Blogs / Cosmic Variance

I’m in the middle of jetting hither and yon, talking to people about the arrow of time. (Wouldn’t it be great if I had a book to sell them?) Right now, as prophesyed, I’m at the Quantum To Cosmos Festival at the Perimeter Institute. They’re extremely on the ball over here, so every event is being recorded by the ultra-professional folks at TVO, and instantly available on the web. So here is the talk I gave on Saturday night — a public-level discussion of entropy and how it connects to the history of our universe.

Yes, that’s a pretty suave picture of me on the image capture. What can I say? I’m just one of those lucky folks with an effortless magic in front of the camera.

If you prefer to get your talks about entropy unadulterated by voice and motion, and don’t mind a more technical presentation, I’ve put the slides from my recent Caltech colloquium online. These are aimed basically at grad students in physics, so there is an equation or two, and the caveats are spelled out more clearly. But the punchline is the same.

Not sure which blogs the editors of the Onion have been reading, but I have to approve of their proposed model for explaining the low entropy at the beginning of a football game by recourse to an infinite series of downs before “first down.”

NEW YORK — Citing the extremely low level of entropy present before a normal set of football downs, scientists from the NFL’s quantum mechanics and cosmology laboratories spoke Monday of a theoretical proto-down before the first. “Ultimately, we believe there are an infinite number of proto-downs played before the first visible snap,” lead NFL scientist Dr. Oliver Claussen said during a press conference, adding that the very last yocto-down is a by-product of leftover fourth downs from this universe, as well as those from a theoretical universe running along an arrow of time concurrent to our own.

Probably some enthusiastic football coach is going to try to cash in by writing a book about the idea, while others fulminate on the sidelines about how such irresponsible speculation is destroying the game. (Thanks to Ahmet Toker and Tom Fishman.)

The book ended up with a pretty fun collection of epigraphs for each chapter. But there are a lot more good quotes about time than chapters in the book. Here are some of the quotes I did not end up using. Further examples are hereby solicited — who knows when they might turn out to be useful?

“Everything happens to everybody sooner or later if there is time enough.” — George Bernard Shaw, Back to Methuselah

“Time is the longest distance between two places.” — Tennessee Williams, The Glass Menagerie

“The future’s not ours to see.” — Doris Day

“Time rushes toward us with its hospital tray of infinitely varied narcotics, even while it is preparing us for its inevitably fatal operation.” — Tennessee Williams, The Rose Tattoo

“Time, you old gypsy man,
Will you not stay,
Put up your caravan
Just for one day?”
– Ralph Hodgeson

“Time present and time past
Are both perhaps present in time future,
And time future contained in time past.
If all time is eternally present
All time is unredeemable.”
– T.S. Eliot, “Burnt Norton” (Four Quartets)

“Time is the substance from which I am made. Time is a river that carries me along, but I am the river; it is a tiger that devours me, but I am the tiger; it is a fire that consumes me, but I am the fire.” — Jorge Luis Borges, Labyrinths.

Apparently you have to be extremely careful when it comes to poetry; fair use doesn’t necessarily extend very far.

It is a truth universally acknowledged that a provocative scientific idea will, before too long, end up in the hands of villains that must be fought by superheroes. Witness Boltzmann brains. Sure, they’ve already made a cameo in Dilbert, but the stakes were pretty low. Now Jim Kakalios (author of the excellent The Physics of Superheroes) sends along sends along a couple of snippets from The Incredible Hercules #133 — in which our intrepid protagonists are attacked by freak observers fluctuated out of thermal equilibrium!

Actually here they are described as “freaky observers,” rather than the more conventional “freak observers.” That description brings to mind Smoove B rather than Ludwig Boltzmann, but who knows? Maybe unlikely thermal fluctuations tend to be pretty kinky.

