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

by Sean M. Carroll

  The Roman poet-philosopher Lucretius (c. 50 B.C.E.) was an avid atomist and follower of Epicurus; he was a primary inspiration for Virgil’s poetry. His poem “On the Nature of Things (De Rerum Natura)” is a remarkable work, concerned with elucidating Epicurean philosophy and applying it to everything from cosmology to everyday life. He was especially interested in dispelling superstitious beliefs; imagine Carl Sagan writing in Latin hexameter. A famous section of “On the Nature of Things” counsels against the fear of death, which he sees as simply a transitional event in the endless play of atoms.

  Lucretius applied atomism, and in particular the idea of the swerve, to the question of the origin of the universe. Here is what he imagines happening:

  For surely the atoms did not hold council, assigning

  Order to each, flexing their keen minds with

  Questions of place and motion and who goes where.

  But shuffled and jumbled in many ways, in the course

  Of endless time they are buffeted, driven along,

  Chancing upon all motions, combinations.

  At last they fall into such an arrangement

  As would create this universe.184

  The opening lines here should be read in a semi-sarcastic tone of voice. Lucretius is mocking the idea that the atoms somehow planned the cosmos; rather, they just jumbled around chaotically. But through those random motions, if we wait long enough we will witness the creation of our universe.

  The resemblance to Boltzmann’s scenario is striking. We should always be careful, of course, not to credit ancient philosophers with a modern scientific understanding; they came from a very different perspective, and worked under a different set of presuppositions than we do today. But the parallelism between the creation scenarios suggested by Lucretius and Boltzmann is more than just a coincidence. In both cases, the task was to explain the emergence of the apparent complexity we see around us without appealing to an overall design, but simply by considering the essentially random motions of atoms. It is no surprise that a similar conclusion is reached: the idea that our observable universe is a random fluctuation in an eternal cosmos. It’s perfectly fair to call this the “Boltzmann-Lucretius scenario” for the origin of the universe.

  Can the real world possibly be like that? Can we live in an eternal universe that spends most of its time in equilibrium, with occasional departures that look like what we see around us? Here we need to rely on the mathematical formalism developed by Boltzmann and his colleagues, to which Lucretius didn’t have recourse.

  UN - BREAKING AN EGG

  The problem with the Boltzmann-Lucretius scenario is not that you can’t make a universe that way—in the context of Newtonian spacetime, with everlasting atoms bumping against one another and occasionally giving rise to random downward fluctuations of entropy, it’s absolutely going to happen that you create a region of the size and shape of our universe if you wait long enough.

  The problem is that the numbers don’t work. Sure, you can fluctuate into something that looks like our universe—but you can fluctuate into a lot of other things as well. And the other things win, by a wide margin.

  Rather than weighing down our brains with the idea of a huge collection of particles fluctuating into something like the universe we see around us (or even just our galaxy), let’s keep things simple by considering one of our favorite examples of entropy in action: a single egg. An unbroken egg is quite orderly and has a very low entropy; if we break the egg the entropy increases, and if we whisk the ingredients together the entropy increases even more. The maximum-entropy state will be a soup of individual molecules; details of the configuration will depend on the temperature, the presence of a gravitational field, and so on, but none of that will matter for our present purposes. The point is that it won’t look anything like an unbroken egg.

  Imagine we take such an egg and seal it in an absolutely impenetrable box, one that will last literally forever, undisturbed by the rest of the universe. For convenience, we put the egg-in-a-box out in space, far away from any gravity or external forces, and imagine that it floats undisturbed for all eternity. What happens inside that box?

  Even if we initially put an unbroken egg inside the box, eventually it would break, just through the random motions of its molecules. It will spend some time as a motionless, broken egg, differentiated into yolk and white and shell. But if we wait long enough, further random motions will gradually cause the yolk and white and even the shell to disintegrate and mix, until we reach a truly high-entropy state of undifferentiated egg molecules. That’s equilibrium, and it will last an extraordinarily long time.

  But if we continue to wait, the same kind of random motions that caused the egg to break in the first place will stir those molecules into lower-entropy configurations. All of the molecules may end up on one side of the box, for example. And after a very long time indeed, random motions will re-create something that looks like a broken egg (shell, yolk, and white), or even an unbroken egg! That seems preposterous, but it’s the straightforward implication of Poincaré’s recurrence theorem, or of taking seriously the ramifications of random fluctuations over extraordinarily long timescales.

  Most of the time, the process of forming an egg through random fluctuations of the constituent molecules will look just like the time-reverse of the process by which an unbroken egg decays into high-entropy goop. That is, we will first fluctuate into the form of a broken egg, and then the broken pieces will by chance arrange themselves into the form of an unbroken egg. That’s just a consequence of time-reversal symmetry; the most common evolutions from high entropy to low entropy look exactly like the most common evolutions from low entropy to high entropy, just played in reverse.

