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

by Sean M. Carroll

  So if we care about what actually happens in the world, we have to supplement the Principle of Indifference with the Past Hypothesis. When it comes to picking out microstates within our macrostate, we do not assign every one equal probability: We choose only those microstates that are compatible with a much lower-entropy past (a very tiny fraction), and take all of those to have equal probability.143

  But this strategy leaves us with a question: Why is the Past Hypothesis true? In Boltzmann’s time, we didn’t know anything about general relativity or the Big Bang, much less quantum mechanics or quantum gravity. But the question remains with us, only in a more specific form: Why did the universe have a low entropy near the Big Bang?

  9

  INFORMATION AND LIFE

  You should call it entropy, for two reasons. In the first place, your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage.

  —John von Neumann, to Claude Shannon144

  In a celebrated episode in Swann’s Way, Marcel Proust’s narrator is feeling cold and somewhat depressed. His mother offers him tea, which he reluctantly accepts. He is then pulled into an involuntary recollection of his childhood by the taste of a traditional French teatime cake, the madeleine.

  And suddenly the memory appeared. That taste was the taste of the little piece of madeleine which on Sunday mornings at Combray . . . when I went to say good morning to her in her bedroom, my aunt Léonie would give me after dipping it in her infusion of tea or lime blossom . . . And as soon as I had recognized the taste of the piece of madeleine dipped in lime-blossom tea that my aunt used to give me . . . immediately the old gray house on the street, where her bedroom was, came like a stage set to attach itself to the little wing opening onto a garden that had been built for my parents behind it . . . ; and with the house the town, from morning to night and in all weathers, the Square, where they sent me before lunch, the streets where I went on errands, the paths we took if the weather was fine.145

  Swann’s Way is the first of the seven volumes of À la recherche du temps perdu, which translates into English as In Search of Lost Time. But C. K. Scott Moncrieff, the original translator, borrowed a line from Shakespeare’s Sonnet 30 to render Proust’s novel as Remembrance of Things Past.

  The past, of course, is a natural thing to have remembrances of. What else would we be remembering, anyway? Surely not the future. Of all the ways in which the arrow of time manifests itself, memory—and in particular, the fact that it applies to the past but not the future—is the most obvious, and the most central to our lives. Perhaps the most important difference between our experience of one moment and our experience of the next is the accumulation of memories, propelling us forward in time.

  My stance so far has been that all the important ways in which the past differs from the future can be traced to a single underlying principle, the Second Law of Thermodynamics. This implies that our ability to remember the past but not the future must ultimately be explained in terms of entropy, and in particular by recourse to the Past Hypothesis that the early universe was in a very low-entropy state. Examining how that works will launch us on an exploration of the relationship between entropy, information, and life.

  PICTURES AND MEMORIES

  One of the problems in talking about “memory” is that there’s a lot we don’t understand about how the human brain actually works, not to mention the phenomenon of consciousness.146 For our present purposes, however, that’s not a significant handicap. When we talk about remembering the past, we’re interested not specifically in the human experience of memory, but in the general notion of reconstructing past events from the present state of the world. We don’t lose anything by considering well-understood mechanical recording devices, or even such straightforward artifacts as photographs or history books. (We are making an implicit assumption that human beings are part of the natural world, and in particular that our minds can in principle be understood in terms of our brains, which obey the laws of physics.)

  So let’s imagine you have in your possession something you think of as a reliable record of the past: for example, a photograph taken of your tenth birthday party. You might say to yourself, “I can be confident that I was wearing a red shirt at my tenth birthday party, because this photograph of that event shows me wearing a red shirt.” Put aside any worries that you might have over whether the photo has been tampered with or otherwise altered. The question is, what right do we have to conclude something about the past from the existence of this photo in the present?

  In particular, let’s imagine that we did not buy into this Past Hypothesis business. All we have is some information about the current macrostate of the universe, including the fact that it has this particular photo, and we have certain memories and so on. We certainly don’t know the current microstate—we don’t know the position and momentum of every particle in the world—but we can invoke the Principle of Indifference to assign equal probability to every microstate compatible with the macrostate. And, of course, we know the laws of physics—maybe not the complete Theory of Everything, but enough to give us a firm handle on our everyday world. Are those—the present macrostate including the photo, plus the Principle of Indifference, plus the laws of physics—enough to conclude with confidence that we really were wearing a red shirt at our tenth birthday party?

  Not even close. We tend to think that they are, without really worrying about the details too much as we get through our lives. Roughly speaking, we figure that a photograph like that is a highly specific arrangement of its constituent molecules. (Likewise for a memory in our brain of the same event.) It’s not as if those molecules are just going to randomly assemble themselves into the form of that particular photo—that’s astronomically unlikely. If, however, there really was an event in the past corresponding to the image portrayed in the photo, and someone was there with a camera, then the existence of the photo becomes relatively likely. It’s therefore very reasonable to conclude that the birthday party really did happen in the way seen in the photo.

