The not-so-nonlinear nonlinearity of Einstein's equation

TL;DR

This paper introduces a variable choice in Einstein's equations that reduces their nonlinearity, simplifying perturbation theory and providing new tools for analyzing solutions in general relativity.

Contribution

It presents a novel formulation of Einstein's equations with reduced nonlinearity, applicable to various spacetime scenarios, and offers a new method for finding exact solutions.

Findings

Reduced nonlinear form of Einstein's equations using specific variables

Simplified perturbation theory for Kerr and plane wave spacetimes

A geometrical interpretation linking local and background causal structures

Abstract

Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate all such nonlinearities beyond a particular order: Both Landau-Lifshitz and tetrad formulations of Einstein's equation are obtained which involve only finite products of the unknowns and their derivatives. Considerable additional simplifications arise in physically-interesting cases where metrics becomes approximately Kerr or, e.g., plane waves, suggesting that the variables described here can be used to efficiently reformulate perturbation theory in a variety of contexts. In all cases, these variables are shown to have simple geometrical interpretations which directly relate the local causal structure associated with the metric of interest to the causal structure associated with a prescribed background. A new method to search for exact solutions is outlined as well.