And yes, before you all start in: we know that Boltzmann Brains don’t really make for a credible alien menace, if you insist on being persnickety about what they supposedly really represent. It’s not that they “perceive” a universe more chaotic than ours — it’s that they would dominate the total number of observers if the universe really were more chaotic than ours. (Which it isn’t!) Also, they would tend to dissolve back into the chaos from which they came, rather than staging a coordinated attack on our homeland. Still! What a novel challenge for the Allies’ greatest hero.

A paper just appeared in Physical Review Letters with a provocative title: “A Quantum Solution to the Arrow-of-Time Dilemma,” by Lorenzo Maccone. Actually just “Quantum…”, not “A Quantum…”, because among the various idiosyncrasies of PRL is that paper titles do not begin with articles. Don’t ask me why.

But a solution to the arrow-of-time dilemma would certainly be nice, quantum or otherwise, so the paper has received a bit of attention (Focus, Ars Technica). Unfortunately, I don’t think this paper qualifies.

The arrow-of-time dilemma, you will recall, arises from the tension between the apparent reversibility of the fundamental laws of physics (putting aside collapse of the wave function for the moment) and the obvious irreversibility of the macroscopic world. The latter is manifested by the growth of entropy with time, as codified in the Second Law of Thermodynamics. So a solution to this dilemma would be an explanation of how reversible laws on small scales can give rise to irreversible behavior on large scales.

The answer isn’t actually that mysterious, it’s just unsatisfying. Namely, the early universe was in a state of extremely low entropy. If you accept that, everything else follows from the nineteenth-century work of Boltzmann and others. The problem then is, why should the universe be like that? Why should the state of the universe be so different at one end of time than at the other? Why isn’t the universe just in a high-entropy state almost all the time, as we would expect if its state were chosen randomly? Some of us have ideas, but the problem is certainly unsolved.

So you might like to do better, and that’s what Maccone tries to do in this paper. He forgets about cosmology, and tries to explain the arrow of time using nothing more than ordinary quantum mechanics, plus some ideas from information theory.

I don’t think that there’s anything wrong with the actual technical results in the paper — at a cursory glance, it looks fine to me. What I don’t agree with is the claim that it explains the arrow of time. Let’s just quote the abstract in full:

The arrow of time dilemma: the laws of physics are invariant for time inversion, whereas the familiar phenomena we see everyday are not (i.e. entropy increases). I show that, within a quantum mechanical framework, all phenomena which leave a trail of information behind (and hence can be studied by physics) are those where entropy necessarily increases or remains constant. All phenomena where the entropy decreases must not leave any information of their having happened. This situation is completely indistinguishable from their not having happened at all. In the light of this observation, the second law of thermodynamics is reduced to a mere tautology: physics cannot study those processes where entropy has decreased, even if they were commonplace.

So the claim is that entropy necessarily increases in “all phenomena which leave a trail of information behind” — i.e., any time something happens for which we can possibly have a memory of it happening. So if entropy decreases, we can have no recollection that it happened; therefore we always find that entropy seems to be increasing. Q.E.D.

But that doesn’t really address the problem. The fact that we “remember” the direction of time in which entropy is lower, if any such direction exists, is pretty well-established among people who think about these things, going all the way back to Boltzmann. (Chapter Nine.) But in the real world, we don’t simply see entropy increasing; we see it increase by a lot. The early universe has an entropy of 10⁸⁸ or less; the current universe has an entropy of 10¹⁰¹ or more, for an increase of more than a factor of 10¹³ — a giant number. And it increases in a consistent way throughout our observable universe. It’s not just that we have an arrow of time — it’s that we have an arrow of time that stretches coherently over an enormous region of space and time.

This paper has nothing to say about that. If you don’t have some explanation for why the early universe had a low entropy, you would expect it to have a high entropy. Then you would expect to see small fluctuations around that high-entropy state. And, indeed, if any complex observers were to arise in the course of one of those fluctuations, they would “remember” the direction of time with lower entropy. The problem is that small fluctuations are much more likely than large ones, so you predict with overwhelming confidence that those observers should find themselves in the smallest fluctuations possible, freak observers surrounded by an otherwise high-entropy state. They would be, to coin a pithy phrase, Boltzmann brains. Back to square one.

