From Eternity to Here: The Quest for the Ultimate Theory of Time - Sean Carroll (2010)



The eternal silence of these infinite spaces fills me with dread.

—Blaise Pascal, Pensées272

Over the course of this book, we’ve explored the meaning of the arrow of time, as embodied in the Second Law of Thermodynamics, and its relationship with cosmology and the origin of the universe. Finally we have enough background to put it all together and address the question once and for all: Why was the entropy of our observable universe low at early times? (Or even better, so as not to succumb to asymmetric language right from the start: Why do we live in the temporal vicinity of an extremely low-entropy state?)

We’ll address the question, but we don’t know the answer. There are ideas, and some ideas seem more promising than others, but all of them are somewhat vague, and we certainly haven’t yet put the final pieces together. That’s science for you. In fact, that’s the exciting part of science—when you have some clues assembled, and some promising ideas, but are still in the process of nailing down the ultimate answers. Hopefully the prospects sketched in this chapter will serve as a useful guide to wherever cosmologists go next in their attempts to address these deep issues.273

At the risk of being repetitive, let’s review the conundrum one last time, so that we can establish what would count as an acceptable solution to the problem.

All of the macroscopic manifestations of the arrow of time—our ability to turn eggs into omelets but not vice versa, the tendency of milk to mix into coffee but never spontaneously unmix, the fact that we can remember the past but not the future—can be traced to the tendency of entropy to increase, in accordance with the Second Law of Thermodynamics. In the 1870s, Boltzmann explained the microscopic under pinnings of the Second Law: Entropy counts the number of microstates corresponding to each macrostate, so if we start (for whatever reason) in a relatively low-entropy state, it’s overwhelmingly likely that the entropy will increase toward the future. However, due to the fundamental reversibility of the laws of physics, if the only thing we have to go on is the fact that the current state is low entropy, we would with equal legitimacy expect the entropy to have been larger in the past. The real world doesn’t seem to work that way, so we need something else to go on. That something else is the Past Hypothesis: the assumption that the very early universe found itself in an extremely low-entropy state, and we are currently witnessing its relaxation to a state of high entropy. The question of why the Past Hypothesis is true belongs to the realm of cosmology. The anthropic principle is woefully inadequate for the task, since we could easily find ourselves constituted as random fluctuations (Boltzmann brains) in an otherwise empty de Sitter space. Likewise, inflation by itself doesn’t address the question, as it requires an even lower-entropy starting point than the conventional Big Bang cosmology. So the question remains: Why does the Past Hypothesis hold within our observable patch of the universe?

Let’s see if we can’t make some headway on this.


Start with the most obvious hypothesis: Deep down, the fundamental laws of physics simply aren’t reversible. I’ve tried to be careful to allude to the existence of this possibility all along, but have always spoken as if it’s a long shot, or not worthy of our serious attention. There are good reasons for that, though they are not airtight.

A reversible system is one that has a space of states, fixed once and for all, and a rule for evolving those states forward in time that conserves information. Two different states, beginning at some initial time, will evolve predictably into two different states at some specific later time—never into the same future state. That way we can reverse the evolution, since every state the system could currently be in has a unique predecessor at every moment in time.

One way to violate reversibility would be to let the space of states itself actually evolve with time. Perhaps there were simply fewer possible states the universe could have been in at early times, so the small entropy is not so surprising. In that case, many possible microstates that live in the same macrostate as the current universe simply have no possible past state from which they could have come.

Indeed, this is how many cosmologists implicitly speak about what happens in an expanding universe. If we restrict ourselves to “states that look like gentle vibrations of quantum fields around a smooth background,” it’s certainly true that this particular part of the space of states grows with time, as space itself (in the old-fashioned three-dimensional sense of “space”) becomes larger. But that’s very different from imagining that the entire space of states is actually changing with time. Almost nobody would claim to support such a position, if they sat down and thought through what it really meant. I explicitly rejected this possibility when I argued that the early universe was finely tuned—among all the states it could have been in, we included states that look like the universe today, as well as various choices with even higher entropy.

The weirdest thing about the idea that the space of states changes with time is that it requires an external time parameter—a concept of “time” that lives outside the actual universe, and through which the universe evolves. Ordinarily, we think of time as part of the universe—a coordinate on spacetime, measured by various sorts of predictably repetitive clocks. The question “What time is it?” is answered by reference to things going on within the universe—that is to say, to features of the state the universe is currently in. (“The little hand is on the three, and the big hand is on the twelve.”) But if the space of states truly changes with time, that conception becomes insufficient. At any one moment, the universe is actually in one specific state. It makes no sense to say, “The space of states is smaller when the universe is in state X than when it is in state Y.” The space of states, by definition, includes all of the states the universe could hypothetically be in.


