Post-Newtonian Expansions for Perfect Fluids

Abstract

We prove the existence of a large class of dynamical solutions to the Einstein-Euler equations that have a first post-Newtonian expansion. The results here are based on the elliptic-hyperbolic formulation of the Einstein-Euler equations used in [15], which contains a singular parameter $${\epsilon = v_T/c}$$ , where v T is a characteristic velocity associated with the fluid and c is the speed of light. As in [15], energy estimates on weighted Sobolev spaces are used to analyze the behavior of solutions to the Einstein-Euler equations in the limit $${\epsilon\searrow 0}$$ , and to demonstrate the validity of the first post-Newtonian expansion as an approximation.