Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

TL;DR

This paper develops a formalism for understanding the equilibrium states of self-gravitating fermionic systems in general relativity, connecting thermodynamics, stability, and equations of state, with applications to astrophysical objects.

Contribution

It extends previous work by providing a comprehensive formalism for self-gravitating systems in general relativity, explicitly considering fermions and their thermodynamic properties.

Findings

Derived equations for self-gravitating fermions in general relativity.

Showed the nonrelativistic limit recovers Newtonian results.

Discussed ensemble inequivalence and stability criteria.

Abstract

We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi-Dirac entropy for fermions. The maximization of entropy at fixed mass-energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium and the Tolman-Klein relations. Our paper provides all the necessary equations that are needed to construct the caloric curves of self-gravitating fermions in general relativity as done in recent works. We consider the nonrelativistic limit $c\rightarrow +\infty$ and recover the equations obtained within the framework of Newtonian gravity. We also discuss the inequivalence of statistical ensembles as well as the relation between the dynamical and thermodynamical stability of self-gravitating systems in Newtonian gravity and general relativity.