Figure 81: On the left we have reversible laws of physics: The system evolves within a fixed space of states, such that different initial states evolve uniquely to different final states. The middle example is irreversible, because the space of states grows with respect to some external time parameter; some states at later times have no predecessors from which they could have come. On the right we have another form of irreversibility, where the space of states remains fixed, but different initial states evolve into the same final state.

So for the space of states to change with time, we would have to posit a notion of time that is not merely measured by features of the state of the universe, but exists outside the universe as we conventionally understand it. Then it would make sense to say, “When this external time parameter was equal to a certain value, the space of states of the universe was relatively small, and when it had progressed to some other value the space of states had grown larger.”

There’s not much to say about this idea. It’s possible, but very few people advocate it as an approach to the arrow-of-time problem.274 It would require a dramatic rethinking of the way we currently understand the laws of physics; nothing about our current framework suggests the existence of a time parameter lurking outside the universe itself. So for now, we can’t rule it out, but it doesn’t give us a warm and fuzzy feeling.


The other way to invent laws of physics that are intrinsically irreversible is to stick once and for all with some space of states, but posit dynamical laws that don’t conserve information. That’s what we saw with checkerboard D back in Chapter Seven; when diagonal lines of gray squares bumped into the vertical line, they simply ceased to exist. There was no way of knowing, from the state at one particular moment in time, precisely what state it came from in the past, since there was no way of reconstructing what diagonals had been lurking around before running afoul of a vertical column.

It’s not hard to come up with slightly more realistic versions of the same idea. In Chapter Eight we contemplated an irreversible game of billiards: a conventional billiards table, where the balls moved forever without losing any energy through friction, except that whenever a ball hit a particular one of the walls of the table, it came perfectly to rest and stayed there forever. The space of states of this system never changes; it always consists of all the possible positions and momenta of the balls on the table. The entropy is defined in the completely conventional way, as the logarithm of the number of states with certain macroscopic properties. But the dynamics are irreversible: Given any one ball stuck to the special wall, we have no way of knowing how long it’s been there. And the entropy of this system flouts the Second Law with impunity; gradually, as more balls get stuck, the system takes up a smaller and smaller portion of the space of states, and the entropy decreases without any intervention from the outside world.

The laws of physics as we know them—putting aside the important question of wave function collapses in quantum mechanics—seem to be reversible. But we don’t know the final laws of physics; all we have are very good approximations. Is it conceivable that the real laws of physics are fundamentally irreversible, and that explains the arrow of time?

First let’s untangle a potential misconception about what that would really mean. To “explain” the arrow of time means to come up with a set of laws of physics, and an “initial” state of the universe, so that we naturally (without fine-tuning) witness a change in entropy over time of the sort we observe around us. In particular, if we simply assume that the initial conditions have low entropy, there is nothing to be explained—the entropy will tend to go up, in accordance with Boltzmann, and we’re done. In that case there’s simply no need to posit irreversible laws of physics; the reversible ones are up to the task. But the problem is that such a low-entropy boundary condition seems unnatural.

So if we wish to explain the arrow of time in a natural way by invoking irreversible fundamental laws, the idea would be to postulate a high-entropy condition—a “generic” state of the universe—and imagine that the laws of physics, when acting on that state, naturally work to decrease its entropy. That would count as a real explanation of the arrow. It might seem that this setup gets it backward—it predicts that entropy goes down, rather than up. But the essence of the arrow of time is simply that entropy changes in a consistent direction. As long as that is true, observers who lived in such a world would always “remember” the direction of time that had a lower entropy; likewise, relationships of cause and effect would always put the causes on the lower-entropy side of things, as that is the direction with fewer allowed choices. In other words, such observers would call the high-entropy end of time the “future,” and the low-entropy end “the past,” even though the fundamental laws of physics in this world would only precisely reconstruct the past from the future, and not vice versa.

Such a universe is certainly conceivable. The problem is, it seems like it would be dramatically different from our universe.

Think carefully about what would have to happen for this scenario to work. The universe, for whatever reason, finds itself in a randomly chosen high-entropy state, which looks like empty de Sitter space. Now our postulated irreversible laws of physics act on that state to decrease the entropy. The result—if all this is to have any chance of working out—should be the history of our actual universe, just reversed in time compared to how we traditionally think about it. In other words: Out of the initial emptiness, some photons miraculously focus on a point in space to create a white hole. That white hole gradually grows in mass through the accretion of additional photons (Hawking radiation in reverse). Gradually a collection of additional white holes come into view from far away, arrayed almost uniformly through space. All of these white holes start belching out gas into the universe, which implodes to make stars, which spiral gently away from the white holes to form galaxies. These stars absorb more radiation from the outside world, and use the energy to break down heavy elements into lighter ones. As the galaxies continue to move toward one another in an increasingly rapid contraction of space, the stars disperse into a uniform distribution of gas. Ultimately the universe collapses to a Big Crunch, as matter and radiation form an extremely smooth and uniform distribution near the end of time.

