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

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 $1\mathrm{/}{c}^7$–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.