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

by Sean M. Carroll

  Some evidence that time doesn’t need to have a beginning comes from quantum gravity, and in particular from the holographic principle we talked about in Chapter Twelve.285 Maldacena showed that a particular theory of gravity in five-dimensional anti-de Sitter space is exactly equivalent to a “dual” four-dimensional theory that doesn’t include gravity. There are plenty of questions that are hard to answer in the five-dimensional gravity theory, just like any other model of quantum gravity. But some of these issues become very straightforward from the dual four-dimensional perspective. For example: Does time have a beginning? Answer: no. The four-dimensional theory doesn’t involve gravity at all; it’s just a field theory that lives in some fixed spacetime, and that spacetime extends infinitely far into the past and the future. That’s true even if there are singularities in the five-dimensional gravity theory; somehow, the theory finds a way to continue on beyond them. So we have an explicit example of a complete theory of quantum gravity, where there exists at least one formulation of the theory in which time never begins or ends, but stretches for all eternity. Admittedly, our own universe does not look much like five-dimensional anti-de Sitter space—it has four macroscopic dimensions, and the cosmological constant is positive, not negative. But Maldacena’s example demonstrates that it’s certainly not necessary that spacetime have a beginning, once quantum gravity is taken into account.

  We can also take a less abstract approach to what might have come before the Big Bang. The most obvious strategy is to replace the Bang by some sort of bounce. We imagine that the universe before what we call the Big Bang was actually collapsing and growing denser. But instead of simply continuing to a singular Big Crunch, the universe—somehow—bounced into a phase of expansion, which we experience as the Big Bang.

  The question is, what causes this bounce? It wouldn’t happen under the usual assumptions made by cosmologists—classical general relativity, plus some reasonable restrictions on the kind of matter and energy in the universe. So we have to somehow change those rules. We could simply wave our hands and say “quantum gravity does it,” but that’s a little unsatisfying.

  Figure 83: A bouncing-universe cosmology replaces the singularity of the standard Big Bang by a (more or less) smooth crossover between a contracting phase and an expanding phase.

  Quite a bit of effort in recent years has gone into developing models that smooth out the Big Bang singularity into a relatively gentle bounce.286 Each of these proposals offers the possibility of extending the history of the universe beyond the Big Bang, but in every case it’s still hard to tell whether the model in question really hangs together. That’s life when you’re trying to understand the birth of the universe in the absence of a full theory of quantum gravity.

  But the crucial point is worth keeping in mind: Even if we don’t have one complete and consistent story to tell about how to extend the universe before the Big Bang, cosmologists are hard at work on the problem, and it’s very plausible that they will eventually succeed. And the possibility that the Big Bang wasn’t really the beginning of the universe has serious consequences for the arrow of time.

  AN ARROW FOR ALL TIME

  If the Big Bang was the beginning of time, we have a very clear puzzle: why was the entropy so small at that beginning? If the Big Bang was not the beginning, we still have a puzzle, but a very different one: why was the entropy small at the bounce, which wasn’t even the beginning of the universe? It was just some moment in an eternal history.

  For the most part, modern discussions of bouncing cosmologies don’t address the question of entropy directly.287 But it’s pretty clear that the addition of a contracting phase before the bounce leaves us with two choices: Either the entropy is increasing as the universe approaches the bounce, or it’s decreasing.

  At first glance, we might expect that the entropy should increase as the universe approaches the bouncing phase from the past. After all, if we started with an initial condition in the ultra-far past, we expect entropy to increase as time goes on, even if space is contracting; that’s just the Second Law as it is ordinarily understood, and it would make the arrow of time consistent through the whole history of the universe. This possibility is illustrated in the bottom left plot of Figure 84. Implicitly or explicitly, that’s what many people have in mind when they discuss bouncing cosmologies.

  Figure 84: At the top, the size of a bouncing universe through time; at bottom, two possible scenarios for the evolution of entropy. The entropy could simply rise forever, as shown at bottom left, giving rise to a consistent arrow of time through all eternity. Or it could decrease during the contracting phase before beginning to increase in the expanding phase, as shown at bottom right.

