Self-gravitating systems of ideal gases in the 1PN approximation

TL;DR

This paper derives the first-order post-Newtonian distribution function for ideal gases in general relativity, computes related physical profiles, and models galaxy rotation curves with static solutions.

Contribution

It introduces the Maxwell-Jüttner distribution at 1PN order and applies it to static, spherically symmetric systems, including galaxy rotation curve modeling.

Findings

Derived density, pressure, and gravitational potential profiles.

Obtained galaxy rotation curves consistent with observed flattening.

Presented a combined static solution model for galaxy cores and outskirts.

Abstract

We obtain the Maxwell-J\"uttner distribution function at first order in the post-Newtonian approximation within the framework of general relativity. Taking into account the aforesaid distribution function, we compute the particle four-flow and energy-momentum tensor. We focus on the search of static solutions for the gravitational potentials with spherical symmetry. In doing so, we obtain the density, pressure and gravitational potential energy profiles in terms of dimensionless radial coordinate by solving the aforesaid equations numerically. In particular, we find the parametric profile for the equation of state $p/\rho$ in terms of the dimensionless radial coordinate. Due to its physical relevance, we also find the galaxy rotation curves using the post-Newtonian approximation. We join two different kinds of static solutions in order to account for the linear regime near the center and the typical flatten behavior at large radii as well.