  All of those statements are reasonable, but the problem is that they are not nearly enough to justify the final conclusion. The reason is simple, and precisely analogous to our discussion of the box of gas at the end of the last chapter. Yes, the photograph is a very specific and unlikely arrangement of molecules. However, the story we are telling to “explain” it—an elaborate reconstruction of the past, involving birthday parties and cameras and photographs surviving essentially undisturbed to the present day—is even less likely than the photo all by itself. At least, if “likely” is judged by assuming that all possible microstates consistent with our current macrostate have an equal probability—which is precisely what we assumed.

  Think of it this way: You would never think to appeal to some elaborate story in the future in order to explain the existence of a particular artifact in the present. If we ask about the future of our birthday photo, we might have some plans to frame it or whatnot, but we’ll have to admit to a great deal of uncertainty—we could lose it, it could fall into a puddle and decay, or it could burn in a fire. Those are all perfectly plausible extrapolations of the present state into the future, even with the specific anchor point provided by the photo here in the present. So why are we so confident about what the photo implies concerning the past?

  Figure 48: Trajectories through (part of) state space, consistent with our present macrostate. We can reconstruct the past accurately only by assuming a Past Hypothesis, in addition to knowledge of our current macrostate.

  The answer, of course, is the Past Hypothesis. We don’t really apply the Principle of Indifference to the current macrostate of the world—we only consider those microstates that are compatible with a very low-entropy past. And that makes all the difference when drawing inferences about the meaning of photographs or memories or other sorts o
f records. If we ask, “What is the most likely way, in the space of all possible evolutions of the universe, to get this particular photograph?” the answer is that it is most likely to evolve as a random fluctuation from a higher-entropy past—by exactly the same arguments that convince us it is likely to evolve toward a high-entropy future. But if instead we ask, “What is the most likely way, in the space of all evolutions of the universe from a very low-entropy beginning, to get this particular photograph?” then we find very naturally that it is most likely to go through the intermediate steps of an actual birthday party, a red shirt, a camera, and all the rest. Figure 48 illustrates the general principle—by demanding that our history stretch from a low-entropy beginning to here, we dramatically restrict the space of allowed trajectories, leaving us with those for which our records are (for the most part) reliable reflections of the past.

  COGNITIVE INSTABILITY

  I know from experience that not everyone is convinced by this argument. One stumbling block is the crucial assertion that what we start with is knowledge of our present macrostate, including some small-scale details about a photograph or a history book or a memory lurking in our brains. Although it seems like a fairly innocent assumption, we have an intuitive feeling that we don’t know something only about the present; we know something about the past, because we see it, in a way that we don’t see the future. Cosmology is a good example, just because the speed of light plays an important role, and we have a palpable sense of “looking at an event in the past.” When we try to reconstruct the history of the universe, it’s tempting to look at (for example) the cosmic microwave background and say, “I can see what the universe was like almost 14 billion years ago; I don’t have to appeal to any fancy Past Hypothesis to reason my way into drawing any conclusions.”

  That’s not right. When we look at the cosmic microwave background (or light from any other distant source, or a photograph of any purported past event), we’re not looking at the past. We’re observing what certain photons are doing right here and now. When we scan our radio telescope across the sky and observe a bath of radiation at about 2.7 Kelvin that is very close to uniform in every direction, we’ve learned something about the radiation passing through our present location, which we then need to extrapolate backward to infer something about the past. It’s conceivable that this uniform radiation came from a past that was actually highly non-uniform, but from which a set of finely tuned conspiracies between temperatures and Doppler shifts and gravitational effects produced a very smooth-looking set of photons arriving at us today. You may say that’s very unlikely, but the time-reverse of that is exactly what we would expect if we took a typical microstate within our present macrostate and evolved it toward a Big Crunch. The truth is, we don’t have any more direct empirical access to the past than we have to the future, unless we allow ourselves to assume a Past Hypothesis.

  Indeed, the Past Hypothesis is more than just “allowed”; it’s completely necessary, if we hope to tell a sensible story about the universe. Imagine that we simply refused to invoke such an idea and stuck solely with the data given to us by our current macrostate, including the state of our brains and our photographs and our history books. We would then predict with strong probability that the past as well as the future was a higher-entropy state, and that all of the low-entropy features of our present condition arose as random fluctuations. That sounds bad enough, but the reality is worse. Under such circumstances, among the things that randomly fluctuated into existence are all of the pieces of information we traditionally use to justify our understanding of the laws of physics, or for that matter all of the mental states (or written-down arguments) we traditionally use to justify mathematics and logic and the scientific method. Such assumptions, in other words, give us absolutely no reason to believe we have justified anything, including those assumptions themselves.

  David Albert has referred to such a conundrum as cognitive instability—the condition we face when a set of assumptions undermines the reasons we might have used to justify those very assumptions.147 It is a kind of helplessness that can’t be escaped without reaching beyond the present moment. Without the Past Hypothesis, we simply can’t tell any intelligible story about the world; so we seem to be stuck with it, or stuck with trying to find a theory that actually explains it.

