Steady heat conduction in general relativity

TL;DR

This paper explores steady heat conduction in general relativity, deriving conditions for thermal equilibrium, and analyzing heat flow and temperature behavior in curved spacetime, including near horizons.

Contribution

It formulates the equations for steady heat conduction in general relativity using a variational approach and introduces a binormal equilibrium condition for temperature gradients.

Findings

Heat cannot propagate faster than light in thermal equilibrium.

The total heat diffusion on a sphere follows a red-shifted constant flux.

Local temperature remains finite near an event horizon.

Abstract

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.