Gravitational Contribution to the Heat Flux in a Simple Dilute Fluid: An Approach Based on General Relativistic Kinetic Theory to First Order in the Gradients

Richard C. Tolman analyzed the relation between a temperature gradient and a gravitational field in an equilibrium situation. In 2012, Tolman’s law was generalized to a non-equilibrium situation for a simple dilute relativistic fluid. The result in that scenario, obtained by introducing the gravitational force through the molecular acceleration, couples the heat flux with the metric coefficients and the gradients of the state variables. In the present paper it is shown, by explicitly describing the single particle orbits as geodesics in Boltzmann’s equation, that a gravitational field drives a heat flux in this type of system. The calculation is devoted solely to the gravitational field contribution to this heat flux in which a Newtonian limit to the Schwarzschild metric is assumed. The corresponding transport coefficient, which is obtained within a relaxation approximation, corresponds to the dilute fluid in a weak gravitational field. The effect is negligible in the non-relativistic regime, as evidenced by the direct evaluation of the corresponding limit.