Steady heat conduction in general relativity

We investigate the steady state of heat conduction in general relativity using a variational approach for two-fluid dynamics. We adopt coordinates based on the Landau-Lifschitz observer because it allows us to describe thermodynamics with heat, formulated in the Eckart decomposition, on a static geometry. Through our analysis, we demonstrate that the stability condition of a thermal equilibrium state arises from the fundamental principle that heat cannot propagate faster than the speed of light. We then formulate the equations governing steady-state heat conduction and introduce a binormal equilibrium condition that the Tolman temperature gradient holds for the directions orthogonal to the heat flow. As an example, we consider radial heat conductions in a spherically symmetric spacetime. We find that the total diffusion over a spherical surface satisfies a red-shifted form, $J(r) \sqrt{-g_{tt}} =$ constant. We also discuss the behavior of local temperature around an event horizon and specify the condition that the local temperature is finite there.