I recently came across some posts in Stack Exchange, which say that simultaneity doesn't have any meaning in General Relativity (see Is simultaneity well-defined in general relativity?).
I came up with a thought experiment which shows that the second law of thermodynamics is compatible with special relativity:
Considering a rod which has two temperatures $T_1>T_2$, where $T_1$ is the temperature in the left half and $T_2$ is the temperature in the right half. If I intentionally hypothesize that the speed of light can be surpassed by heat, then there will exist a point $A$ on the right end of the rod which has higher temperature $T_3>T_2$.
However since this is outside the light cone emanating from the middle, one can then construct a linear simultaneity hypersurface which goes through the point $A$ and say just to the right of the middle point where the temperature is $T_2$.
However since the heat is a physical flow its direction must be preserved in different coordinate frames (the heat vector can have different components in different frames, but the physical flow will be in the same direction). In particular the heat flow will be from left to right.
However, that means that heat is flowing from cold to hot and thereby the 2nd law will be violated. Thus, the violation of SR (Special Relativity) has led to a violation of TD (Thermodynamics). Maybe this thought experiment is wrong. Please point out if so.
But if my thought experiment is correct and simultaneity really doesn't have a meaning in GR (General Relativity) and one can really make “almost anything” simultaneous then one can't really formulate the 2nd law properly in GR. Or does this have to do something with the fact that in GR we are not dealing with inertial frames of reference?