The fast Newtonian limit for perfect fluids

TL;DR

This paper demonstrates the existence of solutions to Einstein-Euler equations that approximate Newtonian gravity in the limit of the speed of light, extending previous results to broader initial conditions.

Contribution

It generalizes earlier work by allowing a larger class of initial data and employs energy and dispersive estimates in a non-local hyperbolic framework for the singular limit.

Findings

Convergence of relativistic solutions to Newtonian Poisson-Euler solutions as c→∞

Extension of previous results to more general initial data

Use of weighted Sobolev space estimates in the analysis

Abstract

We prove the existence of a large class of dynamical solutions to the Einstein-Euler equations for which the fluid density and spatial three-velocity converge to a solution of the Poisson-Euler equations of Newtonian gravity. The results presented here generalize those of \cite{Oli06} to allow for a larger class of initial data. As in \cite{Oli06}, the proof is based on a non-local symmetric hyperbolic formulation of the Einstein-Euler equations which contain a singular parameter $\ep=v_T/c$ with $v_T$ a characteristic speed associated to the fluid and $c$ the speed of light. Energy and dispersive estimates on weighted Sobolev spaces are the main technical tools used to analyze the solutions in the singular limit $\ep\searrow 0$.