Generating exact solutions to Einstein's equation using linearized approximations

TL;DR

This paper demonstrates a method to transform certain linearized solutions of Einstein's equations into exact solutions using gauge transformations, enabling derivation of metrics like Kerr and gravitational waves from approximations.

Contribution

It introduces a gauge transformation technique that converts linearized Einstein solutions into exact solutions, simplifying the process of obtaining complex metrics.

Findings

Exact Kerr metric derived from linear approximation

Gravitational plane wave metric obtained exactly

Method eliminates nonlinearities in Einstein's equations

Abstract

We show that certain solutions to the linearized Einstein equation can---by the application of a particular type of linearized gauge transformation---be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations.