Entropy and time travel

Question

Suppose that time travel is possible.

Can then we circumvent the law of increasing entropy (second law of thermodynamics) transferring entropy from the future to the past?

Answer 1

As a general rule, the second law is used to rule out physical theories; if your physical theory allows you to circumvent the second law then the problem is with your theory, not the second law. This is summed up in a famous quote by Sir Arthur Stanley Eddington:

The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

What does this mean for time travel? Well, it means that if your theory of time travel allows you to circumvent the second law, then your theory is almost certainly wrong.

Does this make time travel impossible? I'm not sure. It could be that time travel is inherently energetically costly, so that whatever you can gain by sending something back in time is outweighed by the cost of sending it. But since we don't know of any physical way in which time travel could be implemented, it's hard to speculate.

Answer 2

I gave a plus for Nathaniel's answer, for I agree at least for large macroscopic systems. The issue is a bit subtle with smaller systems and quantum systems. It is known that some systems sufficiently small can evolve in a way that reverses the flow of entropy so that $dS/dt~<~0$ for some interval of time. The increase in entropy is largely a statistical result, and as a result it can fluctuate in the opposite direction.

Eddington was right about large systems. What about small systems? A similar statement could be made about quantum mechanics. Any theory that violates quantum mechanics in a fundamental way is either completely wrong or at best an approximation that ignores unitary evolution of quantum mechanics.

A quantum state $|\psi\rangle$ that enters a closed timelike curve means that at the point the path loops there is the duplication of the state $|\psi>~\rightarrow~|\psi\rangle|\psi\rangle$. This is known to be not unitary. The reason is that if the state $|\psi\rangle~=~a|\alpha\rangle~+~b|\beta\rangle$ is duplicated we have $$ |\psi>~\rightarrow~|\psi\rangle|\psi\rangle~=~a^2|\alpha\rangle|\alpha\rangle~+~2ab|\alpha\rangle|\beta\rangle~+~b^2|\beta\rangle|\beta\rangle. $$ However this process is basis specific because we could just duplicate $|\alpha\rangle~\rightarrow$ $|\alpha\rangle|\alpha\rangle$ and $|\beta\rangle~\rightarrow$ $|\beta\rangle|\beta\rangle$. And the state $|\psi\rangle$ transforms into $$ |\psi>~\rightarrow~~a^2|\alpha\rangle|\alpha\rangle~+~b^2|\beta\rangle|\beta\rangle. $$ This means potentially a number of things, such as the two states, even given the existence of closed timelike curves, can't interfer in any way. This means the time traveling curve can't self intersect. Hence if time travel is possible it requires that quantum paths or amplitudes not mutually interfere.

This may not rule out time travel, but it imposes a condition on nature that is difficult to uphold. It is in a way similar to the classical condition Kip Thorne imposes to prevent paradoxical time loops. It is then plausible that on scales larger than the Planck scale time loops are not possible and this further requires that time machines or wormholes made into time gates etc are forbidden. Beneath the Planck scale, or the transPlanckian scale, it could be that time loops about in various ways. Yet by some means these are probably removed by renormalization and do not contribute to measurable physics.