From Eternity to Here: The Quest for the Ultimate Theory of Time

by Sean M. Carroll

  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.

  Figure 67: Roger Penrose, who has done more than anyone to emphasize the puzzle of the low-entropy early universe.

  One of the fields in which Penrose likes to poke holes in cherished beliefs is theoretical cosmology. When I was a graduate student in the late 1980s, theoretical particle physicists and cosmologists mostly took it for granted that some version of inflationary cosmology (discussed in the next chapter) must be true; astronomers tended to be more cautious. Today, this belief is even more common, as evidence from the cosmic microwave background has shown that the small variations in density from place to place in the early universe match very well with what inflation would predict. But Penrose has been consistently skeptical, primarily on the basis of inflation’s failure to explain the low entropy of the early universe. I remember reading one of his papers as a student, and appreciating that Penrose was saying something important but feeling that he had missed the point. It took two decades of thinking about entropy before I became convinced that he has mostly been right all along.

  We don’t have a full picture of the space of microstates in quantum gravity, and correspondingly lack a rigorous understanding of entropy. But there is a simple strategy for dealing with this obstacle: We consider what actually happens in the universe. Most of us believe that the evolution of the observable universe has always been compatible with the Second Law, and entropy has been increasing since the Big Bang, even if we’re hazy on the details. If entropy tends to go up, and if there is some process that happens all the time in the universe, but its time-reverse never happens, it probably represents an increase in entropy.

  An example of this is “gravitational instability” in the late universe. We’ve been tossing around the notion of “when gravity is important” and “when gravity is not important,” but what is the criterion to decide whether gravity is important? Generally, given some collection of particles, their mutual gravitational forces will always act to pull the particles together—the gravitational force between particles is universally attractive. (In contrast, for example, with electricity and magnetism, which can be either attractive or repulsive depending on what kinds of electric charges you are dealing with.238) But there are other forces, generally collected together under the rubric of “pressure,” that prevent everything from collapsing to a black hole. The Earth or the Sun or an egg doesn’t collapse under its own gravitational pull, because each is supported by the pressure of the material inside it. As a rule of thumb, “gravity is important” means “the gravitational pull of a collection of particles overwhelms the pressure that tries to keep them from collapsing.”

  In the very early universe, the temperature is high and the pressure is enormous. 239 The local gravity between nearby particles is too weak to bring them together, and the initial smoothness of the matter and radiation is preserved. But as the universe expands and cools, the pressure drops, and gravity begins to take over. This is the era of “structure formation,” where the initially smooth distribution of matter gradually begins to condense into stars, galaxies, and larger groups of galaxies. The initial distribution was not perfectly featureless; there were small deviations in density from place to place. In the denser regions, gravity pulled particles even closer together, while the less dense regions lost particles to their denser neighbors and became even emptier. 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 spread out, so that the gravitational contributions become insignificant), whereas high entropy is achieved when the gravitating bodies clump together.240

  All of that is completely correct, and represents an important insight. Under certain conditions, such as those that pertain in the universe on large scales today, even though we don’t have a cut-and-dried formula for the entropy of a system including gravity, we can say with confidence that the entropy increases as structure forms and the universe becomes lumpier.

  There is another way of reaching a similar conclusion, through the magic of thought experiments. Take the current macrostate of the universe—some collection of galaxies, dark matter, and so forth, distributed in a certain way through space. But now let’s make a single change: Imagine that the universe is contracting rather than expanding. What should happen?

  It should be clear that what won’t happen is a simple time-reversal of the actual history of the universe from its smooth initial state to its lumpy present—at least, not for the overwhelming majority of microstates within our present macrostate. (If we precisely time-reversed the specific microstate of our present universe, then of course that is exactly what would happen.) If the distribution of matter in our present universe were to start contracting together, individual stars and galaxies would not begin to disperse and smooth out. Instead, the gravitational force between heavy objects would draw them together, and the amount of lumpy structure would actually increase, even as the universe contracted. Black holes would form, and coalesce together to create bigger black holes. There would ultimately be a sort of Big Crunch, but (as Penrose emphasizes) it wouldn’t look anything like the smooth Big Bang from which our universe came. Places where the density was high and black holes formed would crash into a future singularity relatively quickly, while places that were emptier would survive for longer.