  CAUSE AND EFFECT

  There is a dramatic temporal asymmetry in this story of how we use memories and records: We invoke a Past Hypothesis but not a future one. In making predictions, we do not throw away any microstates consistent with our current macrostate on the grounds that they are incompatible with any particular future boundary condition. What if we did? In Chapter Fifteen we will examine the Gold cosmology, in which the universe eventually stops expanding and begins to re-collapse, while the arrow of time reverses itself and entropy begins to decrease as we approach the Big Crunch. In that case there would be no overall difference between the collapsing phase and the expanding phase we find ourselves in today—they are identical (at least statistically). Observers who lived in the collapsing phase wouldn’t think anything was funny about their universe, any more than we do; they would think that we were evolving backward in time.

  It’s more illuminating to consider the ramifications of a minor restriction on allowed trajectories into our nearby future. This is essentially the situation we would face if we had a reliable prophecy of future events. When Harry Potter learns that either he will kill Voldemort or Voldemort will kill him, that places a very tight restriction on the allowed space of states.148

  Craig Callender tells a vivid story about what a future boundary condition would be like. Imagine that an oracle with an impeccable track record (much better than Professor Trelawney from the Harry Potter books) tells you that all of the world’s Imperial Fabergé eggs will end up in your dresser drawer, and that when they get there your life will end. Not such a believable prospect, really—you’re not even especially fond of Russian antiques, and now you know better than to let any into your bedroom. But somehow, through a series of unpredictable and unlikely fluke occurrences, those eggs keep finding a way into your drawer. You lock it, but the lock jiggles open; you inform the eggs’ owners to keep them where they are, but thieves and random accidents conspire to gradually collect them all in your room. You get a package that was mistakenly delivered to your address—it was supposed to go to the museum—and you open it to find an egg inside. In a panic, you throw it out the window, but the egg bounces off a street lamp at a crazy angle and careens back into your room to land precisely in your dresser drawer. And then you have a heart attack and die.149

  Throughout this chain of events, no laws of physics are broken along the way. At every step, events occur that are not impossible, just extremely unlikely. As a result, our conventional notions of cause and effect are overturned. We operate in our daily lives with a deep-seated conviction that causes precede effects: “There is a broken egg on the floor because I just dropped it,” not “I just dropped that egg because there was going to be a broken egg on the floor.” In the social sciences, where the causal relationship between different features of the social world can be hard to ascertain, this intuitive feeling has been elevated to the status of a principle. When two properties are highly correlated with each other, it’s not always obvious which is the cause and which is the effect, or whether both are caused by a different effect altogether. If you find that people who are happier in their marriages tend to eat more ice cream, is that because ice cream improves marriage, or happiness leads to more ice-cream eating? But there is one case where you know for sure: When one of the properties comes before the other one in time. Your grandparents’ level of educational attainment may affect your adult income, but your income doesn’t change your grandparents’ education.150

  Future boundary conditions overturn this understanding of cause and effect by insisting that some specific otherwise-unlikely things are necessarily going to happen. The same holds for the idea of free will. Ulti
mately, our ability to “choose” how to act in the future is a reflection of our ignorance concerning the specific microstate of the universe; if Laplace’s Demon were around, he would know exactly how we are going to act. A future boundary condition is a form of predestination.

  All of which may seem kind of academic and not worth dwelling on, for the basic reason that we don’t think there is any kind of future boundary condition that restricts our current microstate, and therefore we believe that causes precede effects. But we have no trouble believing in a past condition that restricts our current microstate. The microscopic laws of physics draw no distinction between past and future, and the idea that one event “causes” another or that we can “choose” different actions in the future in a way that we can’t in the past is nowhere to be found therein. The Past Hypothesis is necessary to make sense of the world around us, but it has a lot to answer for.

  MAXWELL’S DEMON

  Let’s shift gears a bit to return to the thought-experiment playground of nineteenth-century kinetic theory. Ultimately this will lead us to the connection between entropy and information, which will circle back to illuminate the question of memory.

  Perhaps the most famous thought experiment in all of thermodynamics is Maxwell’s Demon. James Clerk Maxwell proposed his Demon—more famous than Laplace’s, and equally menacing in its own way—in 1867, when the atomic hypothesis was just beginning to be applied to problems of thermodynamics. Boltzmann’s first work on the subject wasn’t until the 1870s, so Maxwell didn’t have recourse to the definition of entropy in the context of kinetic theory. But he did know about Clausius’s formulation of the Second Law: When two systems are in contact, heat will tend to flow from the hotter to the cooler, bringing both temperatures closer to equilibrium. And Maxwell knew enough about atoms to understand that “temperature” measures the average kinetic energy of the atoms. But with his Demon, he seemed to come up with a way to increase the difference in temperature between two systems, without injecting any energy—in apparent violation of the Second Law.