On Spherically Symmetric Solutions of the Einstein-Euler Equations

TL;DR

This paper constructs spherically symmetric solutions to the Einstein-Euler equations modeling gaseous stars in general relativity, using Nash-Moser theorem to handle boundary conditions near equilibrium states.

Contribution

It develops a method to find true solutions near time-periodic linearized solutions for the Einstein-Euler system, extending techniques from non-relativistic models.

Findings

Construction of solutions satisfying physical boundary conditions

Application of Nash-Moser theorem to relativistic stellar models

Establishment of solutions near equilibrium states

Abstract

We construct spherically symmetric solutions to the Einstein-Euler equations, which give models of gaseous stars in the framework of the general theory of relativity. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler equations are given by the Tolman-Oppenheimer-Volkoff equations, and time periodic solutions around the equilibrium of the linearized equations can be considered. Our aim is to find true solutions near these time-periodic approximations. Solutions satisfying so called physical boundary condition at the free boundary with the vacuum will be constructed using the Nash-Moser theorem. This work also can be considered as a touchstone in order to estimate the universality of the method which was originally developed for the non-relativistic Euler-Poisson equations.