Thermodynamical instabilities of perfect fluid spheres in General Relativity

TL;DR

This paper demonstrates the equivalence of thermodynamical and dynamical stability conditions for static perfect fluid spheres in General Relativity, and explores the effects of the cosmological constant and Newtonian limit.

Contribution

It establishes the equivalence between thermodynamical and dynamical stability in relativistic fluid spheres and analyzes the Newtonian limit and cosmological constant effects.

Findings

Thermodynamical stability matches dynamical stability in relativistic fluid spheres.

In the Newtonian limit, the microcanonical ensemble becomes equivalent to the canonical ensemble.

The cosmological constant influences the stability of fluid spheres.

Abstract

For a static, perfect fluid sphere with a general equation of state, we obtain the relativistic equation of hydrostatic equilibrium, namely the Tolman-Oppenheimer-Volkov equation, as the thermodynamical equilibrium in the microcanonical, as well as the canonical, ensemble. We find that the stability condition determined by the second variation of entropy coincides with the dynamical stability condition derived by variations to first order in the dynamical Einstein's equations. Thus, we show the equivalence of microcanonical thermodynamical stability with linear dynamical stability for a static, spherically symmetric field in General Relativity. We calculate the Newtonian limit and find the interesting property, that the microcanonical ensemble in General Relativity transforms to the canonical ensemble for non-relativistic dust particles. Finally, for specific kinds of systems, we study the effect of the cosmological constant to the microcanonical thermodynamical stability of fluid spheres.