This is the real history of our observable universe, just played backward in time. It’s a perfectly good solution to the laws of physics as we currently understand them; all we have to do is start with the state near the Big Bang, evolve it forward in time to whatever high-entropy microstate it eventually becomes, and then time-reverse that state. But the hypothesis we’re currently considering is very different: It says that an evolution of this form would happen for almost any high-entropy state of empty de Sitter space. That’s a lot to ask of some laws of physics. It’s one thing to imagine entropy going down as a result of irreversible laws, but it’s another thing entirely to imagine it going down in precisely the right way to produce a time-reversed history of our universe.

We can be more specific about where our discomfort with this scenario comes from. We don’t need to think about the whole universe to experience the arrow of time: It’s right here in our kitchen. We drop an ice cube into a glass of warm water, and the ice melts as the water cools off, eventually reaching a uniform temperature. The fundamental-irreversibility hypothesis claims that this can be explained by the deep laws of physics, starting with the uniformly cool glass of liquid water. In other words, the laws of physics purportedly act on the water to separate out different molecules into the form of an ice cube sitting in a glass of warm water, in precisely the way we would expect had we started with the ice cube and water, only backward in time.

But that’s crazy. For one thing: How does it know? Some glasses of cool water were, five minutes ago, glasses of warm water with ice cubes; but others were just glasses of cool water even five minutes ago. Even though there are relatively few microstates corresponding to each low-entropy macrostate, there are a lot more individual low-entropy macrostates than there are high-entropy ones. (More formally, each low-entropy state contains more information than a high-entropy one.)

The problem is closely tied to the issue of complexity I talked about at the end of Chapter Nine. In the real world, as the universe evolves from a low-entropy Big Bang to a high-entropy future, it creates delicate complex structures along the way. The initially uniform gas doesn’t simply disperse as the universe expands; it first collapses into stars and planets, which increase the entropy locally, and sustain intricate ecosystems and information-processing subsystems along the way.

It’s extremely hard, bordering on impossible, to imagine all of that arising from an initially high-entropy state that gets evolved according to some irreversible laws of physics. This is not an airtight argument, but it seems likely that we will have to look somewhere else for an explanation of the arrow of time in the real world.


From here on, we’ll be operating under the hypothesis that the fundamental laws of physics are truly reversible: The space of allowed states remains fixed, and the dynamical rules of time evolution conserve the information contained in each state. So how can we possibly hope to account for the low-entropy condition in our observable universe?

For Boltzmann, thinking in the context of an absolute Newtonian space and time, this was quite a puzzle. But general relativity and the Big Bang model offer a new possibility, namely: There was a beginning to the universe, including to time itself, and that the beginning state was one with very low entropy. And you’re not allowed to ask why.

Sometimes, the condition “you’re not allowed to ask why” is rephrased as follows: “We posit a new law of nature, which holds that the initial state of the universe had a very low entropy.” It’s not clear why these two formulations are really any different. In our usual understanding of the laws of physics, two ingredients are required to completely specify the evolution of a physical system: a set of dynamical laws that can be used to evolve the system from one state to another through time, and a boundary condition that fixes which state the system is in at some particular moment. But, even though both the laws and the boundary condition are necessary, they seem like very different things; it’s not clear what is to be gained by thinking of the boundary condition as a “law.” A dynamical law demonstrates its validity over and over again; at every moment, the law takes the current state and evolves it into the next state. But the boundary condition is just imposed once and for all; its nature is more like an empirical fact about the universe than an additional law of physics. There isn’t any substantive distinction between the statements “the early universe had a low entropy” and “it is a law of physics that the early universe had a low entropy” (unless we imagine that there are many universes, all with the same boundary condition).275

Be that as it may, it’s undoubtedly possible that this is the most we’ll ever be able to say: The low entropy of the early universe is not to be explained via a better understanding of the dynamical laws of physics, but is simply a brute fact, or (if you prefer) an independent law of nature. An example of this approach has been explicitly advocated by Roger Penrose, who has suggested what he calls the “Weyl curvature hypothesis”—a new law of nature that distinguishes explicitly between spacetime singularities that are in the past and those that are in the future. The basic idea is that past singularities have to be smooth and featureless, while future singularities can be arbitrarily messy and complicated.276 This is an explicit violation of time-reversal symmetry, which would ensure that the Big Bang had a low entropy.