Again, everything about Maccone’s paper seems right to me, except for the grand claims about the arrow of time. It looks like a perfectly reasonable and interesting result in quantum information theory. But if you assume a low-entropy initial condition for the universe, you don’t really need any such fancy results — everything follows the path set out by Boltzmann years ago. And if you don’t assume that, you don’t really explain our universe. So the dilemma lives on.

The internets have spoken, and it’s a good thing I listened. A few months ago I had the idea to organize a session at the upcoming meeting of the American Association for the Advancement of Science, in San Diego next February. It’s a giant cross-scientific-disciplinary meeting, offering a great chance for journalists and scientists in diverse fields to catch up on what’s happening in other areas.

But I couldn’t decide between two possible topics, both of which are close to my heart: “The Origin of the Universe” or “The Arrow of Time.” (My original book subtitle was “The Origin of the Universe and the Arrow of Time,” before that was squelched by the marketing department and replaced with “The Quest for the Ultimate Theory of Time.” Quests are big these days, apparently.) So I did the natural thing: I Tweeted the question. And the internet spoke with a fairly unambiguous voice: “Arrow of Time” sounded more interesting. So that’s what I proposed.

And now we’ve just been accepted, so it’s on for San Diego 2010. We have a fantastic line-up of speakers (and also me), spanning quite a range of topics:

• Me, setting the stage about the arrow of time and the Second Law, and drawing some connections to cosmology.
• Huw Price, director of the Centre for Time, Department of Philosophy, University of Sydney, and author of Time’s Arrow and Archimedes’ Point. Huw will be talking about the arrow of time in philosophy.
• Anthony Leggett, winner of the Nobel Prize in 2003 for his work on superfluidity. Tony will be talking about the arrow of time in quantum mechanics and condensed matter physics.
• Michael Lässig, University of Cologne. Michael specializes in statistical physics and quantitative biology, and will be speaking on the arrow of time in biology, in particular in evolution. He recently gave a talk on this topic at the KITP.
• Daniel Schacter, Harvard, author of The Seven Sins of Memory. Dan was the leader of the studies showing that the brain predicts the future in the same way that it remembers the past. He’ll be talking about the arrow of time in psychology.

That’s the fun part about this topic; it ranges naturally from the birth of the universe to the operation of your brain. Should be a good symposium.

Update: Unfortunately, Daniel Schacter won’t be able to make the symposium. Instead, we are very fortunate to have Kathleen McDermott of Washington University in St. Louis. Her research involves how we remember the past and forecast the future.

After the FQXi Essay Contest, I was asked to comment on some of the essays besides my own, but I never did. Mostly because I didn’t take the time to read them all (there were an awful lot), but also because I just don’t know what to say about many of them. In her essay (which I liked), Fotini Markopoulou divides the world in two:

There are two kinds of people in quantum gravity. Those who think that timelessness is the most beautiful and deepest insight in general relativity, if not modern science, and those who simply cannot comprehend what timelessness can mean and see evidence for time in everything in nature. What sets this split of opinions apart form any other disagreement in science is that almost no one ever changes their mind…

That’s just about right (although perhaps there are also other splits with the same quality). Julian Barbour, whose essay finished first in the judging, has famously championed the view that time does not exist, even writing quite a successful book about it. In a recent Bloggingheads discussion with Craig Callender, Barbour talks a bit more about his view.

To which all I can muster is: I don’t get it. There are a set of technical arguments, which for the most part I do get, that can be used to make it seem as if time does not exist. In ordinary classical mechanics, we can perform some formal tricks to remove the time variable from the conventional equations of physics. More dramatically, in general relativity or quantum gravity we can express Einstein’s equation (at least in certain circumstances) in a form where time does not appear. On the other hand, we can usually re-write any of these equations in a form where time does appear (at least, again, in certain circumstances).

