From Eternity to Here

by Sean Carroll

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.

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.”

IRREVERSIBLE MOTIONS

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?

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.

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.

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.

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.)

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.

A SPECIAL BEGINNING

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.

A SYMMETRIC UNIVERSE

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.

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.

BEFORE THE B...

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.

AN ARROW FOR ALL TIME

A MIDDLE HYPOTHESIS

BABY UNIVERSES

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 reliably calculate the relative frequencies of different unreliable events. If we introduce complications into that picture, all bets are off. (Most bets, anyway.)

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.

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.)

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 inflation-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 u...

A RESTLESS MULTIVERSE

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.

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.

BRINGING IT HOME

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.

16

EPILOGUE

What happens here happens everywhere—as Richard Feynman put it, “The entire universe is in a glass of wine, if we look at it closely enough.”

WHAT’S THE ANSWER?

Our universe isn’t a fluctuation around an equilibrium background, or it would look very different. And it doesn’t seem likely that the fundamental laws of physics are irreversible at a microscopic level—or, if they are, it’s very hard to see how that could actually account for the evolution of entropy and complexity we observe in our universe. A boundary condition stuck at the beginning of time is impossible to rule out, but also seems to be avoiding the question more than answering it. It may ultimately be the best we can do, but I strongly suspect that the low entropy of our early universe is a clue to something deeper, not just a brute fact we can do no more than accept. We’re left with the possibility that our observable universe is part of a much larger structure, the multiverse. By situating what we see inside a larger ensemble, we open the possibility of explaining our apparently finely tuned beginning without imposing any fine-tuning on the multiverse as a whole. That move isn’t sufficient, of course; we need to show why there should be a consistent entropy gradient, and why that gradient should be manifested in a universe that looks like our own, rather than in some other way. We discussed a specific model of which I am personally fond: a universe that is mostly high-entropy de Sitter space, but which gives birth to disconnected baby universes, allowing the entropy to increase without bound and creating patches of spacetime like the one around us along the way. T...

There is one other approach lurking in the background, which we occasionally acknowledged but never granted our undivided attention: the idea that “time” itself is simply an approximation that is occasionally useful, including in our local universe, but doesn’t have any fundamental meaning. This is a perfectly legitimate possibility. Lessons from the holographic principle, as well as a general feeling that the underlying ingredients of a quantum mechanical theory may appear very different from what shows up in the classical regime, make it quite reasonable to imagine that time might be an emergent phenomenon rather than a necessary part of our ultimate description of the world.

THE EMPIRICAL CIRCLE

Scientists are fiercely proud of the empirical nature of what they do. Scientific theories do not become accepted because they are logical or beautiful, or fulfill some philosophical goal cherished by the scientist. Those might be good reasons why a theory is proposed—but being accepted is a much higher standard. Scientific theories must, at the end of the day, fit the data. No matter how intrinsically compelling a theory might be, if it fails to fit the data, it’s a curiosity, not an achievement.

THE MULTIVERSE IS NOT A THEORY

The correct way to think about the multiverse is as a prediction . The theory—such as it is, in its current underdeveloped state—is the marriage of the principles behind quantum field theory to our basic understanding of how curved spacetime works. Starting from those inputs, we don’t simply theorize that the universe could have undergone an early period of superfast acceleration; we predict that inflation should occur, if a quantum inflaton field with the right properties finds itself in the right state. Likewise, we don’t simply say, “Wouldn’t it be cool if there were an infinite number of different universes?” Rather, we predict on the basis of reasonable extrapolations of gravity and quantum field theory that a multiverse really should exist.

THE SEARCH FOR MEANING IN A PREPOSTEROUS UNIVERSE

We don’t know exactly what happened 14 billion years ago, but there’s no reason whatsoever to doubt that we will eventually figure it out.

NEXT STEPS

Predicting the future isn’t easy. (Curse the absence of a low-entropy future boundary condition!) But the pieces are assembled for science to take dramatic steps toward answering the ancient questions we have about the past and the future. It’s time we understood our place within eternity.

APPENDIX:

MATH

EXPONENTIALS

BIG NUMBERS

LOGARITHMS

NOTES

BIBLIOGRAPHY