Boltzmann equation in the $2{\frac12}$-post-Newtonian approximation

TL;DR

This paper develops a relativistic kinetic theory for gases under gravity within the 2.5 post-Newtonian approximation, deriving the Boltzmann equation and hydrodynamic equations up to order 1/c^7.

Contribution

It introduces the Boltzmann equation and equilibrium distribution function in the 2.5 post-Newtonian framework, extending previous models.

Findings

Derived the Boltzmann equation up to 1/c^7 order.

Calculated the energy-momentum tensor components.

Formulated hydrodynamic equations for relativistic gases in gravitational fields.

Abstract

Within the framework of the post-Newtonian $2\frac12$ approximation theory, a kinetic theory for relativistic gases in the presence of gravitational fields is developed. The Boltzmann equation and the equilibrium Maxwell-J\"uttner distribution function are determined up to $1/c^7$--order, which are used to calculate the components of the particle four-flow and energy-momentum tensor and to find the Eulerian hydrodynamic equations for the mass, mass-energy, and momentum densities in the $2\frac12$--post-Newtonian approximation. The energy conservation law follows from the hydrodynamic equation for the total energy density, which is a combination of the hydrodynamic equations for the mass and the mass-energy densities.