  Here is the rub. Let’s imagine that we have such an egg sealed in an impenetrable box, and we peek inside after it’s been left to its own devices for an absurdly long time—much greater than the recurrence time. It’s overwhelmingly likely that what we will see is something very much like equilibrium: a homogeneous mixture of egg molecules. But suppose we get extremely lucky, and we find what looks like a broken egg—a medium-entropy state, with some shards of eggshell and a yolk running into the egg whites. A configuration, in other words, that looks exactly what we would expect if there had recently been a pristine egg, which for some reason had been broken.

  Figure 56: An egg trapped for all eternity in an impenetrable box. Most of the time the box will contain egg molecules in high-entropy equilibrium. Occasionally it will fluctuate into the medium-entropy configuration of a broken egg, as in the top row. Much more rarely, it will fluctuate all the way to the low-entropy form of an unbroken egg, and then back again, as in the bottom row.

  Could we actually conclude, from this broken egg, that there had recently been an unbroken egg in the box? Not at all. Remember our discussion at the end of Chapter Eight. Given a medium-entropy configuration, and no other knowledge or assumptions such as a Past Hypothesis (which would clearly be inappropriate in the context of this ancient sealed box), it is overwhelmingly likely to have evolved from a higher-entropy past, just as it is overwhelmingly likely to evolve toward a higher-entropy future. Said conversely, given a broken egg, it is no more likely to have evolved from an unbroken egg than it is likely to evolve to an unbroken egg. Which is to say, not bloody likely.

  BOLTZMANN BRAINS

  The egg-in-a-box example illustrates the fundamental problem with the Boltzmann-Lucretius scenario: We can’t possibly appeal to a Past Hypothesis that asserts the existence of a low-entropy past state, because the universe (or the egg) simply cycles through every possible configuration it can have, with a predictable frequency. There is no such thing as an “initial condition” in a universe that lasts forever.

  The idea that the universe spends most of its time in thermal equilibrium, but we can appeal to the anthropic principle to explain why our local environment isn’t in equilibrium, makes a strong prediction—and that prediction is dramatically falsified by the dat
a. The prediction is simply that we should be as close to equilibrium as possible, given the requirement that we (under some suitable definition of “we”) be allowed to exist. Fluctuations happen, but large fluctuations (such as creating an unbroken egg) are much more rare than smaller fluctuations (such as creating a broken egg). We can see this explicitly back in Figure 54, where the curve exhibits many small fluctuations and only a few larger ones. And the universe we see around us would have to be a large fluctuation indeed.185

  We can be more specific about what the universe would look like if it were an eternal system fluctuating around equilibrium. Boltzmann invoked the anthropic principle (although he didn’t call it that) to explain why we wouldn’t find ourselves in one of the very common equilibrium phases: In equilibrium, life cannot exist. Clearly, what we want to do is find the most common conditions within such a universe that are hospitable to life. Or, if we want to be a bit more careful, perhaps we should look for conditions that are not only hospitable to life, but hospitable to the particular kind of intelligent and self-aware life that we like to think we are.

  Maybe this is a way out? Maybe, we might reason, in order for an advanced scientific civilization such as ours to arise, we require a “support system” in the form of an entire universe filled with stars and galaxies, originating in some sort of super-low-entropy early condition. Maybe that could explain why we find such a profligate universe around us.

  No. Here is how the game should be played: You tell me the particular thing you insist must exist in the universe, for anthropic reasons. A solar system, a planet, a particular ecosystem, a type of complex life, the room you are sitting in now, whatever you like. And then we ask, “Given that requirement, what is the most likely state of the rest of the universe in the Boltzmann-Lucretius scenario, in addition to the particular thing we are asking for?”

  And the answer is always the same: The most likely state of the rest of the universe is to be in equilibrium. If we ask, “What is the most likely way for an infinite box of gas in equilibrium to fluctuate into a state containing a pumpkin pie?,” the answer is “By fluctuating into a state that consists of a pumpkin pie floating by itself in an otherwise homogeneous box of gas.” Adding anything else to the picture, either in space or in time—an oven, a baker, a previously existing pumpkin patch—only makes the scenario less likely, because the entropy would have to dip lower to make that happen. By far the easiest way to get a pumpkin pie in this context is for it to gradually fluctuate all by itself out of the surrounding chaos.186

  Sir Arthur Eddington, in a lecture from 1931, considered a perfectly reasonable anthropic criterion:

  A universe containing mathematical physicists [under these assumptions] will at any assigned date be in the state of maximum disorganization which is not inconsistent with the existence of such creatures.187

  Eddington presumes that what you really need to make a good universe is a mathematical physicist. Sadly, if the universe is an eternally fluctuating collection of molecules, the most frequently occurring mathematical physicists will be all by themselves, surrounded by randomness.

