Plane wave analysis of the second post-Newtonian hydrodynamic equations

TL;DR

This paper analyzes the second post-Newtonian hydrodynamic equations using plane wave solutions, revealing how relativistic corrections influence gravitational instability and the Jeans mass in astrophysical contexts.

Contribution

It introduces a detailed analysis of second post-Newtonian effects on hydrodynamic stability and gravitational collapse, extending previous Newtonian and first post-Newtonian models.

Findings

Second post-Newtonian corrections alter the Jeans mass.

Relativistic effects can reduce the mass needed for collapse.

The second post-Newtonian Jeans mass is larger than the first post-Newtonian one.

Abstract

The second post-Newtonian hydrodynamic equations are analyzed within the framework of a plane wave solution. The hydrodynamic equations for the mass and momentum density are coupled with six Poisson equations for the Newtonian and post-Newtonian gravitational potentials. Perturbations of the basic fields and gravitational potentials from a background state by assuming plane wave representations lead to a dispersion relation where the Jeans instability condition emerges. The influence of the first and second post-Newtonian approximations on the Jeans mass is determined. It was shown that the relative difference of the first post-Newtonian and the Newtonian Jeans masses is negative while the one of the second post-Newtonian approximation is positive. The two contributions imply a smaller mass needed for an overdensity to initiate the gravitational collapse than the one given by the Newtonian theory.