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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