The real problem with a proposal like this is its essentially ad hoc nature.277 Asserting that past singularities had to be very smooth doesn’t help you understand anything else about the universe. It “explains” time asymmetry by putting it in by hand. Nevertheless, one could think of it as a placeholder for a more fundamental understanding: If some deeper principles were uncovered that led to a fundamental distinction between initial and final singularities, such that the curvature of the former were constrained but not the latter, we would certainly have made substantial progress toward understanding the origin of the arrow of time. But even this formulation suggests that the real agenda is to keep looking for something deeper.


If the fundamental laws of physics are reversible, and we don’t allow ourselves to simply impose time-asymmetric boundary conditions, the remaining possibility seems to be that the evolution of the universe actually is time-symmetric itself, despite appearances to the contrary. It’s not hard to imagine how that might happen, if we are open to the possibility that the universe will eventually stop expanding and re-collapse. Before the discovery of dark energy, many cosmologists found a re-collapsing universe philosophically attractive: Einstein and Wheeler, among others, were drawn to the notion of a universe that was finite in both space and time. A future Big Crunch would provide a pleasing symmetry to the history of a universe that began with a Big Bang.

In the conventional picture, however, any such symmetry would be dramatically marred by the Second Law. Everything we know about the evolution of the entropy of the universe is readily explained by assuming that the entropy was very low near the beginning; from there, it naturally increases with time. If the universe were to re-collapse, there is nothing in the known laws of physics that would prevent the entropy from continuing to increase. The Big Crunch would be a messy, high-entropy place, in stark contrast to the pristine smoothness of the Big Bang.

In an attempt to restore the overall symmetry of the history of the universe, people have occasionally contemplated the need for an additional law of physics: a boundary condition in the future (a “Future Hypothesis,” in addition to the Past Hypothesis), which would guarantee that entropy was low near the Crunch as well as the Bang. This idea, suggested by Thomas Gold (better known as a pioneer of the Steady State model) and others, would imply that the arrow of time would reverse at the moment the universe hit its maximum size, and it would always be true that entropy increased in the direction of time toward which the universe was expanding.278

The Gold universe never really caught on among cosmologists, for a simple reason: There’s no good reason for there to be a future boundary condition of any particular sort. Sure, it restores the overall symmetry of time, but nothing we have experienced in the universe demands such a condition, nor does it follow from any other underlying principles.


Figure 82: At the top, the size of a re-collapsing universe through time; at bottom, two possible scenarios for the evolution of entropy. By conventional lights, we would expect the entropy to increase even as the universe collapsed, as shown at bottom left. In the Gold universe, the entropy is constrained to decrease by a low-entropy future boundary condition.

On the other hand—there’s no good reason for there to be a past boundary condition, either, except for the stubborn fact that we need to invoke one to explain the universe we actually see.279 Huw Price has championed the Gold universe as something that cosmologists should take seriously—at least at the level of a thought experiment, if not as a model for the real world—for just this reason.280 We don’t know why entropy was low near the Big Bang, but it was; therefore, the fact that we don’t know why the entropy should be low near a Big Crunch is not a sufficient reason to discard the possibility. Indeed, without introducing time asymmetry by hand, it stands to reason that whatever unknown principle of physics enforces the low entropy at the Bang could also do so at the Crunch.

It’s interesting to approach this scenario like real scientists, and ask whether there could be any testable consequences of a low-entropy future condition. Even if such a condition existed, it would be easy enough to avoid any prospective consequences, just by putting the Big Crunch very far in the future. But if it were relatively near in time (a trillion years from now, say, rather than a googol years), we might be able to see the effects of the future decrease in entropy.281

Imagine, for example, that there was a bright source of light (which we’ll call a “star” for convenience) that lived in the future collapsing phase. How might we detect it? The way we detect an ordinary star is that it emits photons, which travel on light cones radially outward from the star; we absorb the photon in the future of the emitting event, and declare that we see the star. Now let’s run this backward in time.282 We find photons moving radially toward the star in the future; instead of shining, the star sucks light out of the universe.

So you might think that we could “see” the future star by looking in the opposite direction from where the star actually is, and detecting one of the photons that was headed its way. But that’s not right—if we absorb the photon, it never makes it to the star. There is a future boundary condition, which requires that photons be absorbed by the star—not merely that they are headed its way. What we would actually see is our telescopes emitting light out into space, in the direction of the future star.283 If the telescope is pointed in the direction of a future star, it emits light; if it’s not, it remains dark. That’s the time-reverse of the more conventional idea: “If the telescope is pointed in the direction of a past star, it sees light; if it’s not, it doesn’t see anything.”

