Time

The important event this Dec. 25 isn’t celebrating the birthday of Isaac Newton or other historical figures, it’s the release of The Curious Case of Benjamin Button, a David Fincher film starring Brad Pitt and based on the story by F. Scott Fitzgerald. As you all know, it’s a story based on the device of incompatible arrows of time: Benjamin is born old and ages backwards into youth (physically, not mentally), while the rest of the world behaves normally. Some have pretended that scientific interest in the movie centers on issues of aging and longevity, but of course it’s thermodynamics and entropy that take center stage. While entropy increases and the Second Law is respected in the rest of the world, Benjamin Button’s body seems to be magically decreasing in entropy. (Which does not, strictly speaking, violate the Second Law, since his body isn’t a closed system, but it sure is weird.)

It’s a great opportunity to address an old chestnut: why do arrows of time have to be compatible? Why can’t we imagine ever discovering another galaxy in which entropy increased toward (what we call) the past instead of the future, as in Greg Egan’s story, “The Hundred Light-Year Diary”? Or why can’t a body age backwards in time?

First we need to decide what the hell we mean. Let’s put aside for the moment sticky questions about collapsing wave functions, and presume that the fundamental laws of physics are perfectly reversible. In that case, given the precise state of the entire universe (or any closed system) at any one moment in time, we can use those laws to determine what the state will be at any future time, or what it was at any past time. That’s just how awesome the laws of physics are. (Of course we don’t know the laws, nor the state of the entire universe, nor could we actually carry out the relevant calculation even if we did, but we’re doing thought experiments here.) We usually take that time to be the “initial” time, but in principle we could choose any time — and in the present context, when we’re worried about arrows of time pointing in different directions, there is no time that is initial for everything. So what we mean is: Why is it difficult/impossible to choose a state of the universe with the property that, as we evolve it forward in time, some parts of it have increasing entropy and some parts have decreasing entropy?

Notice that we can choose conditions that reverse the arrow of time for some individual isolated system. Entropy counts the “typicalness” of the system’s microscopic state, from the point of view of macroscopic observers. And it tends to go up, because there are many more ways to be high-entropy than low entropy. Consider a box of gas, in which the gas molecules are (by some means) all bunched together in the middle of the box, in a low-entropy configuration. If we just let it evolve, the molecules will move around, colliding with each other and with the walls of the box, and ending up (with overwhelmingly probability) in a much higher-entropy configuration.

It’s easy to convince ourselves that there exists some configurations from which the entropy would spontaneously go down. For example, take the state of the above box of gas at any moment after it has become high-entropy, and consider the state in which all of the molecules have exactly the same positions but precisely reversed velocities. From there, the motion of the molecules will precisely re-trace the path that they took from the previous low-entropy state. To an external observer, it will look as if the entropy is spontaneously decreasing. (Of course we know that it took a lot of work to so precisely reverse all of those velocities, and the process of doing so increased the entropy of the wider world, so the Second Law is safe.)

But a merely reversed arrow of time is not the point; we want incompatible arrows of time. That means entropy increasing in some part of the universe while it is decreasing in others.

At first it would seem simple enough. Take two boxes, and prepare one of them in the low entropy state with gas in the middle, and the other in the delicately constructed state with reversed velocities. (That is, the two boxes on the left side of the two figures above.) The entropy will go up in one box, and down in the other, right? That’s true, but it’s kind of trivial. We need to have systems that interact — one system can somehow communicate with the other.

And that ruins everything, of course. Imagine we started with these two boxes, one of which had an entropy that was ready to go up and the other ready to go down. But now we introduced a tiny coupling — say, a few photons moving between the boxes, bouncing off a molecule in one before returning to the other. Certainly the interaction of Benjamin Button’s body with the rest of the world is much stronger than that. (Likewise Egan’s time-reversed galaxy, or Martin Amis’s narrator in Time’s Arrow.)