  We can take this logic to its ultimate conclusion. If what we want is a single planet, we certainly don’t need a hundred billion galaxies with a hundred billion stars each. And if what we want is a single person, we certainly don’t need an entire planet. But if in fact what we want is a single intelligence, able to think about the world, we don’t even need an entire person—we just need his or her brain.

  So the reductio ad absurdum of this scenario is that the overwhelming majority of intelligences in this multiverse will be lonely, disembodied brains, who fluctuate gradually out of the surrounding chaos and then gradually dissolve back into it. Such sad creatures have been dubbed “Boltzmann brains” by Andreas Albrecht and Lorenzo Sorbo.188 You and I are not Boltzmann brains—we are what one might call “ordinary observers,” who did not fluctuate all by ourselves from the surrounding equilibrium, but evolved gradually from an earlier state of very low entropy. So the hypothesis that our universe is a random fluctuation around an equilibrium state in an eternal spacetime seems to be falsified.

  You may have been willing to go along with this line of reasoning when only an egg was involved, but draw up short when we start comparing the number of disembodied brains to the number of ordinary observers. But the logic is exactly the same, if (and it’s a big “if ”) we are considering an eternal universe full of randomly fluctuating particles. In such a universe, we know what kinds of fluctuations there are, and how often they happened; the more the entropy changes, the less likely that fluctuation will be. No matter how many ordinary observers exist in our universe today, they would be dwarfed by the total number of Boltzmann brains to come. Any given observer is a collection of particles in some particular state, and that state will occur infinitely often, and the number of times it will be surrounded by high-entropy chaos is enormously higher than the number of times it will arise as part of an “ordinary” universe.

  Just to be careful, though—are you really sure you are not a Boltzmann brain? You might respond that you feel the rest of your body, you see other objects around you, and for that matter you have memories of a lower-entropy past: All things that would appear inconsistent with the idea that you are a disembodied brain recently fluctuated out of the surrounding molecules. The problem is, these purported statements about the outside world are actually just statements about your brain. Your feelings, visions, and memories are all contained within the state of your brain. We could certainly imagine that a brain with exactly these sensations fluctuated out of the surrounding chaos. And, as we have argued, it’s much more likely for that brain to fluctuate by itself than to be part of a giant universe. In the Boltzmann-Lucretius scenario, we don’t have recourse to the Past Hypothesis, so it is overwhelmingly likely that all of our memories are false.

  Nevertheless, we are perfectly justified in dismissing this possibility, as long as we think carefully about what precisely we are claiming. It’s not right to say, “I know I am not a Boltzmann brain, so clearly the universe is not a random fluctuation.” The right thing to say is, “If I were a Boltzmann brain, there would be a strong prediction: Everything else about the universe should be in equilibrium. But it’s not. Therefore the universe is not a random fluctuation.” If we insist on being strongly skeptical, we might wonder whether not only our present mental states, but also all of the additional sensory data we are apparently accumulating, represent a random fluctuation rather than an accurate reconstruction of our surroundings. Strictly speaking, that certainly is possible, but it’s cognitively unstable in the same sense that we discussed in the last chapter. There is no sensible way to live and think and behave if that is the case, so there is no warrant for believing it. Better to accept the universe around us as it appears (for the most part) to be.

  This point was put with characteristic clarity by Richard Feynman, in his famous Lectures on Physics:

  [F]rom the hypothesis that the world is a fluctuation, all of the predictions are that if we look at a part of the world we have never seen before, we will find it mixed up, and not like the piece we just looked at. If our order were due to a fluctuation, we would not expect order anywhere but where we have just noticed it . . .

  We therefore conclude that the universe is not a fluctuation, and that the order is a memory of conditions when things started. This is not to say that we understand the logic of it. For some reason, the universe at one time had a very low entropy for its energy content, and since then the entropy has increased. So that is the way toward the future. That is the origin of all irreversibility, that is what makes the processes of growth and decay, that makes us remember the past and not the future, remember the things which are closer to that moment in history of the universe when the order was higher than now, and why we are not able to remember things where the disorder is higher than now, which we call the future.189

  WHO ARE WE IN THE MULTIVERSE?

 
There is one final loophole that must be closed before we completely shut the door on the Boltzmann-Lucretius scenario. Let’s accept the implications of conventional statistical mechanics, that small fluctuations in entropy happen much more frequently than large fluctuations, and that the overwhelming majority of intelligent observers in a universe fluctuating eternally around equilibrium will find themselves alone in an otherwise high-entropy environment, not evolving naturally from a prior configuration of enormously lower entropy.

  One might ask: So what? Why should I be bothered that most observers (under any possible definition of “observers”) find themselves alone as freak fluctuations in a high-entropy background? All I care about is who I am, not what most observers are like. As long as there is one instance of the universe I see around me somewhere in the eternal lifetime of the larger world (which there will be), isn’t that all I need to proclaim that this picture is consistent with the data?