  But a scenario in which the entropy of our comoving patch increases consistently through a universal bounce faces an incredible problem. In conventional Big Bang cosmology, we have the problem that the entropy is relatively small in the current observable universe, and was substantially smaller in the past. This implies a great deal of hidden fine-tuning in the present microstate of the universe, so that entropy would decrease if we used the laws of physics to run it backward in time. But in the bouncing scenario, where we have pushed the “beginning of the universe” infinitely far away, the amount of fine-tuning needed to make this happen becomes infinitely bad. If we believe in reversible laws of physics, we need to imagine a state of the universe today with the property that it could be evolved backward in time forever, with the entropy continually decreasing all the way. That’s a lot to ask.288

  We should also mention a closely related problem. We know that the entropy of our comoving patch immediately after the bounce has to be small—that is, much smaller than it might have been. (From the estimates we made in Chapter Thirteen, it had to be 1088 or smaller, while it might have been as large as 10120.) Which implies that the entropy was as small, or smaller, just before the bounce. If the entropy were large, you wouldn’t get a bounce; you would get a chaotic mess that would have no hope of coming out the other side as the nice smooth universe from which we emerged. So what we have to imagine is that this comoving patch of space had been contracting for an infinitely long time (from the far past to the moment of the bounce), and in that time the entropy was increasing all along, but managed to increase only a tiny bit. That’s not impossible to imagine, but it strikes us as unusual, to say the least.289

  Even if we do allow ourselves to contemplate the possibility of the extraordinary amount of fine-tuning necessary to let entropy increase consistently for all time, we are left with absolutely no good reason why our universe should actually be that way. We have so far provided no justification for why our universe should be finely tuned at all, and now we are suggesting an infinite amount of fine-tuning. This doesn’t really sound like progress.

  A MIDDLE HYPOTHESIS

  So we are led to consider the alternative, portrayed at bottom right in Figure 84: a bouncing universe where entropy decreases during the contracting phase, reaches a minimum value at the bounce, and begins to increase thereafter. Now, perhaps, we are getting somewhere. An explicit model of such a bouncing cosmology was proposed by Anthony Aguirre and Steven Gratton in 2003. They based their construction on inflation and showed that by clever cutting and pasting we could take an inflationary universe that was expanding forward in time and glue it at the beginning to an inflationary universe expanding backward in time, to obtain a smooth bounce.290

  This alternative comes with a dramatic advantage: The behavior of the universe is symmetric in time. Both the size of the universe, and its entropy, would have a minimum value at the bounce, and increase in either direction. Conceptually, that’s a big improvement over any of the other models we’ve contemplated; the underlying time-reversal symmetry of the laws of physics is reflected in the large-scale behavior of the universe. In particular, we avoid the pitfall of temporal chauvinism—the temptation to treat the “initial” state of the universe differently from the “final” state. It was our wish to si
destep that fallacy that led us to contemplate the Gold universe, which was also symmetric about one moment in time. But now that we allow ourselves to think about a possible universe before the Big Bang, the solution seems more acceptable: The universe is symmetric, not because entropy is low at either end of time, but because it’s high at either end.

  Nevertheless, this is a funny universe. The evolution of entropy is responsible for all the various manifestations of the arrow of time, including our ability to remember the past and our feeling that we move through time. In the bouncing-entropy scenario, the arrow of time reverses direction at the bounce. From the perspective of our observable universe, portrayed on the right-hand side of the plots in Figure 84, the past is the low-entropy direction of time, toward the bounce. But observers on the other side of the bounce, which we have (given our own perspective) labeled “contraction” in the plots, would also define the “past” as the direction of time in which entropy was lower—that is, the direction of the bounce. The arrow of time always points in the direction in which entropy is increasing, from the point of view of a local observer. On either side of the bounce, the arrow points toward a “future” in which the universe is expanding and emptying out. To observers on either side, observers on the other side experience time “running backward.” But this mismatch of arrows is completely unobservable—people on one side of the bounce can’t communicate with people on the other, any more than we can communicate with anyone else in our past. Everyone sees the Second Law of Thermodynamics operating normally in his or her observable part of the universe.

  Unfortunately, a bouncing-entropy cosmos is not quite enough to allow us to declare in good conscience that we have solved the problem we set out at the beginning of this chapter. Sure, allowing for a cosmological bounce that is also a minimum point for the entropy of the universe avoids the philosophical pitfall of placing initial conditions and final conditions on a different footing. But it does so at the cost of a new puzzle: Why is the entropy so low in the middle of the history of the universe?

  In other words, the bouncing-entropy model doesn’t, by itself, actually explain anything at all about the arrow of time. Rather, it takes the need for a Past Hypothesis and replaces it with the need for a “Middle Hypothesis.” There is just as much fine-tuning as ever; we are still stuck trying to explain why the configuration of our comoving patch of space found itself in such a low-entropy state near the cosmological bounce. So it would appear that we still have some work to do.

  BABY UNIVERSES

  To make an honest attempt at providing a robust dynamical explanation of the low entropy of our early universe, let’s take it backward. Put aside for a moment what we know about our actual universe, and return to the question we asked in Chapter Thirteen: What should the universe look like? In that discussion, I argued that a natural universe—one that didn’t rely on finely tuned low-entropy boundary conditions at any point, past, present, or future—would basically look like empty space. When we have a small positive vacuum energy, empty space takes the form of de Sitter space.