That extra little interaction will slightly alter the velocities of the molecules with which it interacts. (Momentum is conserved, so it has no choice.) That’s no problem for the box that starts with low entropy, as there is no delicate tuning required to make the entropy go up. But it completely ruins our attempt to set up conditions in the other box so that entropy goes down. Just a tiny change in velocity will quickly propagate through the gas, as one affected molecule hits another molecule, and then they hit two more, and so on. It was necessary for all of the velocities to be very precisely aligned to make the gas miraculously conspire to decrease its entropy, and any interaction we might want to introduce will destroy the required conspiracy. The entropy in the first box will very sensibly go up, while the entropy in the other will just stay high. You can’t have incompatible arrows of time among interacting subsystems of the universe.

The Foundational Questions Institute is sponsoring an essay competition on “The Nature of Time.” Needless to say, I’m in. It’s as if they said: “Here, you keep talking about this stuff you are always talking about anyway, except that we will hold out the possibility of substantial cash prizes for doing so.” Hard to resist.

The deadline for submitting an entry is December 1, so there’s still plenty of time (if you will), for anyone out there who is interested and looking for something to do over Thanksgiving. They are asking for essays under 5000 words, on any of various aspects of the nature of time, pitched “between the level of Scientific American and a review article in Science or Nature.” That last part turns out to be the difficult one — you’re allowed to invoke some technical concepts, and in fact the essay might seem a little thin if you kept it strictly popular, but hopefully it should be accessible to a large range of non-experts. Most entries seem to include a few judicious equations while doing their best to tell a story in words.

All of the entries are put online here, and each comes with its own discussion forum where readers can leave comments. A departure from the usual protocols of scientific communication, but that’s a good thing. (Inevitably there is a great deal of chaff along with the wheat among the submitted essays, but that’s the price you pay.) What is more, in addition to a judging by a jury of experts, there is also a community vote, which comes with its own prizes. So feel free to drop by and vote for mine if you like — or vote for someone else’s if you think it’s better. There’s some good stuff there.

My essay is called “What if Time Really Exists?” A lot of people who think about time tend to emerge from their contemplations and declare that time is just an illusion, or (in modern guise) some sort of semi-classical approximation. And that might very well be true. But it also might not be true; from our experiences with duality in string theory, we have explicit examples of models of quantum gravity which are equivalent to conventional quantum-mechanical systems obeying the time-dependent Schrödinger equation with the time parameter right there where Schrödinger put it.

And from that humble beginning — maybe ordinary quantum mechanics is right, and there exists a formulation of the theory of everything that takes the form of a time-independent Hamiltonian acting on a time-dependent quantum state defined in some Hilbert space — you can actually reach some sweeping conclusions. The fulcrum, of course, is the observed arrow of time in our local universe. When thinking about the low-entropy conditions near the Big Bang, we tend to get caught up in the fact that the Bang is a singularity, forming a boundary to spacetime in classical general relativity. But classical general relativity is not right, and it’s perfectly plausible (although far from inevitable) that there was something before the Bang. If the universe really did come into existence out of nothing 14 billion years ago, we can at least imagine that there was something special about that event, and there is some deep reason for the entropy to have been so low. But if the ordinary rules of quantum mechanics are obeyed, there is no such thing as the “beginning of time”; the Big Bang would just be a transitional stage, for which our current theories don’t provide an adequate spacetime interpretation. In that case, the observed arrow of time in our local universe has to arise dynamically according to the laws of physics governing the evolution of a wave function for all eternity.

Interestingly, that has important implications. If the quantum state evolves in a finite-dimensional Hilbert space, it evolves ergodically through a torus of phases, and will exhibit all of the usual problems of Boltzmann brains and the like (as Dyson, Kleban, and Susskind have emphasized). So, at the very least, the Hilbert space (under these assumptions) must be infinite-dimensional. In fact you can go a bit farther than that, and argue that the spectrum of energy eigenvalues must be arbitrarily closely spaced — there must be at least one accumulation point.

Sexy, I know. The remarkable thing is that you can say anything at all about the Hilbert space of the universe just by making a few simple assumptions and observing that eggs always turn into omelets, never the other way around. Turning it into a respectable cosmological model with an explicit spacetime interpretation is, admittedly, more work, and all we have at the moment are some very speculative ideas. But in the course of the essay I got to name-check Parmenides, Heraclitus, Lucretius, Augustine, and Nietzsche, so overall it was well worth the effort.