All this seems crazy; but that’s only because we’re not used to thinking about a world with a future boundary condition. “How does the telescope know to emit light when it’s pointed in the direction of a star that won’t even exist for another trillion years?” That’s what future boundary conditions are all about—they pick out the fantastically tiny fraction of microstates within our current macrostate in which such a seemingly unlikely event happens.284 Deep down, there is nothing stranger about this than there is about the past boundary condition in our actual universe, other than we’re used to one but not the other. (By the way, so far nobody has found any experimental evidence for future stars, or any other evidence of a future low-entropy boundary condition. If they had, you probably would have heard about it.)

Meanwhile, the example of the Gold universe serves more as a cautionary tale than as a serious candidate to account for the arrow of time. If you think you have some natural explanation for why the early universe had such a low entropy, but you claim not to invoke any explicit violations of time-reversal symmetry, why shouldn’t the late universe look the same way? This thought experiment drives home just how puzzling the low-entropy configuration of the Big Bang really is.

The smart money these days is that the universe won’t actually re-collapse. The universe is accelerating; if the dark energy is an absolutely constant vacuum energy (which is the most straightforward possibility), the acceleration will continue forever. We don’t know enough to say for sure, but it’s most likely that our future is absolutely unlike our past. Which, again, places the unusual circumstances surrounding the Big Bang front and center as a puzzle we would like to solve.


We almost seem to have run out of options. If we don’t put in time asymmetry by hand (either in the dynamical laws or in a boundary condition), and the Big Bang has a low entropy, but we don’t insist on a low-entropy future condition—what is left? We seem to be caught in a viselike grip of logic, with no remaining avenues to reconcile the evolution of entropy in our observable universe with the reversibility of the fundamental laws of physics.

There is a way out: We can accept that the Big Bang had a low entropy, but deny that the Big Bang was the beginning of the universe.

This sounds a bit heretical to anyone who has read about the success of the Big Bang model, or who knows that the existence of an initial singularity is a firm prediction of general relativity. We are often told that there is no such thing as “before the Big Bang”—that time itself (as well as space) doesn’t exist prior to the initial singularity. That is, the concept of “prior to the singularity” just doesn’t make any sense.

But as I mentioned briefly in Chapter Three, the idea that the Big Bang is truly the beginning of the universe is simply a plausible hypothesis, not a result established beyond reasonable doubt. General relativity doesn’t predict that space and time didn’t exist before the Big Bang; it predicts that the curvature of spacetime in the very early universe became so large that general relativity itself ceases to be reliable. Quantum gravity, which we can happily ignore when we’re talking about the curvature of spacetime in the relatively placid context of the contemporary universe, absolutely must be taken into account. And, sadly, we don’t understand quantum gravity well enough to say for sure what actually happens at very early times. It might very well be true that space and time “come into existence” in that era—or not. Perhaps there is a transition from a phase of an irredeemably quantum wave function to the classical spacetime we know and love. But it is equally conceivable that space and time extend beyond the moment that we identify as “the Big Bang.” Right now, we simply don’t know; researchers are investigating different possibilities, with an open mind about which will eventually turn out to be right.

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.


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.


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 sidestep 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.


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 fluctuations 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.

Every once in a while, however, we could get lucky. The process of getting lucky is portrayed in Figure 85. What we see is a simultaneous fluctuation of the inflaton field, creating a bubble of false vacuum, and of space itself, creating a region that pinches off from the rest of the universe. The tiny throat that connects the two is a wormhole, as we discussed way back in Chapter Six. But this wormhole is unstable and will quickly collapse to nothing, leaving us with two disconnected spacetimes: the original parent universe and the tiny baby.

Now we have a baby universe, dominated by false vacuum energy, all set up to undergo inflation and expand to a huge size. If the properties of the false vacuum are just right, the energy will eventually be converted into ordinary matter and radiation, and we’ll have a universe that evolves according to the standard i nflation-plus-Big-Bang story. The baby universe can grow to an arbitrarily large size; there is no limitation imposed, for example, by energy conservation. It is a curious feature of general relativity that the total energy of a closed, compact universe is exactly zero, once we account for the energy of the gravitational field as well as everything else. So inflation can take a microscopically tiny ball of space and blow it up to the size of our observable universe, or much larger. As Guth puts it: “Inflation is the ultimate free lunch.”

Of course the entropy of the baby universe starts out very small. That might seem like cheating—didn’t we go to great lengths to argue that there are many degrees of freedom in our observable universe, and all of them still existed when the universe was young, and if we picked a configuration of them randomly it would be preposterously unlikely to obtain a low-entropy state? All that is true, but the process of making a baby universe is not one where we choose the configuration of our universe randomly. It’s chosen in a very specific way: the configuration that is most likely to emerge as a quantum fluctuation in an empty background spacetime that is able to pinch off and become a disconnected universe. Considered as a whole, the entropy of the multiverse doesn’t go down during this process; the initial state is high-entropy de Sitter space, which evolves into high-entropy de Sitter space plus a little extra universe. It’s not a fluctuation of an equilibrium configuration into a lower-entropy state, but a leakage of a high-entropy state into one with an even higher entropy overall.

