The stability of relativistic stars and the role of the adiabatic index

TL;DR

This paper analyzes the stability of three analytical solutions to Einstein's equations for fluid spheres, focusing on the role of the adiabatic index and its implications for the stability of various compact stars.

Contribution

It re-examines the stability ranges of the Tolman VII, Buchdahl, and Nariai IV solutions and investigates how the adiabatic index influences stability conditions.

Findings

All solutions are stable over extensive compactness ranges.

The critical adiabatic index varies linearly with the ratio of central pressure to energy density.

Constraints on the adiabatic index can inform stable equations of state for compact objects.

Abstract

We study the stability of three analytical solutions of the Einstein's field equations for spheres of fluid. These solutions are suitable to describe compact objects including white dwarfs, neutron stars and supermassive stars and they have been extensively employed in the literature. We re-examine the range of stability of the Tolman VII solution, we focus on the stability of the Buchdahl solution which is under contradiction in the literature and we examine the stability of the Nariai IV solution. We found that all the mentioned solutions are stable in an extensive range of the compactness parameter. We also concentrate on the effect of the adiabatic index on the instability condition. We found that the critical adiabatic index, depends linearly on the ratio of central pressure over central energy density $P_c/{\cal E}_c$, up to high values of the compactness. Finally, we examine the possibility to impose constraints, via the adiabatic index, on realistic equations of state in order to ensure stable configurations of compact objects.