  Figure 68: When gravity is unimportant, increasing entropy tends to smooth things out; when gravity does become important, matter tends to clump together as entropy increases.

  This story fits in well with the idea that the space of states within our comoving patch remains fixed, but when the universe is small most of the states cannot be described as vibrating quantum fields in an otherwise smooth space. Such a picture would be completely inadequate to describe the chaotic black-hole-filled mess that we would generically expect from a collapsing universe. But such a messy configuration is just as much an allowed state of the universe as the relatively smooth backgrounds we traditionally deal with in cosmology. Indeed, such a configuration has a higher entropy than a smooth universe (which we know because a collapsing universe would generically evolve into something messy), which means that there are many more microstates of that form than of the form where everything is relatively smooth. Why our actual universe is so atypical is, of course, at the heart of our mystery.

  THE EVOLUTION OF ENTROPY

  We’ve now assembled enough background knowledge to follow Penrose and take a stab at quantifying how the entropy of our universe changes from early times to today. We know the basic story of how our comoving patch evolves—at early times it was small, and full of hot, dense gas that was very close to perfectly smooth, and at later times it is larger and cooler and more dilute, and contains a distribution of stars and galaxies that is quite lumpy
on small scales, although it’s still basically smooth over very large distances. So what is its entropy?

  At early times, when things were smooth, we can calculate the entropy by simply ignoring the influence of gravity. This might seem to go against the philosophy I’ve thus far been advocating, but we’re not saying that gravity is irrelevant in principle—we’re simply taking advantage of the fact that, in practice, our early universe was in a configuration where the gravitational forces between particles didn’t play a significant role in the dynamics. Basically, it was just a box of hot gas. And a box of hot gas is something whose entropy we know how to calculate.

  The entropy of our comoving patch of space when it was young and smooth is:

  Searly ≈ 1088.

  The “≈” sign means “approximately equal to,” as we want to emphasize that this is a rough estimate, not a rigorous calculation. This number comes from simply treating the contents of the universe as a conventional gas in thermal equilibrium, and plugging in the formulas worked out by thermodynamicists in the nineteenth century, with one additional feature: 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.)

  Today, gravity has become important. It is not very accurate to think of the matter in the universe as a gas in thermal equilibrium with negligible gravity; ordinary matter and dark matter have condensed into galaxies and other structures, and the entropy has increased considerably. Unfortunately, we don’t have a reliable formula that tracks the change in entropy during the formation of a galaxy.

  What we do have is a formula for the circumstance in which gravity is at its most important: in a black hole. As far as we know, very little of the total mass of the universe is contained in the form of black holes.241 In a galaxy like the Milky Way, there are a number of stellar-sized black holes (with maybe 10 times the mass of the Sun each), but the majority of the total black hole mass is in the form of a single supermassive black hole at the galactic center. While supermassive black holes are certainly big—often over a million times the mass of the Sun—that’s nothing compared to an entire galaxy, where the total mass might be 100 billion times the mass of the Sun.

  But while only a tiny fraction of the mass of the universe appears to be in black holes, they contain a huge amount of entropy. A single supermassive black hole, a million times the mass of the Sun, has an entropy according to the Bekenstein-Hawking formula of 1090. That’s a hundred times larger than all of the nongravitational entropy in all the matter and radiation in the observable universe.242

  Even though we don’t have a good understanding of the space of states of gravitating matter, it’s safe to say that the total entropy of the universe today is mostly in the form of these supermassive black holes. Since there are about 100 billion (1011 ) galaxies in the universe, it’s reasonable to approximate the total entropy by assuming 100 billion such black holes. (They might be missing from some galaxies, but other galaxies will have larger black holes, so this is not a bad approximation.) With an entropy of 1090 per million-solar-mass black hole, that gives us a total entropy within our comoving patch today of

  Stoday ≈ 10101.