But none of these technical arguments are really the point. What I don’t understand — and this is a sincere lack of understanding on my part, not an indirect claim that this perspective is wrong — is what’s supposed to be so great about timelessness. What are we supposed to gain from thinking in this way? What problems is it supposed to solve?

Put it this way: clearly time appears to exist, at first glance. Even the timelessness crowd somehow manages to submit their essay competition entries by the deadline, and finish their Bloggingheads dialogues within an hour. So the claim “time does not exist” certainly doesn’t mean the same kind of thing as “unicorns do not exist.” It must mean (I suppose) that, while we all find time very useful in our everyday lives, there is a deeper level of description in which time doesn’t appear at all; it only emerges in some sort of approximate description of reality. But that approximate description seems extremely valid and useful, including all of the phenomena in the observable universe. Surely it behooves us to take this purportedly-non-fundamental notion seriously, and attempt to understand some of its puzzling features? Moreover, even if “time” doesn’t turn out to be fundamental, why would that tempt you into saying that it doesn’t exist? Protons are made of quarks, but you don’t hear particle physicists going around claiming that protons don’t exist.

The problem is not that I disagree with the timelessness crowd, it’s that I don’t see the point. I am not motivated to make the effort to carefully read what they are writing, because I am very unclear about what is to be gained by doing so. If anyone could spell out straightforwardly what I might be able to understand by thinking of the world in the language of timelessness, I’d be very happy to re-orient my attitude and take these works seriously.

With the new Star Trek out, it’s long past time (as it were) that we laid out the rules for would-be fictional time-travelers. (Spoiler: Spock travels to the past and gets a sex change and becomes Kirk’s grandfather lover.*) Not that we expect these rules to be obeyed; the dramatic demands of a work of fiction will always trump the desire to get things scientifically accurate, and Star Trek all by itself has foisted half a dozen mutually-inconsistent theories of time travel on us. But time travel isn’t magic; it may or may not be allowed by the laws of physics — we don’t know them well enough to be sure — but we do know enough to say that if time travel were possible, certain rules would have to be obeyed. And sometimes it’s more interesting to play by the rules. So if you wanted to create a fictional world involving travel through time, here are 10+1 rules by which you should try to play.

0. There are no paradoxes.

This is the overarching rule, to which all other rules are subservient. It’s not a statement about physics; it’s simply a statement about logic. In the actual world, true paradoxes — events requiring decidable propositions to be simultaneously true and false — do not occur. Anything that looks like it would be a paradox if it happened indicates either that it won’t happen, or our understanding of the laws of nature is incomplete. Whatever laws of nature the builder of fictional worlds decides to abide by, they must not allow for true paradoxes.

1. Traveling into the future is easy.

We travel into the future all the time, at a fixed rate: one second per second. Stick around, you’ll be in the future soon enough. You can even get there faster than usual, by decreasing the amount of time you experience elapsing with respect to the rest of the world — either by low-tech ways like freezing yourself, or by taking advantage of the laws of special relativity and zipping around near the speed of light. (Remember we’re talking about what is possible according to the laws of physics here, not what is plausible or technologically feasible.) It’s coming back that’s hard.

2. Traveling into the past is hard — but maybe not impossible.

If Isaac Newton’s absolute space and time had been the correct picture of nature, we could simply say that traveling backwards in time was impossible, and that would be the end of it. But in Einstein’s curved-spacetime universe, things are more flexible. From your own personal, subjective point of view, you always more forward in time — more technically, you move on a timelike curve through spacetime. But the large-scale curvature of spacetime caused by gravity could, conceivably, cause timelike curves to loop back on themselves — that is to say, become closed timelike curves — such that anyone traveling on such a path would meet themselves in the past. That’s what respectable, Einstein-approved time travel would really be like. Of course, there’s still the little difficulty of warping spacetime so severely that you actually create closed timelike curves; nobody knows a foolproof way of doing that, or even whether it’s possible, although ideas involving wormholes and cosmic strings and spinning universes have been bandied about.

