Second Law of Thermodynamics in General Relativity

Question

I recently came across some posts in StackExchange, 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?

user4552 · Answer

I came up with a thought experiment which shows that the second law of thermodynamics is compatible with special relativity

You claim to have come up with a thought experiment showing that for a certain type of violation of SR, there will be problems with thermodynamics as well. That isn't the same thing as showing that thermodynamics is compatible with SR.

The title of the question refers to general relativity, but there is nothing in the text of the question that has anything to do with general relativity.

However since the heat is a physical flow its direction must be preserved in different coordinate frames

This is not true even in Galilean relativity. Say a car is driving to the east, and heat is flowing from the engine compartment into the passenger compartment, in the car's frame. In the frame of the earth, heat is flowing to the west.

Michele Grosso · Answer

Here a formulation of the second law of thermodynamics in GR.

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. It remains constant in ideal cases where the system is in thermodynamic equilibrium or increases in case of irreversible processes. However the statistical interpretation of entropy, as the number of microstates accessible to a macroscopic state, requires it to be an invariant, that is independent of the reference frame.

To state this in GR, you firstly define the thermodynamic macrovariables transformation law in SR and secondly via the Einstein equivalence principle you extend it to GR.

1)
In SR given the laboratory reference frame $O$ and the macroscopic object reference frame $O'$ moving with uniform relative velocity $v$ vs. the laboratory, you define:
$dQ = gamma (v) dQ'$
where:
$dQ$ amount of heat supplied to the macroscopic system in the laboratory frame
$gamma (v)$ Lorentz factor
$dQ'$ amount of heat supplied to the macroscopic system in its rest frame
Note: it is the same transformation law of the energy, i.e. in compliance with the interpretation of the amount of heat as a form of energy.

As the entropy $S$ is defined as $dS = dQ / T$, where $T$ is the absolute temperature, to have it invariant you define:
$T = gamma (v) T'$

2)
The Einstein equivalence principle states that locally the physical laws can be formulated according to SR. The extension to GR is realized via the tensorial formalism.

Second Law of Thermodynamics in General Relativity

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