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

by Sean M. Carroll

  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

  Figure 71: 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.

  In each of these examples, the crucial underlying feature is the dynamical nature of spacetime in general relativity. In a fixed, absolute spacetime (such as Boltzmann would have presumed), it makes sense to imagine a universe filled with gas at a uniform temperature and density—thermal equilibrium everywhere. That’s a high-entropy state, and a natural guess for what the universe “should” look like. It’s no surprise that Boltzmann suggested that our observable universe could be a statistical fluctuation within such a configuration.

  But general relativity changes everything. A gas of uniform density in a static spacetime is not a solution to Einstein’s equation—the universe would have to be either expanding or contracting. Before Einstein came along it made sense to start your thought experiments by fixing the average density of matter, or the total volume of the region under consideration. But in general relativity, these aren’t things you’re allowed to simply keep fixed; they will tend to change with time. One way of thinking about it is to realize that general relativity always gives you a way to increase the entropy of any particular configuration: Make the universe bigger, and let the stuff expand to fill the new volume. The ultimate end of such a process is, of course, empty space. That’s what counts as a “high-entropy” state once we take gravity into account.

  None of these arguments is airtight, of course. They are suggestive of a result that seems to hang together and make sense, once you think it through, but that’s far short of a 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. Gravity is tricky, and there’s a lot we don’t understand about it, so it’s not a good idea to become too emotionally invested in any particular speculative scenario.

  THE REAL WORLD

  Let’s apply these ideas to the real world. If high-entropy states are those that look like empty space, presumably our actual observable universe should be evolving toward such a state. (It is.)


  We have casually been assuming that when things collapse under the force of gravity, they end up as a black hole before ultimately evaporating away. It’s far from obvious that this holds true in the real world, where we see lots of objects held together by gravity, but that are very far from being black holes—planets, stars, and even galaxies.

  But the reality is, all of these things will eventually “evaporate” if we wait long enough. We can see this most clearly in the case of a galaxy, which can be thought of as a collection of stars orbiting under their mutual gravitational pull. As individual stars pass by other stars, they interact much like molecules in a box of gas, except that the interaction is solely through gravity (apart from those very rare instances when the stars smack right into one another). These interactions can exchange energy from one star to the other.248 Over the course of many such encounters, stars will occasionally pick up so much energy that they reach escape velocity and fly away from the galaxy altogether. The rest of the galaxy has now lost some of its energy, and as a consequence it shrinks, so that its stars are clustered more tightly together. Eventually, the remaining stars are going to be packed so closely that they all fall into a black hole at the center. From that point, we return to our previous story.

  Similar logic works for any other object in the universe, even if the details might differ. The basic point is that, given some rock or star or planet or what have you, that particular physical system wants to be in the highest-entropy arrangement of the constituents from which it is made. That’s a little poetic, as inanimate objects don’t really have desires, but it reflects the reality that an unfettered evolution of the system would naturally bring it into a higher-entropy configuration.

  You might think that the evolution is, in fact, fettered: A planet, for example, might have a higher entropy if its entire mass collapsed into a black hole, but the pressure inside keeps it stable. Here’s where the miracle of quantum mechanics comes in. Remember that a planet isn’t really a collection of classical particles; it’s described by a wave function, just like everything else. That wave function characterizes the probability that we will find the constituents of the planet in any of their possible configurations. One of those possible configurations, inevitably, will be a black hole. In other words, from the point of view of someone observing the planet (or anything else), there is a tiny chance they will find that it has spontaneously collapsed into a black hole. That’s the process known as “quantum tunneling.”

  Do not be alarmed. Yes, it’s true, just about everything in the universe—the Earth, the Sun, you, your cat—has a chance of quantum-tunneling into the form of a black hole at any moment. But the chance is very small. It would be many, many times the age of the universe before there were a decent chance of it happening. But in a universe that lasts for all eternity, that means the chances are quite good that it will eventually happen—it’s inevitable, in fact. No collection of particles can simply sit undisturbed in the universe forever. The lesson is that matter will find a way to transform into a higher-entropy configuration, if one exists; it might be via tunneling into the form of a black hole, or through much more mundane channels. 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

  As we discussed back in Chapter Three, there’s more than matter and radiation in the universe—there’s also dark energy, responsible for making the universe accelerate. We don’t know for sure what the dark energy is, but the leading candidate is “vacuum energy,” also known as the cosmological constant. Vacuum energy is simply a constant amount of energy inherent in every cubic centimeter of space, one that remains fixed throughout space and time.

  The existence of dark energy both simplifies and complicates our ideas about high-entropy states in the presence of gravity. I’ve been suggesting that the natural behavior of matter is to disperse away into empty space, which is therefore the best candidate for a maximum-entropy state. In a universe like ours, with a vacuum energy that is small but greater than zero, this conclusion becomes even more robust. A positive vacuum energy imparts a perpetual impulse to the expansion of the universe, which helps the general tendency of matter and radiation to dilute away. If, within the next few years, human beings perfect an immortality machine and/or drug, cosmologists who live forever will have to content themselves with observing an increasingly empty universe. Stars will die out, black holes will evaporate, and everything will be pushed away by the accelerating effects of vacuum energy.

