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

A transmission problem for quasi-linear wave equations

We prove the local existence and uniqueness of solutions to a system of quasi-linear wave equations involving a jump discontinuity in the lower order terms. A continuation principle is also established.Comment: The results of this article are essential to a local existence proof of solutions to the Einstein field equations coupled to elastic matter that represent gravitating compact bodies. The full existence proof will be presented in a separate articl