The Newtonian limit for perfect fluids

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.Comment: Differs slightly from the published versio

The Newtonian limit on cosmological scales

We establish the existence of a wide class of inhomogeneous relativistic solutions to the Einstein-Euler equations that are well approximated on cosmological scales by solutions of Newtonian gravity. Error estimates measuring the difference between the Newtonian and relativistic solutions are provided.Comment: This version agrees with the published on

Lagrange coordinates for the Einstein-Euler equations

We derive a new symmetric hyperbolic formulation of the Einstein-Euler equations in Lagrange coordinates that are adapted to the Frauendiener-Walton formulation of the Euler equations. As an application, we use this system to show that the densitized lapse and zero shift coordinate systems for the vacuum Einstein equations are equivalent to Lagrange coordinates for a fictitious fluid with a specific equation of state