I gave a talk yesterday at the Center for Inquiry branch here in LA. It was a popular-level spiel on The Origin of the Universe and the Arrow of Time; click for slides. If I had been thinking, I would have advertised the existence of the talk before I had given it, rather than afterward. Either that, or I was trying to smoke out time-travelers.

But the real reason I’m even bringing it up is to give credit to this great YouTube video, found via Swans on Tea.

I was literally zipping through blogs yesterday morning while drinking coffee and preparing for the upcoming talk, when up popped this wonderful illustration of entropy and the arrow of time, which naturally I showed at the talk. And it features a kitty. (Schrodinger has his own cat, why shouldn’t Boltzmann?)

Many of you scoffed last week when I mentioned that Lucretius had been a pioneer in statistical mechanics. (Not out loud, but inwardly, there was scoffing.) But it’s true. Check out this passage from De Rerum Natura, in which Lucretius proposes that the universe arises as a quantum fluctuation:

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…

Lucretius, along with Democritus and Epicurus, was an early champion of atomism — the idea that the tremendous variety of substances we see around us arise from different combinations of a few kinds of underlying particles. He was also a materialist, believing that the atoms obeyed laws, not that they received external guidance. So a problem arose: how could all of that regular atomic motion give rise to the complexity we see around us? In response, Lucretius (actually Epicurus — see below) invented the “swerve” — an occasional, unpredictable deviation from regular atomic behavior. And then, he points out, if you wait long enough you will swerve your way into the universe.

It’s a good idea, and one that has been re-invented since then. Boltzmann, another famous atomist, hit upon the same basic scenario. Here is Boltzmann in 1897:

There must then be in the universe, which is in thermal equilibrium as a whole and therefore dead, here and there relatively small regions of the size of our galaxy (which we call worlds), which during the relatively short time of eons deviate significantly from thermal equilibrium. Among these worlds the state probability increases as often as it decreases. For the universe as a whole the two directions of time are indistinguishable, just as in space there is no up or down.

However, just as at a certain place on the earth’s surface we can call “down” the direction toward the centre of the earth, so a living being that finds itself in such a world at a certain period of time can define the time direction as going from less probable to more probable states (the former will be the “past” and the latter the “future”) and by virtue of this definition he will find that this small region, isolated from the rest of the universe, is “initially” always in an improbable state.

Boltzmann imagines the universe as a whole (or what we would call the “multiverse”) is in thermal equilibrium, about which he knew a lot more than Lucretius. But he also understood that the Second Law was only statistical, not absolute. Eventually there would be statistical fluctuations that took the thermal gas and turned them into something that looks like our universe (which, as far as Boltzmann knew, was just the galaxy).

We are now smart enough to know that this kind of scenario doesn’t work, at least in its unmodified form. The problem is that fluctuations are rare, and large fluctuations are much more rare; a universe-size fluctuation would be rare indeed. Who needs 100 billion galaxies when one will do? Or even just one observer? This objection was forcefully put forward by none other than Sir Arthur Eddington in 1931:

A universe containing mathematical physicists [which is obviously the correct anthropic criterion — ed.] will at any assigned date be in the state of maximum disorganization which is not inconsistent with the existence of such creatures.

These days, we throw away the rest of the mathematical physicist and focus exclusively on the cognitive capacities thereof, and call the resulting thermodynamic monstrosity a Boltzmann Brain. The conclusion of this argument is: the universe we see around us is not eternal in time and bounded in phase space. Because if it is, over the long term we really would just see statistical fluctuations, and we would most likely be lonely brains. So either the universe is not eternal — so that it doesn’t have time to fluctuate ergodically throughout phase space — or its set of states is not bounded — so that it evolves forever, but doesn’t sample every possible configuration.

Sorry about that, Lucretius. You’ll be happy to know that we’re still struggling with these same issues. Except that you’re dead and famously railed against the irrationality of belief in life after death. So probably you don’t care.