  Mathematician Edward Kasner coined the term googol to stand for 10100, a number he used to convey the idea of an unimaginably big quantity. The entropy of the universe today is about ten googols. (The folks from Google used this number as an inspiration for the name of their search engine; now it’s impossible to refer to a googol without being misunderstood.)

  When we write the current entropy of our comoving patch as 10101, it doesn’t seem all that much larger than its value in the early universe, 1088. But that’s just the miracle of compact notation. In fact, 10101 is ten trillion (1013) times bigger than 1088. 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. What is the maximum value the entropy of the observable universe could have? Again, we don’t know enough to say for sure what the right answer is. But we can say that the maximum entropy must be at least a certain number, simply by imagining that all of the matter in the universe were rearranged into one giant black hole. That’s an allowed configuration for the physical system corresponding to our comoving patch of universe, so it’s certainly possible that the entropy could be that large. Using what we know about the total mass contained in the universe, and plugging once again into the Bekenstein-Hawking entropy formula for black holes, we find that the maximum entropy of the observable universe is at least

  Smax ≈ 10120.

  That’s a fantastically big number. A hundred quintillion googols! The maximum entropy the observable universe could have is at least that large.

  These numbers drive home the puzzle of entropy that modern cosmology presents to us. If Boltzmann is right, and entropy characterizes the number of possible microstates of a system that are macroscopically indistinguishable, it’s clear that the early universe was in an extremely special state. Remember that the entropy is the logarithm of the number of equivalent states, so a state with entropy S is one of 10S indistinguishable states. So the early universe was in one of

  101088

  different states. But it could have been in any of the

  1010120

  possible states that are accessible to the universe. Again, the miracle of typography makes these numbers look superficially similar, but in fact the latter number is enormously, inconceivably larger than the former. 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 the late universe in a similar state? That’s what we’re here to figure out.

  MAXIMIZING ENTROPY

  We’ve established that the early universe was in a very unusual state, which we think is something that demands explanation. What about the question we started this chapter with: What should the universe look like? What is the maximum-entropy state into which we can arrange our comoving patch?

  Roger Penrose thinks the answer is a black hole.

  What about the maximum-entropy state? Whereas with a gas, the maximum entropy of thermal equilibrium has the gas uniformly spread throughout the region in question, with large gravitating bodies, maximum entropy is achieved when all the mass is concentrated in one place—in the form of an entity known as a black hole.243

  You can see why this is a tempting answer. As we’ve seen, in the presence of gravity, entropy increases when things cluster together, rather than smoothing out. A black hole is certainly as densely packed as things can possibly get. As we discussed in the last chapter, a black hole represents the most entropy we can squeeze into a region of spacetime with any fixed size; that was the inspiration behind the holographic principle. And the resulting entropy is undoubtedly a big number, as we’ve seen in the case of a supermassive black hole.

  But in the final analysis, that’s not the best way to think about it.244 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. One of the wonders of general relativity—and a crucial distinction with the absolute spacetime of Newtonian mechanics—is that sizes are never fixed. Even without a complete understanding of entropy, we can get a handle on the answer by following in Penrose’s footsteps, and simply examining the natural evolution of systems toward higher-entropy states.

  Consider a simple example: a collection of matter gathered in one region of an otherwise empty universe, without even any vacuum energy. In other words, a spacetime that is empty almost everywhere, except for some particular place where some matter particles are congregated. Because most of space has no energy in it at all, the universe won’t be expanding or contracting, so nothing really happens outside the region where the matter is located. The particles will contract together under their own gravitational force.

  Let’s imagine that they collapse all the way to a black hole. Along the way, there’s no question that the entropy increases. However, the black hole doesn’t just sit there for all of eternity—it gives off Hawking radiation, gradually shrinking as it loses energy, and eventually evaporating away completely.

  Figure 69: A black hole has a lot of entropy, but it evaporates into radiation that has even more entropy.