You might think that the birth of a new universe is a dramatic and painful event, just like the birth of a new person. But it’s actually not so. Inside the bubble, of course, things are pretty dramatic—there’s a new universe where there was none before. But from the point of view of an outside observer in the parent universe, the entire process is almost unnoticeable. What it looks like is a fluctuation of thermal particles that come together to form a tiny region of very high density—in fact, a black hole. But it’s a microscopic black hole, with a tiny entropy, which then evaporates via Hawking radiation as quickly as it formed. The birth of a baby universe is much less traumatic than the birth of a baby human.

Indeed, if this story is true, a baby universe could be born right in the room where you’re reading this book, and you would never notice. It’s not very likely; in all the spacetime of the universe we can currently observe, chances are it never happened. If it did, the action would all be on a microscopic scale. The new universe could grow to a tremendous size, but it would be completely disconnected from the original spacetime. Like some other children, there is absolutely no communication between the baby universe and its parent—once they split, they remain separate forever.


So it’s possible that, even when de Sitter space is in a high-entropy true-vacuum state, it’s not completely stable. Rather, it can give birth to new baby universes, which grow up into large universes in their own right (and could very well give rise to new babies themselves). The original de Sitter space continues on its way, essentially unperturbed.

The prospect of baby universes makes all the difference in the world to the question of the arrow of time. Remember the basic dilemma: The most natural universe to live in is de Sitter space, empty space with a positive vacuum energy, which acts like an eternal box of gas at a fixed temperature. The gas spends most of its time in thermal equilibrium, with rare fluctuations into states of lower entropy. With that kind of setup, we could fairly reliably quantify what kinds of fluctuations there will be, and how often they will happen. Given any particular thing you would want the fluctuation to contain—a person, or a galaxy, or even a hundred billion galaxies—this scenario strongly predicts that most such fluctuations will look like they are in equilibrium, apart from the presence of the fluctuation itself. Furthermore, most such fluctuations will arise from higher-entropy states, and evolve back into higher-entropy states. So most observers will find themselves alone in the universe, having arisen as random arrangements of molecules out of the surrounding high-entropy gas of particles; likewise most galaxies, and so on. You could potentially fluctuate into something that looks just like the history of our Big Bang cosmology; but the number of observers within such a fluctuation is much smaller than the number of observers who are otherwise alone.

Baby universes change this picture in a crucial way. Now it’s no longer true that the only thing that can happen is a thermal fluctuation away from equilibrium and then back again. A baby universe is a kind of fluctuation, but it’s one that never comes back—it grows and cools off, but it doesn’t rejoin the original spacetime.

What we’ve done is given the universe a way that it can increase its entropy without limit. In a de Sitter universe, space grows without bound, but the part of space that is visible to any one observer remains finite, and has a finite entropy—the area of the cosmological horizon. Within that space, the fields fluctuate at a fixed temperature that never changes. It’s an equilibrium configuration, with every process occurring equally as often as its time-reverse. Once baby universes are added to the game, the system is no longer in equilibrium, for the simple reason that there is no such thing as equilibrium. In the presence of a positive vacuum energy (according to this story), the entropy of the universe never reaches a maximum value and stays there, because there is no maximum value for the entropy of the universe—it can always increase, by creating new universes. That’s how the paradox of the Boltzmann-Lucretius scenario can be avoided.

Consider a simple analogy: a ball rolling on a hill. Not a quantum field moving in a potential, an actual down-to-Earth ball. But a very special hill: one that doesn’t ever reach a particular bottom, but rolls smoothly away to infinity. And one on which there is absolutely no friction, so the ball can roll forever with the same amount of total energy.

Now let’s ask ourselves: What should the ball be doing? That is, if we imagine finding such a ball, which has miraculously been operating as an isolated system for all of eternity, undisturbed by the rest of the universe, what kind of state would we expect the ball to be in?


Figure 86: A ball rolling on a hill that doesn’t have a bottom. There is only one kind of trajectory such a ball could have: coming in from infinitely far away in the infinite past, rolling up to a turning point, reversing direction, and rolling back out to infinity in the infinite future.

That may or may not be a sensible question, but it’s not that hard to answer, because there aren’t that many things the ball possibly could be doing. Every allowed trajectory for the ball looks basically the same: It rolls in from infinity, turns around, and rolls back out again. Depending on the total energy that the ball has, the turning point will reach different possible heights up the hill, but the qualitative behavior will be the same. So there will be precisely one moment in the lifetime of the ball when it isn’t moving: the point where it turns around. At every other moment, it’s either moving to the left, or moving to the right. Therefore, when we observe the ball at some random time, it seems very likely that it will be moving in one direction or the other.