Greetings from Paris! Just checking in to do a bit of self-promotion, from which no blog-vacation could possibly keep me. I’ve written an article in this month’s Scientific American about the arrow of time and cosmology. It’s available for free online; the given title is “Does Time Run Backward in Other Universes?”, which wasn’t my choice, but these happenings are team events.

As a teaser, here is a timeline of the history of the universe according to the standard cosmology:

• Space is empty, featuring nothing but a tiny amount of vacuum energy and an occasional long-wavelength particle formed via fluctuations of the quantum fields that suffuse space.
• High-intensity radiation suddenly sweeps in from across the universe, in a spherical pattern focused on a point in space. When the radiation collects at that point, a “white hole” is formed.
• The white hole gradually grows to billions of times the mass of the sun, through accretion of additional radiation of ever decreasing temperature.
• Other white holes begin to approach from billions of light-years away. They form a homogeneous distribution, all slowly moving toward one another.
• The white holes begin to lose mass by ejecting gas, dust and radiation into the surrounding environment.
• The gas and dust occasionally implode to form stars, which spread themselves into galaxies surrounding the white holes.
• Like the white holes before them, these stars receive inwardly directed radiation. They use the energy from this radiation to convert heavy elements into lighter ones.
• Stars disperse into gas, which gradually smooths itself out through space; matter as a whole continues to move together and grow more dense.
• The universe becomes ever hotter and denser, eventually contracting all the way to a big crunch.

Despite appearances, this really is just the standard cosmology, not some fairy tale. I just chose to tell it from the point of view of a time coordinate that is oriented in the opposite direction from the one we usually use. Given that the laws of physics are reversible, this choice is just as legitimate as the usual one; nevertheless, one must admit that the story told this way seems rather unlikely. So why does the universe evolve this way? That’s the big mystery, of course.

Today’s Bloggingheads dialogue features me and writer John Horgan — I will spare you a screen capture of our faces, but here is a good old-fashioned link.

John is the author of The End of Science, in which he argues that much of modern physics has entered an era of “ironic science,” where speculation about unobservable things (inflation, other universes, extra dimensions) has replaced the hard-nosed empiricism of an earlier era. Most of our discussion went over that same territory, focusing primarily on inflation but touching on other examples as well.

You can judge for yourself whether I was persuasive or not, but the case I tried to make was that attitudes along the lines of “that stuff you’re talking about can never be observed, so you’re not doing science, it’s just theology” are woefully simplistic, and simply don’t reflect the way that science works in the real world. Other branches of the wavefunction, or the state of the universe before the Big Bang, may by themselves be unobservable, but they are part of a larger picture that remains tied to what we see around us. (Inflation is a particularly inappropriate example to pick on; while it has by no means been established, and it is undeniably difficult to distinguish definitively between models, it keeps making predictions that are tested and come out correct — spatial flatness of the universe, density fluctuations larger than the Hubble radius, correlations between perturbations in matter and radiation, fluctuation amplitudes on different scales that are almost equal but not quite…)

If you are firmly convinced that talking about the multiverse and other unobservable things is deeply unscientific and a leading indicator of the Decline of the West, nothing I say will change your mind. In particular, you may judge that the question which inflation tries to answer — “Why was the early universe like that?” — is a priori unscientific, and we should just accept the universe as it is. That’s an intellectually consistent position that you are welcome to take. The good news is that the overwhelming majority of interesting science being done today remains closely connected to tangible phenomena just as it (usually!) has been through the history of modern science. But if you instead ask in good faith why sensible people would be led to hypothesize all of this unobservable superstructure, there are perfectly good answers to be had.

The most important point is that the underlying goal of science is not simply making predictions — it’s developing an understanding of the mechanisms underlying the operation of the natural world. This point is made very eloquently by David Deutsch in his book The Fabric of Reality. As I mention in the dialogue, Deutsch chooses this quote by Steven Weinberg as an exemplar of hard-boiled instrumentalism:

The important thing is to be able to make predictions about images on the astronomers’ photographic plates, frequencies of spectral lines, and so on, and it simply doesn’t matter whether we ascribe these predictions to the physical effects of gravitational fields on the motion of planets and photons or to a curvature of space and time.

