For non-viscous fluids in normal flow on static or stationary spacetimes, the redshifted heat current is conserved and the redshifted temperature obeys a curved-space Laplace-type heat equation.
Thermodynamic equilibrium condition and the first law of thermodynamics for charged perfect fluids in electromagnetic and gravitational fields
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abstract
We provide a proof of the necessary and sufficient condition on the profile of the temperature, chemical potential, and angular velocity for a charged perfect fluid in dynamic equilibrium to be in thermodynamic equilibrium not only in fixed but also in dynamical electromagnetic and gravitational fields. In passing, we also present the corresponding expression for the first law of thermodynamics for such a charged star.
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General relativistic heat flow from first order hydrodynamics
For non-viscous fluids in normal flow on static or stationary spacetimes, the redshifted heat current is conserved and the redshifted temperature obeys a curved-space Laplace-type heat equation.