From Eternity to Here (56 page)

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

S
_max
≈ 10
¹²⁰
.

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 10
S
indistinguishable states. So the early universe was in one of

10
¹⁰⁸⁸

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

10
¹⁰¹²⁰

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.

The natural behavior of black holes in an otherwise empty universe is to radiate away into a dilute gas of particles. Because such behavior is natural, we expect it to represent an increase in entropy—and it does. 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.
²⁴⁵

Now, the
density
of entropy has clearly gone down—when we had a black hole, all that entropy was packed into a small volume, whereas the Hawking radiation is emitted gradually and gets spread out over a huge region of space. But again, the density of entropy isn’t what we care about; it’s just the total amount.

EMPTY SPACE

The lesson of this thought experiment is that the rule of thumb “when we take gravity into consideration, higher-entropy states are lumpy rather than smooth” is not an absolute law; it’s merely valid under certain circumstances. The black hole is lumpier (higher contrast) than the initial collection of particles, but the eventual dispersing radiation isn’t lumpy at all. In fact, as the radiation scurries off to the ends of the universe, we approach a configuration that grows ever smoother, as the density everywhere approaches zero.

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.

That’s a counterintuitive claim, worth examining from different angles.
²⁴⁶
The case of a collection of matter that all falls together to form a black hole is a relatively straightforward one, where we can actually plug in numbers and verify that the entropy increases when the black hole evaporates away. But that’s a far cry from proving that the result (an increasingly dilute gas of particles moving through empty space) is really the highest-entropy configuration possible. We should try to contemplate other possible answers. The guiding principles are that we want a configuration that other kinds of configurations naturally evolve into, and that itself persists forever.

What if, for example, we had an array of many black holes? We might imagine that black holes filled the universe, so that the radiation from one black hole eventually fell into another one, which kept them from evaporating away. However, general relativity tells us that such a configuration can’t last. By sprinkling objects throughout the universe, we’ve created a situation where space has to either expand or contract. If it expands, the distance between the black holes will continually grow, and eventually they will simply evaporate away. As before, the long-term future of such a universe simply looks like empty space.

If space is contracting, we have a different story. When the entire universe is shrinking, the future is likely to end in a Big Crunch singularity. That’s a unique case; on the one hand, the singularity doesn’t really last forever (since time ends there, at least as far as we know), but it doesn’t evolve into something else, either. We can’t rule out the possibility that the future evolution of some hypothetical universe ends in a Big Crunch, but our lack of understanding of singularities in quantum gravity makes it difficult to say very much useful about that case. (And our real world doesn’t seem to be behaving that way.)

One clue is provided by considering a collapsing collection of matter (black holes or otherwise) that looks exactly like a contracting universe, but one where the matter only fills a finite region of space, rather than extending throughout it. If the rest of the universe is empty, this local region is exactly like the situation we considered before, where a group of particles collapsed to make a black hole. So what looks from the inside like a universe collapsing to a Big Crunch looks from the outside like the formation of a giant black hole. In that case, we know what the far future will bring: It might take a while, but that black hole will radiate away into nothing. The ultimate state is, once again, empty space.

Figure 70:
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).

We can be a little more systematic about this. Cosmologists are used to thinking of universes that are doing the same thing all throughout space, because the observable part of our own universe seems to be like that. But let’s not take that for granted; let’s ask what could be going on throughout the universe, in perfect generality.

The notion that the space is “expanding” or “contracting” doesn’t have to be an absolute property of the entire universe. If the matter in some particular region of space is moving apart and diluting away, it will look locally like an expanding universe, and likewise for contraction when matter moves together. So if we imagine sprinkling particles throughout an infinitely big space, most of the time we will find that some regions are expanding and diluting, while other regions are contracting and growing more dense.

But if that’s true, a remarkable thing happens: Despite the apparent symmetry between “expanding” and “contracting,” pretty soon the expanding regions begin to win. And the reason is simple: The expanding regions are growing in volume, while the contracting ones are shrinking. Furthermore, the contracting regions don’t stay contracted forever. In the extreme case where matter collapses all the way to a black hole, eventually the black holes just radiate away. So starting from initial conditions that contain both expanding and contracting regions, if we wait long enough we’ll end up with empty space—entropy increasing all the while.
²⁴⁷