  In particular, if the dark energy is really a cosmological constant (rather than something that will ultimately fade away), we can be sure that the universe will never re-collapse into a Big Crunch of any sort. After all, the universe is not only expanding but also accelerating, and that acceleration will continue forever. This scenario—which, let’s not forget, is the most popular prognosis for the real world according to contemporary cosmologists—vividly highlights the bizarre nature of our low-entropy beginnings. We’re contemplating a universe that has existed for a finite time in the past but will persist forever into the future. The first few tens of billions of years of its existence are a hot, busy, complex and interesting mess, which will be followed by an infinite stretch of cold, empty quietness. (Apart from the occasional statistical fluctuation; see next section.) Although it’s not much more than a gut feeling, it just seems like a waste to face the prospect of an endless duration of dark loneliness after a relatively exciting few years in our universe’s past.

  The existence of a positive cosmological constant allows us to actually prove a somewhat rigorous result, rather than just spinning through a collection of thought experiments. The cosmic no-hair theorem states that, under the familiar set of “reasonable assumptions,” a universe with a positive vacuum energy plus some matter fields will, if it lasts long enough for the vacuum energy to take over, eventually evolve into empty universe with nothing but vacuum energy. The cosmological constant always wins, in other words.249

  The resulting universe—empty space with a positive vacuum energy—is known as de Sitter space, after Dutch physicist Willem de Sitter, one of the first after Einstein to study cosmology within the framework of general relativity. As we mentioned back in Chapter Three, empty space with zero vacuum energy is known as Minkowski space, while empty space with a negative vacuum energy is anti- de Sitter space. Even though spacetime is empty in de Sitter space, it’s still curved, because of the positive vacuum energy. The vacuum energy, as we know, imparts a perpetual impulse to the expansion of space. If we consider two particles initially at rest in de Sitter space, they will gradually be pulled apart by the expansion. Likewise, if we trace their motion into the past, they would have been coming toward each other, but ever more slowly as the space between them was pushed apart. Anti-de Sitter space is the reverse; particles are pulled toward each other.

  Figure 72: Three different versions of “empty space,” with different values of the vacuum energy: Minkowski space when the vacuum energy vanishes, de Sitter when it is positive, and anti-de Sitter when it is negative. In Minkowski space, two particles initially at rest will stay motionless with respect to each other; in de Sitter space they are pushed apart, while in anti-de Sitter space they are pulled together. The larger the magnitude of the vacuum energy, the stronger the pushing or pulling.

  Everything we’ve been arguing points to the idea that de Sitter space is the ultimate endpoint of cosmological evolution when the vacuum energy is positive, and hence the highest-entropy state we can think of in the presence of gravity. That’s not a definitive statement—the state of the art isn’t sufficiently advanced to allow for definitive statements along these lines—but it’s suggestive.

  You might wonder how empty space can have a large entropy—entropy is supposed to count the number of ways we can rearrange microstates, and what is there to rearrange if all we have is empty space? But this is just
the same puzzle that faced us with black holes. The answer must be that there are a large number of microstates describing the quantum states of space itself, even when it’s empty. Indeed, if we believe in the holographic principle, we can assign a definite value to the entropy contained within any observable patch of de Sitter space. The answer is a huge number, and the entropy is larger when the vacuum energy is smaller.250 Our own universe is evolving toward de Sitter space, and the entropy of each observable patch will be about 10120. (The fact that this is the same as the entropy we would get by collapsing all the matter in the observable universe into a black hole is a coincidence—it’s the same coincidence as the fact that the matter density and vacuum energy are approximately equal at the present time, even though the matter dominated in the past and the vacuum energy will dominate in the future.)

  While de Sitter space provides a sensible candidate for a high-entropy state, the idea of vacuum energy complicates our attempts to understand entropy in the context of quantum gravity. The basic problem is that the effective vacuum energy—what you would actually measure as the energy of the vacuum at any particular event in spacetime—can certainly change, at least temporarily. Cosmologists talk about the “true vacuum,” in which the vacuum energy takes on its lowest possible value, but also various possible “false vacua,” in which the effective vacuum energy is higher. Indeed, it’s possible that we might be in a false vacuum right now. The idea that “high entropy” means “empty space” becomes a lot more complicated when empty space can take on different forms, corresponding to different values of the vacuum energy.

  That’s a good thing—we don’t want empty space to be the highest-entropy state possible, because we don’t live there. In the next couple of chapters we’re going to see whether we can’t take advantage of different possible values of vacuum energy to somehow make sense of the universe. But first we need to assure ourselves that, without some strategy along those lines, it really would be extremely surprising that we don’t live in a universe that is otherwise empty. And that calls for another visit with some of the giants on whose shoulders we are standing, Boltzmann and Lucretius.