Now imagine further that there is an entire tiny civilization living inside the ball, complete with tiny scientists and philosophers. One of their favorite topics of discussion is what they call the “arrow of motion.” These thinkers have noticed that their ball evolves in perfect accord with Newton’s laws of motion. Those laws don’t distinguish between left and right: They are completely reversible. If a ball were to be placed at the bottom of a valley, it would simply sit there forever, motionless. If it were not quite at the bottom, it would start rolling toward the bottom, and then oscillate back and forth in that vicinity. Yet, their ball seems to be rolling consistently in the same direction for very long periods of time! What can be going on?

In case the terms of this somewhat-off-kilter analogy are not immediately clear, the ball represents our universe, and the position from left to right represents entropy. The reason why it’s not surprising to find the ball moving in a consistent direction is that it tends to always be moving in the same direction, with the exception of the one special turnaround point. Despite appearances, the portion of the trajectory where the ball is coming in from right to left is not any different from the portion where it is moving away from left to right; the motion of the ball is time-reversal symmetric around that turning point.

Perhaps the entropy of our universe is like that. The real problem with de Sitter space (without baby universes) is that it’s almost always in equilibrium—any particular observer sees a thermal bath that lasts forever, with predictable fluctuations. More generally, if there exists any such thing as “equilibrium” in the context of cosmology, it’s hard to understand why we don’t find the universe in that state. By suggesting that there is no such thing as equilibrium, we can avoid this dilemma. It becomes natural to observe entropy increasing, simply because entropy can always increase.

This is the scenario suggested by Jennifer Chen and me in 2004.294 We started by assuming that the universe is eternal—the Big Bang is not the beginning of time—and that de Sitter space was a natural high-entropy state for the universe to be in. That means we can “start” with almost any state you like—pick some favorite distribution of matter and energy throughout space, and let it evolve. We put start in quotation marks because we don’t want to prejudice initial conditions over conditions at any other time; respecting the reversibility of the laws of physics, we evolve the state both forward and backward in time. As I’ve argued here, the natural evolution forward in time is for space to expand and empty out, eventually approaching a de Sitter state. But from there, if we wait long enough, we will see the occasional production of baby universes via quantum fluctuations. These baby universes will expand and inflate, and their false vacuum energy will eventually convert into ordinary matter and radiation, which eventually dilutes away until we achieve de Sitter space once again. From there, both the original universe and the new universe can give birth to new babies. This process continues forever. In the parts of spacetime that look like de Sitter, the universe is in equilibrium, and there is no arrow of time. But in baby universes, for the time in between the initial birth and the final cooling off, there is a pronounced arrow of time, as the entropy starts near zero and expands to its equilibrium value.

Most interestingly, the same story can be told backward in time, starting from the initial state, as depicted in Figure 87. If it is not de Sitter already, the universe will empty out backward in time as well as forward. From there it will give birth to baby universes, which expand and cool off. In these baby universes, the arrow of time is oriented in the opposite direction to those in the universes we have put in “the future.” The overall direction of the time coordinate is utterly arbitrary, of course. Observers in the universes at the top of the diagram will think of the bottom of the diagram as “the past,” while observers in the universes at the bottom of the diagram will think of the top as “the past.” Their arrows of time are incompatible, but that doesn’t lead to any Benjamin Button unpleasantness; these baby universes are completely separate from one another in time, and their arrows point away from each other, so no communication between them is possible.

In this scenario, the multiverse on ultra-large scales is symmetric about the middle moment; statistically, at least, the far future and the far past are indistinguishable. In that sense this picture resembles the bouncing cosmologies we discussed earlier; entropy increases forever in both directions of time, around a middle point of lowest entropy. There is a crucial difference, however: The moment of “lowest” entropy is not actually a moment of “low” entropy. That middle moment was not finely tuned to some special very-low-entropy initial condition, as in typical bouncing models. It was as high as we could get, for a single connected universe in the presence of a positive vacuum energy. That’s the trick: allowing entropy to continue to rise in both directions of time, even though it started out large to begin with. There isn’t any state we could possibly have chosen that would have prevented this kind of evolution from happening. An arrow of time is inevitable.295


Figure 87: Baby universes created in a background de Sitter space, both to the past and to the future. Each baby universe starts in a dense, low-entropy state, and exhibits a local arrow of time as it expands and cools. The multiverse manifests overall time-reversal symmetry, as baby universes born in the past have an arrow of time pointing in the opposite direction to those in the future.

