Post-Newtonian Kinetic Theory

TL;DR

This paper develops a relativistic kinetic theory for gases under gravity using second post-Newtonian approximation, deriving Boltzmann, hydrodynamic equations, and energy relations.

Contribution

It introduces a second post-Newtonian kinetic framework for relativistic gases in gravitational fields, including Boltzmann and hydrodynamic equations.

Findings

Derived the Boltzmann equation in second post-Newtonian approximation.

Obtained hydrodynamic equations for mass, energy, and momentum densities.

Connected hydrodynamic equations to internal energy density in first post-Newtonian order.

Abstract

A kinetic theory for relativistic gases in the presence of gravitational fields is developed in the second post-Newtonian approximation. The corresponding Boltzmann equation is determined from the evolution of the one-particle distribution function with respect to the proper time along the world line of the particle. From the knowledge of the equilibrium Maxwell-J\"uttner distribution function in the second post-Newtonian approximation the components of the particle four-flow and energy-momentum tensor are obtained. The Eulerian hydrodynamic equations for the mass density, mass-energy density and momentum density in the second post-Newtonian approximation are determined from the Boltzmann equation. It is shown that the combination of the hydrodynamic equations of mass and mass-energy densities leads to the hydrodynamic equation for the internal energy density in the first post-Newtonian approximation.