3. Traveling through time is like traveling through space.

I’m only going to say this once: there would be no flashing lights. At least, there would only be flashing lights if you brought along some strobes, and decided to start them flashing as you traveled along your closed timelike curve. Likewise, there is no disappearance in a puff of smoke and re-appearing at some other time. Traveling through time is just like traveling through space: you move along a certain path, which (we are presuming) the universe has helpfully arranged so that your travels bring you to an earlier moment in time. But a time machine wouldn’t look like a booth with spinning wheels that dematerializes now and rematerializes some other time; it would look like a rocket ship. Or possibly a DeLorean, in the unlikely event that your closed timelike curve started right here on Earth and never left the road.

Think of it this way: imagine there were a race of super-intelligent trees, who could communicate with each other using abstract concepts but didn’t have the ability to walk. They might fantasize about moving through space, and in their fantasies “space travel” would resemble teleportation, with the adventurous tree disappearing in a puff of smoke and reappearing across the forest. But we know better; real travel from one point to another through space is a continuous process. Time travel would be like that.

4. Things that travel together, age together.

If you travel through time, and you bring along with you some clocks or other objects, all those things experience time in exactly the same way that you do. In particular, both you and the clocks march resolutely forward in time, from your own perspective. You don’t see clocks spinning wildly backwards, nor do you yourself “age” backwards, and you certainly don’t end up wearing the clothes you favored back in high school. Your personal experience of time is governed by clocks in your brain and body — the predictable beating of rhythmic pulses of chemical and biological processes. Whatever flow of time is being experienced by those processes — and thus by your conscious perception — is also being experienced by whatever accompanies you on your journey.

5. Black holes are not time machines.

Sadly, if you fell into a black hole, it would not spit you out at some other time. It wouldn’t spit you out at all — it would gobble you up and grow slightly more corpulent in the process. If the black hole were big enough, you might not even notice when you crossed the point of no return defined by the event horizon. But once you got close to the center of the hole, tidal forces would tug at you — gently at first, but eventually tearing you apart. The technical term is spaghettification. Not a recommended strategy for would-be time adventurers.

Wormholes — tunnels through spacetime, which in principle can connect widely-separated events — are a more promising alternative. Wormholes are to black holes as elevators are to deep wells filled with snakes and poisoned spikes. The problem is, unlike black holes, we don’t know whether wormholes exist, or even whether they can exist, or how to make them, or how to preserve them once they are made. Wormholes want to collapse and disappear, and keeping them open requires a form of negative energies. Nobody knows how to make negative energies, although they occasionally slap the name “exotic matter” on the concept and pretend it might exist.

(more…)

Since no one is blogging around here, and I’m still working on my book, I will cheat and just post an excerpt from the manuscript. Not an especially original one, either; in this section I steal shamelessly from the nice paper that Ted Bunn wrote last year about evolution and entropy (inspired by an previous paper by Daniel Styer).

————————————

Without even addressing the question of how “life” should be defined, we can ask what sounds like a subsequent question: does life make thermodynamic sense? The answer, before you get too excited, is “yes.” But the opposite has been claimed – not by any respectable scientists, but by creationists looking to discredit Darwinian natural selection as the correct explanation for the evolution of life on Earth. One of their arguments relies on a misunderstanding of the Second Law, which they read as “entropy always increases,” and then interpret as a universal tendency toward decay and disorder in all natural processes. Whatever life is, it’s pretty clear that life is complicated and orderly – how, then, can it be reconciled with the natural tendency toward disorder?

There is, of course, no contradiction whatsoever. The creationist argument would equally well imply that refrigerators are impossible, so it’s clearly not correct. The Second Law doesn’t say that entropy always increases. It says that entropy always increases (or stays constant) in a closed system, one that doesn’t interact noticeably with the external world. But it’s pretty obvious that life is not like that; living organisms interact very strongly with the external world. They are the quintessential examples of open systems. And that is pretty much that; we can wash our hands of the issue and get on with our lives.

