According to the increase of entropy principle, the entropy of the universe is always increasing. So, does going back in time violates the second law of thermodynamics? Because entropy of the universe will have to decrease in that case.
Does that make time travel theoretically impossible? I read in Wikipedia that general theory of relativity does allow going back in time, at least theoretically.
The simplest example in those cases (and the oldest, too) is always the timelike cylinder, defined as $M = \mathbb{R} \times S$, with the metric
$$ds^2 = -dt^2 + dx^2$$
Where $t$ is cyclical defined on $[0,T]$. It's a fairly well behaved spacetime with respect to quantum theory (it is F local, mostly has a well defined Cauchy problem and all that). The one big condition that we require on it is that the wavefunction and field operators are all continuous across time, so that
\begin{eqnarray} \Psi(t) &=& \Psi(t+T)\\ \phi(t) &=& \phi(t + T) \end{eqnarray}
As usual we define entropy in a quantum theory as
$$S = -\operatorname{Tr}(\rho \ln (\rho))$$
As $\rho$ depends on the quantum state, it is easy to see that it will itself be cyclical in time, meaning that $S(t) = S(t+T)$. This is not very surprising because we do require all measurable quantities to be the same as identical spacetime points and thus by continuity they have to be thusly along closed timelike curves.
This is one effect of closed timelike curves called retrocausality : the evolution of system is influenced by future events. Not all initial conditions are allowed in a spacetime with closed timelike curves, as otherwise they may not provide any time evolution which is consistent, so in this example any field on that spacetime will be required to evolve with a cyclical entropy (it is quite likely, in a realistic case with interacting fields, that no such configuration exists, hence why closed timelike curves are probably not a big worry).
If such a configuration does exist by the way, the problem isn't really specific to closed timelike curves : the timelike cylinder has a causal universal cover (it's just Minkowski space), in which case we simply have the same field configuration repeating itself over and over in time (a lot of closed timelike curves have that property that they can be unrolled into causal spacetimes like that). This could be the case, it's just extremely unlikely that the initial conditions of the universe would allow such a thing to happen.
As said above, the fact that entropy goes up is just a statistical effect. It is always possible to find contrived examples in which the entropy does not go up or even goes down, but those are just statistically unlikely to happen.
At least one serious physiciat seems to think that closed timelike curves would indeed violate the second law of thermodynamics, although his argument is not as straightforward as you presented it and makes use of something called the generalized second law of thermodynamics.
Time Machines Would Run Afoul of the Second Law of Thermodynamics - Scientific American
