Post-Newtonian expansions for perfect fluids

TL;DR

This paper establishes the existence of a broad class of solutions to the Einstein-Euler equations that admit a valid first post-Newtonian expansion, using elliptic-hyperbolic formulation and energy estimates.

Contribution

It proves the existence of dynamical solutions with a first post-Newtonian expansion for the Einstein-Euler equations, extending previous work with a rigorous mathematical foundation.

Findings

Existence of solutions with first post-Newtonian expansion.

Validation of the expansion as an accurate approximation in the limit.

Application of energy estimates on weighted Sobolev spaces.

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 \cite{Oli06}, which contains a singular parameter $\ep = v_T/c$, where $v_T$ is a characteristic velocity associated with the fluid and $c$ is the speed of light. As in \cite{Oli06}, energy estimates on weighted Sobolev spaces are used to analyze the behavior of solutions to the Einstein-Euler equations in the limit $\ep\searrow 0$, and to demonstrate the validity of the first post-Newtonian expansion as an approximation.