But there’s a more sophisticated version of the argument, which you could imagine being true – although it still isn’t – and it’s illuminating (and fun) to see exactly how it fails. The more sophisticated argument is quantitative: sure, living beings are open systems, so in principle they can decrease entropy somewhere as long as it increases somewhere else. How do you know that the increase in entropy in the outside world is really enough to account for the low entropy of living beings?

As we mentioned way back in Chapter Two, the Earth and its biosphere are systems that are very far away from thermal equilibrium. In equilibrium, the temperature is the same everywhere, whereas when we look up we see a very hot Sun in an otherwise very cold sky. There is plenty of room for entropy to increase, and that’s exactly what’s happening. But it’s instructive to run the numbers.

The energy budget of the Earth, considered as a single system, is pretty simple. We get energy from the Sun, via radiation; we lose the same amount of energy to empty space, also via radiation. (Not exactly the same; processes such as nuclear decays also heat up the Earth and leak energy into space, and the rate at which energy is radiated is not strictly constant. Still, it’s an excellent approximation.) But while the amount is the same, there is a big difference in the quality of the energy we get and the energy we give back. Remember back in the pre-Boltzmann days, entropy was understood as a measurement of the uselessness of a certain amount of energy; low-entropy forms of energy could be put to useful work, such as powering an engine or grinding flour, while high-entropy forms of energy just sat there.

The energy we get from the Sun is of a low-entropy, useful form, while the energy we radiate back out into space has a much higher entropy. The temperature of the Sun is about twenty times the average temperature of the Earth. The temperature of radiation is just the average energy of the photons of which it is made, so the Earth needs to radiate twenty low-energy (long-wavelength, infrared) photons for every one high-energy (short-wavelength, visible) photon it receives. It turns out, after a bit of math, that twenty times as many photons directly translates into twenty times the entropy. The Earth emits the same amount of energy as it receives, but with twenty times higher entropy.

The hard part is figuring out just what we mean when we say that the life forms here on Earth are “low-entropy.” How exactly do we do the coarse-graining? It is possible to come up with reasonable answers to that question, but it’s complicated. Fortunately, there is a dramatic shortcut we can take. Consider the entire biomass of the Earth – all of the molecules that are found in living organisms of any type. We can easily calculate the maximum entropy that collection of molecules could have, if it were in thermal equilibrium; plugging in the numbers (the biomass is 10¹⁵ kilograms, the temperature of the Earth is 255 Kelvin), we find that its maximum entropy is 10⁴⁴. And we can compare that to the absolute minimum entropy it could have – if it were in an exactly unique state, the entropy would be precisely zero.

So the largest conceivable change in entropy that would be required to take a completely disordered collection of molecules the size of our biomass and turn them into absolutely any configuration at all – including the actual ecosystem we currently have – is 10⁴⁴. If the evolution of life is consistent with the Second Law, it must be the case that the Earth has generated more entropy over the course of life’s evolution by converting high-energy photons into low-energy ones than it has decreased entropy by creating life. The number 10⁴⁴ is certainly an overly generous estimate – we don’t have to generate nearly that much entropy, but if we can generate that much, the Second Law is in good shape.

How long does it take to generate that much entropy by converting useful solar energy into useless radiated heat? The answer, once again plugging in the temperature of the Sun and so forth, is: about one year. Every year, if we were really efficient, we could take an undifferentiated mass as large as the entire biosphere and arrange it in a configuration with as small an entropy as we can imagine. In reality, life has evolved over billions of years, and the total entropy of the “Sun + Earth (including life) + escaping radiation” system has increased by quite a bit. So the Second Law is perfectly consistent with life as we know it; not that you were ever in doubt.

His Brains, anyway. (Which he never talked about himself, but that’s neither here nor there.) Random fluctuations make an appearance in Dilbert. (Hat tip Nick Suntzeff.)

One can only wonder what Calvin and Hobbes could have done with this.