Post-Newtonian Jeans Equation for Stationary and Spherically Symmetrical Self-Gravitating System

TL;DR

This paper derives a post-Newtonian version of the Jeans equation for stationary, spherically symmetric self-gravitating systems, incorporating relativistic corrections to better understand galaxy dynamics near massive black holes.

Contribution

It introduces a novel post-Newtonian Jeans equation coupled with Poisson equations, extending classical models with relativistic effects for the first time.

Findings

Post-Newtonian corrections significantly affect velocity dispersion profiles.

The presence of a central black hole alters the impact of relativistic effects.

The model provides improved accuracy for galaxy dynamics near massive objects.

Abstract

The post-Newtonian Jeans equation for stationary self-gravitating systems is derived from the post-Newtonian Boltzmann equation in spherical coordinates. The Jeans equation is coupled with the three Poisson equations from the post-Newtonian theory. The Poisson equations are functions of the energy-momentum tensor components which are determined from the post-Newtonian Maxwell--J\"uttner distribution function. As an application, the effect of a central massive black hole on the velocity dispersion profile of the host galaxy is investigated and the influence of the post-Newtonian corrections are determined.