## Ice water on closed timelike curves

Today, let’s kick off our shoes and relax our standards for what counts as a reasonable spacetime. Let’s talk about spacetimes allowing for a certain kind of time travel, called a closed timelike curve. Being stuck on one would be like being in Nietzsche’s “eternally recurring” world — or more recently, like Bill Murray in the movie Groundhog Day.

Although worlds containing closed timelike curves arise from classical *general relativity*, one can worry about whether such worlds obey the laws of *statistical physics*. For example, what would happen to a glass of ice water on a closed timelike curve?

The state of the ice water would have to be cyclic. So, if the ice were able to melt at all, then it would also have to “unmelt,” and coagulate as ice again when it came back around. Apparently, entropy in our time-traveling ice water has to regularly and dramatically decrease. But that’s supposed to be impossible.

Does that make closed timelike curves incompatible with the principles of statistical physics? Not necessarily. Boltzmann would argue that the reason the ice should always melt is that, among all the possible trajectories that the particles in the ice water might travel, those in which the ice melts form the overwhelming majority.

So, assuming each possible trajectory is equally likely, it follows that the ice is overwhelmingly likely to melt.

But suppose we restrict what counts as a “possible trajectory” for a particle in our time-traveling ice water. After all, we’re on a closed timelike curve. The possible trajectories *are* different, because they are required to be cyclic — every particle has to end up back where it started. This constraint guarantees that, if the ice melts at all, it always also unmelts. Boltzmann’s counting argument then just determines what most commonly happens in between, among these trajectories.

Of course, everything has to happen the same way every time around an individual closed timelike curve. So, “most commonly” would have to be made sense of in terms of an array closed timelike curves, each with a glass of ice water on it. But apparently, no fundamental principles are violated. And so statistical physics might yet live in harmony with closed timelike curves.

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Do you even need closed spacetime curves? Don’t you get pretty much the same result from Poincarre’s recursion theorem?

I agree with Gabriele.

No, I think CTCs introduce something new here.

Poincaré recurrence can only be expected to occur on enormous time scales — so it’s not an objection to Boltzmann’s argument for every-day glasses of water for normal lengths of time.

On the other hand, the state of ice water on a CTC can cycle very rapidly.

Hi Bryan,

If I understand what you are saying in the post, I can’t see what role the time-scale is supposed to be playing in the problem you are discussing.

I thought the problem was that:

Apparently, entropy in our time-traveling ice water has to regularly and dramatically decrease. But that’s supposed to be impossible. Does that make closed timelike curves incompatible with the principles of statistical physics? Not necessarily.First of all, as Craig Callender would put it, you seem to be taking TD too seriously when you say that the decrease is supposed to be

impossiblerather than just overwhelminglyimprobable.Second, since the principle of SM unlike those of TD do not seem to absolutely forbid entropy increase there is no straightforward

incompatibilitybetween CTCs and them.Third, the principles of SM are not fundamental by the theory’s own light. So, why not just say that they would have to give in a CTC?

Fourth, why does timescale matter?

Gabriele,

Thanks for all the thoughtful comments!

1) I didn’t say “decrease,” I said “regular and dramatic decrease” in entropy. That is a statistical impossibility.

2) See above:

regularanddramatic.3) Not sure what you mean here by “not fundamental.”

4) Look, the basic point I’m making here is well-known. Deutsch (1991, p. 3199) attributes it to Tipler & Hawking: “classically there are more constraints on initial data in chronology-violating spacetimes than in chronology respecting ones.” An example would be the following. Here on earth, on the time scale that it takes a cube of ice to melt in a glass of room-temp ice water, SM

forbidsregular dramatic decreases in entropy. In particular, it is impossible for the ice in this system to regularly melt and freeze again. I’m suggesting that for similar time scales on a closed timelike curve, this is not impossible — so long as the initial data is suitably constrained.Hi Bryan,

1) What you mean by ‘statistical impossibility’? Is p statistically impossible if Pr(p)=0?

3) I mean ‘the principles of statistical mechanics supervene on the fundamental laws they are not among them’. Once God creates everything in the world and establishes the laws of nature, She does not need to include the principles of statistical mechanics among them.

4) I still don’t see what the CTC is doing for you.

Ifit’s possible to have such a process in a CTC given certain suitable “initial” constraints, then it is possible to have the same phenomenon without a CTC but the same dynamics. Just consider a system that has the same suitable contraints as your system in a CTC but in a “flat” spacetime and you’ll get the same periodic behaviour. So if, it’s possible at all, it’s possible in both spacetimes and the timscales are going to be the same irrespectively of what the spacetime is like. Isn’t it?Hi Gabriele,

1) I think I see now what you’re worried about. No, that’s not what I mean. Suppose a physical theory (like SM) assigns a probability Pr(p) to an event p. I’m calling a set of experimental outcomes

statistically impossibleif they confirm a probability Pr(p) that is different than the one assigned by the theory. In particular, it’s statistically impossible for ice cubes to regularly (say, once per minute) “unmelt” back into a glass of water. Such a set of experiments would disconfirm the unimaginably low probability that SM assigns to this event.4) It may be true that, as you say, “

Ifit’s possible to have such a process in a CTC given certain suitable “initial” constraints, then it is possible to have the same phenomenon without a CTC”. However, the converse is false. Just imagine someone running in a circle with uniformly increasing speed. That’s possible in Minkowski spacetime, but not in Minkowski spacetime rolled up along the time access. So it’s false that “if, it’s possible at all, it’s possible in both spacetimes”.Hi Bryan,

1) Isn’t that way too weak a condition for statistical

impossibility?4) That’s why I made that claim, not its converse ;-). But I don’t see why that would make “if, it’s possible at all, it’s possible in both spacetimes” false. ‘It’ refers to the sort of process you describe in your post not just to any process.