Having said all that, we may still want to ask why our patch of observable universe has such a low-entropy boundary condition at one end of time—why were our particular degrees of freedom ever found in such an unnatural state? But in this picture, that’s not quite the right question to ask. We don’t start by knowing which degrees of freedom we are, and then asking why they are (or were) in a certain configuration. Rather, we need to look at the multiverse as a whole, and ask what is most often experienced by observers like us. (If our scenario is going to be useful, the specific definition of “like us” shouldn’t matter.)

This version of the multiverse will feature both isolated Boltzmann brains lurking in the empty de Sitter regions, and ordinary observers found in the aftermath of the low-entropy beginnings of the baby universes. Indeed, there should be an infinite number of both types. So which infinity wins? The kinds of fluctuations that create freak observers in an equilibrium background are certainly rare, but the kinds of fluctuations that create baby universes are also very rare. Ultimately, it’s not enough to draw fun pictures of universes branching off in both directions of time; we need to understand things at a quantitative level well enough to make reliable predictions. The state of the art, I have to admit, isn’t up to that task just yet. But it’s certainly plausible that a lot more observers arise as the baby universes grow and cool toward equilibrium than come about through random fluctuations in empty space.


Does it work? Does a multiverse scenario with baby universes offer a satisfactory explanation for the arrow of time?

We’ve covered a lot of possible approaches to the problem of the arrow of time: a space of states that changes with time, intrinsically irreversible dynamical laws, a special boundary condition, a symmetric re-collapsing universe, a bouncing universe with and without overall time-symmetry, an unbounded multiverse, and of course the Boltzmann-Lucretius scenario of fluctuations around an eternal equilibrium state. The re-collapsing Gold universe seems pretty unlikely on empirical grounds, since the expansion of the universe is accelerating; and the Boltzmann-Lucretius universe also seems ruled out by observation, since the Big Bang had a much lower entropy than it had any right to in that picture. But the other possibilities are still basically on the table; we may find them more or less satisfying, but we can’t be confident enough to dismiss them out of hand. Not to mention the very real possibility that the right answer is something we simply haven’t thought of yet.

It’s hard to tell whether baby universes and the multiverse will ultimately play a role in understanding the arrow of time. For one thing, as I’ve taken pains (perhaps too many) to emphasize, there were many steps along the way where we were fearlessly speculative, to say the least. Our understanding of quantum gravity is not good enough to say for sure whether baby universes really do fluctuate into existence from de Sitter space; there seem to be arguments both for and against. We also don’t completely understand the role of the vacuum energy. We’ve been speaking as if the cosmological constant we observe in our universe today is really the minimum possible vacuum energy, but there is little hard evidence for that assumption. In the context of the string theory landscape, for example, it’s easy enough to get states with the right value of the vacuum energy, but it’s also easy to get all kinds of states, including ones with negative vacuum energy or precisely zero vacuum energy. A more comprehensive theory of quantum gravity and the multiverse would predict how all of these possible states fit together, including transitions between different numbers of macroscopic dimensions as well as different values of the vacuum energy. Not to mention that we haven’t really taken quantum mechanics completely seriously—we’ve nodded in the direction of quantum fluctuations but have drawn pictures of what are essentially classical space times. The right answer, whatever it may turn out to be, will more likely be phrased in terms of wave functions, Schrödinger’s equation, and Hilbert spaces.

The important point is not the prospects of any particular model, but the crucial clue that the arrow of time provides us as we try to understand the universe on the largest possible scales. If the universe we see is really all there is, with the Big Bang as a low-entropy beginning, we seem to be stuck with an uncomfortable fine-tuning problem. Embedding our observable patch in a wider multiverse alleviates this problem by changing the context: The goal is not to explain why the whole universe has a low-entropy boundary condition at the beginning of time, but why there exist relatively small regions of spacetime, arising within a much larger ensemble, where the entropy dramatically increases. That question, in turn, can be answered if the multiverse doesn’t have any state of maximum entropy: The entropy increases because it can always increase, no matter what state we are in. The trick is to set things up so that the mechanism by which entropy increases overall is the production of universes that resemble our own.

The nice thing about a multiverse based on de Sitter space and baby universes is that it avoids all of the standard pitfalls that beset many approaches to the arrow of time: It treats the past and future on an equal footing, doesn’t invoke irreversibility at the level of fundamental dynamics, and never assumes an ad hoc low-entropy state for the universe at any moment in time. It serves as a demonstration that such an explanation is at least conceivable, even if we aren’t yet able to judge whether this particular one is sensible, much less part of the ultimately correct answer. There’s every reason to be optimistic that we will eventually settle on an understanding of how the arrow of time arises naturally and dynamically from the laws of physics themselves.