That’s crazy, of course — the dynamics through which we derive those predictions matters enormously. (I suspect that Weinberg was trying to emphasize that there may be formulations of the same underlying theory that look different but are actually equivalent; then the distinction truly wouldn’t matter, but saying “the important thing is to make predictions” is going a bit too far.) Deutsch asks us to imagine an “oracle,” a black box which will correctly answer any well-posed empirical question we ask of it. So in principle the oracle can help us make any prediction we like — would that count as the ultimate end-all scientific theory? Of course not, as it would provide no understanding whatsoever. As Deutsch notes, it would be able to predict that a certain rocket-ship design would blow up on take-off, but offer no clue as to how we could fix it. The oracle would serve as a replacement for experiments, but not for theories. No scientist, armed with an infinite array of answers to specific questions but zero understanding of how they were obtained, would declare their work completed.

If making predictions were all that mattered, we would have stopped doing particle physics some time around the early 1980’s. The problem with the Standard Model of particle physics, remember, is that (until we learned more about neutrino physics and dark matter) it kept making predictions that fit all of our experiments! We’ve been working very hard, and spending a lot of money, just to do experiments for which the Standard Model would be unable to make an accurate prediction. And we do so because we’re not satisfied with predicting the outcome of experiments; we want to understand the underlying mechanism, and the Standard Model (especially the breaking of electroweak symmetry) falls short on that score.

The next thing to understand is that all of these crazy speculations about multiverses and extra dimensions originate in the attempt to understand phenomena that we observe right here in the nearby world. Gravity and quantum mechanics both exist — very few people doubt that. And therefore, we want a theory that can encompass both of them. By a very explicit chain of reasoning — trying to understand perturbation theory, getting anomalies to cancel, etc. — we are led to superstrings in ten dimensions. And then we try to bring that theory back into contact with the observed world around us, compactifying those extra dimensions and trying to match onto particle physics and cosmology. The program may or may not work — it’s certainly hard, and we may ultimately decide that it’s just too hard, or find an idea that works just as well without all the extra-dimensional superstructure. Theories of what happened before the Big Bang are the same way; we’re not tossing out scenarios because we think it’s amusing, but because we are trying to understand features of the world we actually do observe, and that attempt drives us to these hypotheses.

Ultimately, of course, we do need to make contact with observation and experiment. But the final point to emphasize is that not every prediction of every theory needs to be testable; what needs to be testable is the framework as a whole. If we do manage to construct a theory that makes a set of specific and unambiguous testable predictions, and those predictions are tested and the theory comes through with flying colors, and that theory also predicts unambiguously that inflation happened or there are multiple universes or extra dimensions, I will be very happy to believe in the reality of those ideas. That happy situation does not seem to be around the corner — right now the data are offering us a few clues, on the basis of which invent new hypotheses, and we have a long way to go before some of those hypotheses grow into frameworks which can be tested against data. If anyone is skeptical that this is likely to happen, that is certainly their prerogative, and they should feel fortunate that the overwhelming majority of contemporary science is not forced to work that way. Others, meanwhile, will remain interested in questions that do seem to call for this kind of bold speculation, and are willing to push the program forward for a while to see what happens. Keeping in mind, of course, that when Boltzmann was grounding the laws of thermodynamics using kinetic theory, most physicists scoffed at the notion of these “atoms” and rolled their eyes at the invocation of unobservable entities to explain everyday phenomena.

There is also a less rosy possibility, which may very well come to pass: that we develop more than one theory that fits all of the experimental data we know how to collect, such that they differ in specific predictions that are beyond our technological reach. That would, indeed, be too bad. But at the moment, we seem to be in little danger of this embarrassment of theoretical riches. We don’t even have one theory that reconciles gravity and quantum mechanics while matching cleanly onto our low-energy world, or a comprehensive model of the early universe that explains our initial conditions. If we actually do develop more than one, science will be faced with an interesting kind of existential dilemma that doesn’t have a lot of precedent in history. (Can anyone think of an example?) But I’m not losing sleep over this possibility; and in the meantime, I’ll keep trying to develop at least one such idea.