On the Symmetries of the Edgar-Ludwig Metric

TL;DR

This paper analyzes the symmetries of the Edgar-Ludwig metric by solving conformal Killing equations, revealing the existence of conformal symmetries and specific cases with additional homotheties or motions.

Contribution

It provides a detailed classification of the conformal and isometric symmetries of the Edgar-Ludwig metric, including a previously overlooked special case.

Findings

Fifteen independent conformal Killing vectors generally exist.

Most cases admit no Killing or homothetic vectors.

Special cases can have one or two-dimensional groups of symmetries.

Abstract

The conformal Killing equations for the most general (non-plane wave) conformally flat pure radiation field are solved to find the conformal Killing vectors. As expected fifteen independent conformal Killing vectors exist, but in general the metric admits no Killing or homothetic vectors. However for certain special cases a one-dimensional group of homotheties or motions may exist and in one very special case, overlooked by previous investigators, a two-dimensional homethety group exists. No higher dimensional groups of motions or homotheties are admitted by these metrics.