Relativistic gas in a Schwarzschild metric

TL;DR

This paper investigates the behavior of a relativistic gas in a Schwarzschild gravitational field, deriving transport coefficients and analyzing how gravity influences thermal and viscous properties using the Boltzmann equation.

Contribution

It provides explicit expressions for transport coefficients of a relativistic gas under gravity, highlighting their dependence on gravitational potential and temperature regimes.

Findings

Transport coefficients depend on gravitational potential.

Heat flux includes relativistic terms related to inertia of energy and gravitational potential gradient.

Equilibrium requires a balance between temperature and gravitational potential gradients.

Abstract

A relativistic gas in a Schwarzschild metric is studied within the framework of a relativistic Boltzmann equation in the presence of gravitational fields, where Marle's model for the collision operator of the Boltzmann equation is employed. The transport coefficients of bulk and shear viscosities and thermal conductivity are determined from the Chapman-Enskog method. It is shown that the transport coefficients depend on the gravitational potential. Expressions for the transport coefficients in the presence of weak gravitational fields in the non-relativistic (low temperatures) and ultra-relativistic (high temperatures) limiting cases are given. Apart from the temperature gradient the heat flux has two relativistic terms. The first one, proposed by Eckart, is due to the inertia of energy and represents an isothermal heat flux when matter is accelerated. The other, suggested by Tolman, is proportional to the gravitational potential gradient and indicates that -- in the absence of an acceleration field -- a state of equilibrium of a relativistic gas in a gravitational field can be attained only if the temperature gradient is counterbalanced by a gravitational potential gradient.