Gabriele,

1) However you like. :)

4) Ok, that sounds fine then.

Of course, for statistical mechanics, what matters is what the *majority* of possible classical trajectories do. For short time scales on flat Minkowski spacetime, most trajectories lead to melting. On a CTC, they need not.

Thanks — this was fun!

Hello,

It’s funny to see this post, since I was just thinking about this. Last week I told my colleague Chris that I was thinking about one of the silliest spacetimes around. Is macroscopic time travel improbable? In general, yes, for reasons of entropy, as you say. Still, is *any* macroscopic time travel probable? Yes, for it’s easy to see that it’s possible: just put a Pricean “Gold” universe on cylinder spacetime rolled up along a spacelike axis. A Gold universe is one with a future and past low entropy condition. Identity the so-called Past Hypothesis State (low entropy big bang) with the Future Hypothesis State (low entropy big crunch). Systems evolving from the low entropy condition will start to “feel” the future low entropy condition and naturally (probably) have their entropies start to decrease. But it’s a thermodynamic-consistent CTC.

Of course, such time travel would be lame. To get back in time, not only would I have to traverse all of future world history (the time machine is a cheap one, going only as fast as time “flows”), but after the maximum entropy point I might get my time sense turned around, so the second half of my trip wouldn’t seem like the second half of a single trip. One nice thing about the universe: once you realize that the PH=FH, then you can find out the radius of cylinder spacetime when entropy maxes out and starts turning around.

Thanks for the post.

once you realize that the PH=FH, then you can find out the radius of cylinder spacetime when entropy maxes out and starts turning around.Assuming, that is, that the most probable classical trajectories are on average symmetric around the cylinder, so that ice will most likely “unmelt” at the same rate that it melts. Then you could calculate the radius. Indeed, I suppose you could actually try to prove that using Boltzmann’s counting argument.

Another thought — suppose you could furthermore show that the rate of entropy increase were very likely to be uniform on the cylinder. Then you could figure out the radius of the cylinder from within any old local region, by just observing how fast ice melts, and concluding how long it will take to get to the half way point.

Thanks for the note Craig.

Boltzmann’s argument relies on the assumption that the probability of a macrostate is proportional to the Lebesgue measure of the corresponding region in phase space. The usual justification of this assumption is that the system under consideration is ergodic. Apart from a set of measure 0, every point in the phase space of such a system will be on a trajectory for which the time spent in a particular macro-region is proportional to the measure of the region.

The problem with CTCs is that they are (in general) incompatible with ergodic trajectories, since they will require the system to cycle back to its initial state in times much less than the Poincare recurrence time for the system. So they place constraints on phase space that restrict the system to the set of measure 0 which does not contain ergodic trajectories. For points in this set, you cannot appeal to the ergodic theory to connect probability with measure, so applying the Boltzmann counting argument is not so straightforward.

This is a nice thought — I wouldn’t call ergodicity the “usual justification” though. The world doesn’t seem to contain any ergodic systems, and the Kolmogorov-Arnold-Moser theorem suggests they can’t exist. And even if they did exists, it isn’t clear that their ergodicity would help… http://www.jstor.org/pss/1215826

Gabriele seriously won that round.

Hi Bryan,

Sorry if this comes so late (I haven’t had a chance to post that before and it sounded like you didn’t want to continue the conversation).

First of all, I just wanted to make it as clear as possible that none of the above was about “winning rounds” (as Anon 4:55 seems to suggest). I would never do that–that’s not how I do philosophy. If at any point I gave anyone that impression, I apologize.

Second (and more substantively), you say: “Of course, for statistical mechanics, what matters is what the *majority* of possible classical trajectories do. For short time scales on flat Minkowski spacetime, most trajectories lead to melting. On a CTC, they need not.”

I don’t think that anything in what you said shows that. You have given us no reason to think that, on CTCs, most trajectories include glasses of icy water. In fact, the very same reasons why such trajectries would see to be extrmely rare in flat STs would suggest they are extrmely rare on CTCs. Don’t you think so?

Hi Gabrielle — always nice to hear your thoughtful comments, and I honestly enjoy the discussion. No worries and no offense taken about Anon 4:55.

You’re making a fair point here — perhaps I didn’t understand you earlier. It would definitely be hard to put a glass of icewater on a CTC. Indeed, consistency constraints make it hard to put *anything* on a CTC. So I would absolutely agree with you here.

That said: my concern in the passage you quote is with a situation in which we *have* somehow managed to get icewater on a CTC (however difficult this may be). There is a certain immediate response to this which says, if the system is to remain consistent (i.e., the icewater state of affairs will return to its original configuration), then the 2nd law will be violated. My thesis in this post (though it remains to be precisely worked out) is that it need not be, if you properly apply the consistency constraints demanded by the CTC.