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.
(30) for the Schwarzschild case with g00 = −1/g rr = − ( 1 − 2m r ) and assuming α to be constant, we have µ T = A + q0 α ∫ 1 r2( 1 − 2m r ) 3/ 2 dr
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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.