Sachs equations and plane waves II: Isometries and conformal isometries

TL;DR

This paper analyzes the symmetries of plane wave spacetimes, classifying their isometries and conformal automorphisms, and explores implications for spacetime classification and observable encoding.

Contribution

It provides a detailed classification of isometries and conformal automorphisms of plane wave spacetimes, including explicit examples and the structure of symmetry groups.

Findings

Classified all conformal automorphisms of plane waves.

Presented an explicit vacuum spacetime example encoding Bernoulli shift.

Analyzed the structure of isometry groups for homogeneous plane waves.

Abstract

This article describes the symmetries of plane wave spacetimes in dimension four and greater. It begins with a description of the isometric automorphisms, and in particular the homogeneous plane waves. Then the article turns to describing isometries from one plane wave to another. The structure of the isometries is relevant for the problem of classifying vacuum spacetimes by observables, and the article presents an explicit example of a family of vacuum spacetimes encoding the Bernoulli shift, and so not classifiable by observables. Next, the article turns to a description of the conformal isometries. Here it is assumed that the conformal curvature does not vanish identically, the case of Minkowski space being both trivial and very degenerate. The article then classifies all conformal automorphisms and isometries of plane waves of the Rosen and Brinkmann forms.