Sachs equations and plane waves, I: Rosen universes

Abstract

This article, the first in a series, analyzes the general theory of plane wave spacetimes. Following Dmitri Aleekseevsky, these are defined as spacetimes admitting a group of dilations leaving invariant a smooth curve. If this curve is specified as part of the structure, the spacetime is termed a Penrose limit, whose theory was developed first by Roger Penrose. The main result is that every plane wave is a Rosen universe, a generalization of the smooth metrics of Albert Einstein and Nathan Rosen, allowing for certain isolated co-ordinate singularities; the latter are characterized. We conclude with an extended example, using the techniques developed in the article to associate a vacuum plane wave in four dimensions to any hyperbolic billiard trajectory.