On spherically symmetric solutions of the Einstein–Euler equations

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.