that time. Today, the Hubble length is enormously larger—about 10
⁶⁰
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.

Figure 66:
As the universe expands, it can accommodate more kinds of waves. More things can happen, so the space of states would appear to be growing.

But if a space of states changes with time, the evolution clearly can’t be information conserving and reversible. If there are more possible states today than there were yesterday, and two distinct initial states always evolve into two distinct final states, there must be some states today that didn’t come from anywhere. That means the evolution can’t be reversed, in general. All of the conventional reversible laws of physics we are used to dealing with feature spaces of states that are fixed once and for all, not changing with time. The configuration within that space will evolve, but the space of states itself never changes.

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.

The situation is reminiscent of the puzzle of information loss in black holes. There, we (or Stephen Hawking, more accurately) used quantum field theory in curved spacetime to derive a result—the evaporation of black holes into Hawking radiation—that seemed to destroy information, or at least scramble it. Now in the case of cosmology, the rules of quantum field theory in an expanding universe seem to imply fundamentally irreversible evolution.

I am going to imagine that this puzzle will someday be resolved in favor of information conservation, just as Hawking (although not everyone) now believes is true for black holes. The early universe and the late universe are simply two different configurations of the same physical system, evolving according to reversible fundamental laws within precisely the same space of possible states. The right thing to do, when characterizing the entropy of a system as “large” or “small,” is to compare it to the largest possible entropy—not the largest entropy compatible with some properties the system happens to have at the time. If we were to look at a box of gas and find that all of the gas was packed tightly into one corner, we wouldn’t say, “That’s a high-entropy configuration, as long as we restrict our attention to configurations that are packed tightly into that corner.” We would say, “That’s a very low-entropy configuration, and there’s probably some explanation for it.”

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. If the total space of states remains fixed, it must be the case that many of the possible states of the early universe have an irreducibly quantum-gravitational character, and simply can’t be described in terms of quantum fields on a smooth background. Presumably, a better theory of quantum gravity would help us understand what those states might be, but even without that understanding, the basic principle of information conservation assures us that they must be there, so it seems logical to accept that and try to explain why the early universe had such an apparently low-entropy configuration.

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.” That may be right. Until we have the final answers, the best we can do is follow our intuitions and try to come up with testable predictions that we can compare against data. When it comes to the origin of the universe, we’re not there yet, so it pays to keep an open mind.

LUMPINESS

Because we don’t have quantum gravity all figured out, it’s hard to make definitive statements about the entropy of the universe. But we do have some basic tools at our disposal—the idea that entropy has been increasing since the Big Bang, the principle of information conservation, the predictions of classical general relativity, the Bekenstein-Hawking formula for black-hole entropy—that we can use to draw some reliable conclusions.

One obvious question is: What does a high-entropy state look like when gravity is important? If gravity is not important, high-entropy states are states of thermal equilibrium—stuff tends to be distributed uniformly at a constant temperature. (Details may vary in particular systems, such as oil and vinegar.) There is a general impression that high-entropy states are
smooth
, while lower-entropy states can be
lumpy
. Clearly that’s just a shorthand way of thinking about a subtle phenomenon, but it’s a useful guide in many circumstances.
²³⁷
Notice that the early universe is, indeed, smooth, in accordance with the let’s-ignore-gravity philosophy we just examined.

But in the later universe, when stars and galaxies and clusters form, it becomes simply impossible to ignore the effects of gravity. And then we see something interesting: The casual association of “high-entropy” with “smooth” begins to fail, rather spectacularly.

For many years now, Sir Roger Penrose has been going around trying to convince people that this feature of gravity—things get lumpier as entropy increases in the late universe—is crucially important and should play a prominent role in discussions of cosmology. Penrose became famous in the late 1960s and early 1970s through his work with Hawking to understand black holes and singularities in general relativity, and he is an accomplished mathematician as well as physicist. He is also a bit of a gadfly, and takes great joy in exploring positions that run decidedly contrary to the conventional wisdom in various fields, from quantum mechanics to the study of consciousness.