The Newtonian limit for perfect fluids

TL;DR

This paper demonstrates the existence of non-stationary solutions to Einstein-Euler equations that approach Newtonian physics as the characteristic velocity becomes much smaller than the speed of light, using a hyperbolic formulation.

Contribution

It introduces a symmetric hyperbolic formulation with a singular parameter to analyze the Newtonian limit of Einstein-Euler solutions.

Findings

Existence of non-stationary solutions with Newtonian limit.

Energy estimates independent of the small parameter.

Analysis of solution behavior as the parameter approaches zero.

Abstract

We prove that there exists a class of non-stationary solutions to the Einstein-Euler equations which have a Newtonian limit. The proof of this result is based on a symmetric hyperbolic formulation of the Einstein-Euler equations which contains a singular parameter $\ep = v_T/c$ where $v_T$ is a characteristic velocity scale associated with the fluid and $c$ is the speed of light. The symmetric hyperbolic formulation allows us to derive $\ep$ independent energy estimates on weighted Sobolev spaces. These estimates are the main tool used to analyze the behavior of solutions in the limit $\ep \searrow 0$.