Einstein-Rosen Waves and the Geroch Group

TL;DR

This paper demonstrates an algebraic method to derive Einstein-Rosen metrics, describing cylindrical gravitational waves, by leveraging the Geroch group to transform Minkowski space into vacuum solutions with two Killing vectors.

Contribution

It provides a purely algebraic derivation of Einstein-Rosen metrics using the Geroch group, extending the approach to gravitational pulse waves.

Findings

Algebraic derivation of Einstein-Rosen metric

Extension to gravitational pulse waves

Simplification of deriving vacuum metrics with symmetries

Abstract

Under the action of the Geroch group, the Minkowski metric can be transformed into any vacuum metric with two commuting Killing vectors. In principle, this reduces the problem of deriving vacuum metrics with two commuting Killing vectors to pure algebra. In this short note, we use these facts to give a purely algebraic derivation of the Einstein-Rosen metric, which describes a cylindrical gravitational wave. Our derivation has a straightforward extension to gravitational pulse waves.