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.”
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 n
ow 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 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.
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 m
ight 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.
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 37
