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.

Hyper-intelligent fish and black hole thermodynamics

Bill Unruh's recent collection on black hole analogues begins,

Deep beneath the great encircling seas of the Discworld lived a species of hyper-intelligent fish. (Unruh 2007, p.1)

Unusual, but inspiring: Unruh compares Hawking radiation -- the thermal heat bath emitted by black holes -- to a scenario he imagines in Terry Pratchett's Discworld. Pratchett's world is basically a big dish, with water flowing over the edges.

On Unruh's take, the dish-water is filled with little physicist fish, who are trying to determine the laws of physics. The fish are blind, but use sound waves to interpret their environment. And they are mostly successful. However, as water falls off the edge of the world, it reaches speeds faster than the speed of sound. Events beyond this "sound horizon" are thus inaccessible to the fish in the ocean.

One day, a graduate-student-fish goes flying off the edge while the professor-fish observes. (Professor Unruh apparently expects a lot of his students.) The graduate student yells "Help," while falling off. Then he plunges to his doom. But, from the professor's perspective, the sound of the graduate student's scream persists forever, getting ever more bass-shifted, as the student approaches the horizon.

The point is, the unlucky graduate-student-fish is directly analogous to an astronaut falling into a black hole. From the astronaut's perspective, nothing special happens as she crosses the event horizon. But from an outside observer's perspective, the astronaut appears to be forever approaching (but never crossing) the event horizon, and the light she emits getts ever more red-shifted.

Of course, the astronaut will get ripped to shreds by tidal forces, while the fish will not.

And so the "black hole analogue" debate begins. Black holes are widely believed to have a number of thermal properties -- for example, black holes have a temperature proportional to their surface gravity. Analogously, soundless "dumb-holes" (as Unruh calls them) in water can be shown to have interesting thermal properties as well. And -- tantalizingly -- it appears possible to carry out experiments that would actually test the properties of "dumb-holes," even though black holes remain outside our reach.

But does evidence for a sound-based analogue somehow provide us evidence about a real black hole?

I see no plausible way that it can. Although a black hole is mathematically similar to a "dumb hole," it is not the same thing. And history has something to teach us here: gas and fluid vortices are "mathematically similar" to Descartes' aether vortices. But experiments with the former do not provide evidence for the latter. After all, aether vortices don't exist! So, in spite of some interesting recent experiments (see here), we still don't have any new evidence that black holes have thermal properties.

Nevertheless, there might be one thing that sound-based experiments can still teach us about black holes, according to Unruh:

such successful experiments would greatly increase the confidence in the approximation which were being made in both the gravitational and the analogue situations. ... Certainly the suggestions from the sonic case are that Planckian physics is irrelevant to black hole evaporation, and that the radiation emitted by a black hole is due to low energy processes, processes on the length scale set by the black hole, and not by quantum gravity. (Unruh 2007, p.3.)

This to me seems very plausible: an analogy can tell us whether or not scale is relevant to the effect. According to Unruh, sound-based experiments are really teaching us that black hole thermodynamics is about essentially macroscopic effects. So, our prediction of thermal effects like Hawking radiation won't change when a new theory of quantum gravity comes along, and modifies our picture of the (microscopic, high-energy) Planck scale.

It's a bold and intriguing suggestion, but I'll wait for the iron hand of history to decide.
(If you have a Springer subscription, you can see a version of Unruh's article here.)