  The question that any modern theory of cosmology must therefore answer is: Why don’t we live in de Sitter space? It has a high entropy, it lasts forever, and the curvature of spacetime induces a small but nonzero temperature. De Sitter space is empty apart from the thin background of thermal radiation, so for the most part it is completely inhospitable to life; there is no arrow of time, since it’s in thermal equilibrium. There will be thermal fluctuations, just as we would expect in a sealed box of gas in a Newtonian spacetime. Such fluctuations can give rise to Boltzmann brains, or entire galaxies, or whatever other macrostate you have in mind, if you wait long enough. But we do not appear to be such a fluctuation—if we were, the world around us would be as high entropy as it could possibly get, which it clearly is not.

  There is a way out: De Sitter space might not simply stretch on for all eternity, uninterrupted. Something might happen to it. If that were the case, everything we have said about Boltzmann brains would be out the window. That argument made sense only because we knew exactly what kind of system we were dealing with—a gas at a fixed temperature—and we knew that it would last forever, so that even very improbable events would eventually occur, and we could reliable calculate the relative frequencies of different unreliable events. If we introduce complications into that picture, all bets are off. (Most bets, anyway.)

  It’s not hard to imagine ways that de Sitter space could fail to last forever. Remember that the “old inflation” model was basically a period of de Sitter space in the early universe, with a very high energy density provided by an inflaton field stuck in a false vacuum state. As long as there is another vacuum state of lower energy, that de Sitter space will eventually decay via the appearance of bubbles of true vacuum. If bubbles appear rapidly, the false vacuum will completely disappear; if they appear slowly, we’ll end up with a fractal mixture of true-vacuum bubbles in a persistent false-vacuum background.

  In the case of inflation, a crucial point was that the energy density during the de Sitter phase was very high. Here we are interested in the opposite end of the spectrum—where the vacuum energy is extremely low, as it is in our current universe.

  That makes a huge difference. High-energy states naturally like to decay into states of lower energy, but not vice versa. The reason is not because of energy conservation, but because of entropy.291 The entropy associated with de Sitter space is low when the energy density is high, and high when the energy density is low. The decay of high-energy de Sitter space into a state with lower vacuum energy is just the natural evolution of a low-entropy state into a high-entropy one. But we want to know how we might escape from a situation like the one into which our current universe is evolving: empty de Sitter space with a very small vacuum energy, and a very high entropy. Where do we go from there?

  If the correct theory of everything were quantum field theory in a classical de Sitter space background, we’d be pretty much stuck. Space would keep expanding, quantum fields would keep fluctuating, and we’d be more or less in the situation described by Boltzmann and Lucretius. But there is (at least) one possible escape route, courtesy of quantum gravity: the creation of baby universes. If de Sitter space gives birth to a continuous stream of baby universes, each of which starts with a low entropy and expands into a high-entropy de Sitter phase of its own, we could have a natural mechanism for creating more and more entropy in the universe.

  As we’ve reiterated at multiple points, there’s a lot we don’t understand about quantum gravity. But there’s a lot that we do understand about classical gravity, and about quantum mechanics; so we have certain reasonable expectations for what should happen in quantum gravity, even if the details remain to be ironed out. In particular, we expect that spacetime itself should be susceptible to quantum fluctuations. Not only should quantum fields in the de Sitter background be fluctuating, but the de Sitter space itself should be fluctuating.

  One way in which spacetime might fluctuate was studied in the 1990s by Edward Farhi, Alan Guth, and Jemal Guven.292 They suggested that spacetime could not only bend and stretch, as in ordinary classical general relativity, but also split into multiple pieces. In particular, a tiny bit of space could branch off from a larger universe and go its own way. The separate bit of space is, naturally, known as a baby universe. (In contrast to the “pocket universes” mentioned in the last chapter, which remained connected to the background spacetime.)

  We can be more specific than that. The thermal fluctuations in de Sitter space are really fluctuations of the underlying quantum fields; the particles are just what we see when we observe the fields. Let’s imagine that one of those fields has the right properties to be an inflaton—there are places in the potential where the field could sit relatively motionless in a false vacuum valley or a new-inflation plateau. But instead of starting it there, we consider what happens when the field starts at the bottom, where the vacuum energy is very small. Quantum fluctuation
s will occasionally push the field up the potential, from the true vacuum to the false vacuum—not everywhere at once, but in some small region of space.

  What happens when a bubble of false vacuum fluctuates into existence in de Sitter space? To be honest once again, we’re not sure.293 One thing seems likely: Most of the time, the field will simply dissipate away back into its thermal surroundings. Inside, where we’ve fluctuated into the false vacuum, space wants to expand; but the wall separating the inside from the outside of the bubble wants to shrink, and usually it shrinks away quickly before anything dramatic happens.

  Figure 85: Creation of a baby universe via quantum fluctuation of a false-vacuum bubble.