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

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.

Riemann problem and limits of solutions to the isentropic relativistic Euler equations for isothermal gas with flux approximation

We are concerned with the vanishing flux-approximation limits of solutions to the isentropic relativistic Euler equations governing isothermal perfect fluid flows. The Riemann problem with a two-parameter flux approximation including pressure term is first solved. Then, we study the limits of solutions when the pressure and two-parameter flux approximation vanish, respectively. It is shown that, any two-shock-wave Riemann solution converges to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between these two shocks tends to a weighted δ-measure that forms a delta shock wave. By contract, any two-rarefaction-wave solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations, and the intermediate state in between tends to a vacuum state.

Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations

The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.