Boltzmann equation in the $2\frac{1}{2}$-post-Newtonian approximation

Abstract

Within the framework of the post-Newtonian $2\frac{1}{2}$ approximation theory, a kinetic theory for relativistic gases in the presence of gravitational fields is developed. The Boltzmann equation and the equilibrium Maxwell--Jüttner 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 $2\frac{1}{2}$--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. Here, the total energy conservation law is derived in the $1\frac{1}{2}$--post-Newtonan approximation, since we have to know the components of the metric tensor $g_{ij}$ in the $3\frac{1}{2}$--th post-Newtonian approximation to obtain the energy conservation law in the $2\frac{1}{2}$--th post-Newtonian approximation.

arXiv comment14 pages