Existence of families of spacetimes with a Newtonian limit

TL;DR

This paper reviews frame theory, developed by J"urgen Ehlers, which bridges general relativity and Newtonian gravity through a parameter that approaches zero, enabling rigorous analysis of the Newtonian limit and post-Newtonian expansions.

Contribution

It provides a comprehensive review of frame theory and its applications to understanding the Newtonian limit and post-Newtonian expansions in gravitational physics.

Findings

Frame theory unifies general relativity and Newtonian gravity via a parameter.

Rigorous results on the Newtonian limit have been derived from frame theory.

Applications include analysis of post-Newtonian expansions.

Abstract

J\"urgen Ehlers developed \emph{frame theory} to better understand the relationship between general relativity and Newtonian gravity. Frame theory contains a parameter $\lambda$, which can be thought of as $1/c^2$, where $c$ is the speed of light. By construction, frame theory is equivalent to general relativity for $\lambda >0$, and reduces to Newtonian gravity for $\lambda =0$. Moreover, by setting $\ep=\sqrt{\lambda}$, frame theory provides a framework to study the Newtonian limit $\ep \searrow 0$ (i.e. $c\to \infty$). A number of ideas relating to frame theory that were introduced by J\"urgen have subsequently found important applications to the rigorous study of both the Newtonian limit and post-Newtonian expansions. In this article, we review frame theory and discuss, in a non-technical fashion, some of the rigorous results on the Newtonian limit and post-Newtonian expansions that have followed from J\"urgen's work.