# Post-Newtonian Jeans analysis

## TL;DR

This paper extends the classical Jeans instability analysis into the first post-Newtonian regime, revealing relativistic effects that alter the critical mass for gravitational collapse and suggesting pressure can promote instability in high-temperature systems.

## Contribution

It derives a new relativistic Jeans mass in the post-Newtonian limit and compares it to the classical mass, highlighting the impact of relativistic effects on gravitational instability.

## Key findings

- Relativistic Jeans mass is smaller than Newtonian Jeans mass.
- Fractional difference increases with temperature and pressure.
- Pressure can enhance instability in high-temperature relativistic systems.

## Abstract

The Jeans analysis is studied in the first post-Newtonian limit. In other words, the relativistic effects on the local gravitational instability are considered for systems where characteristic velocity of the system and corresponding gravitational field are higher than what permitted in Newtonian limit. The dispersion relation for propagation of small perturbations is found in the post-Newtonian approximation using two different techniques. A new Jeans mass is derived and compared to the standard Jeans mass. In this limit, the relativistic effects make the new Jeans mass to be smaller than the Newtonian Jeans mass. Furthermore, the fractional difference between these two masses increases when temperature/pressure of the system increases. Interestingly, in this limit pressure can help the gravitational instability instead of preventing it. Finally the results are applied to high temperature astrophysical systems and the possibility of local fragmentations in some relativistic systems is investigated.