Static spherical perfect fluid stars with finite radius in general relativity: a review

TL;DR

This review explains the derivation, solutions, and properties of the TOV equation for static, spherical relativistic stars, emphasizing existence, uniqueness, and bounds on stellar compactness.

Contribution

It consolidates and clarifies the derivation, solutions, and bounds of the TOV equation for relativistic stars, providing a comprehensive pedagogical overview.

Findings

Proof of existence and uniqueness of finite radius solutions

Derivation of the relativistic ideal gas equation of state

Validation of the Buchdahl bound for stellar compactness

Abstract

In this article, we provide a pedagogical review of the Tolman-Oppenheimer-Volkoff (TOV) equation and its solutions which describe static, spherically symmetric gaseous stars in general relativity. Our discussion starts with a systematic derivation of the TOV equation from the Einstein field equations and the relativistic Euler equations. Next, we give a proof for the existence and uniqueness of solutions of the TOV equation describing a star of finite radius, assuming suitable conditions on the equation of state characterizing the gas. We also prove that the compactness of the gas contained inside a sphere centered at the origin satisfies the well-known Buchdahl bound, independent of the radius of the sphere. Further, we derive the equation of state for an ideal, classical monoatomic relativistic gas from statistical mechanics considerations and show that it satisfies our assumptions for the existence of a unique solution describing a finite radius star. Although none of the results discussed in this article are new, they are usually scattered in different articles and books in the literature; hence it is our hope that this article will provide a self-contained and useful introduction to the topic of relativistic stellar models.