Integration in the GHP formalism III: Finding all conformally flat radiation metrics as an example of an `optimal situation'

TL;DR

This paper systematically derives all conformally flat, pure radiation metrics using the GHP formalism, demonstrating that the class of such metrics without Killing vectors is broader than previously identified.

Contribution

It provides a complete classification of conformally flat, pure radiation metrics within the GHP formalism, extending prior results by Wils.

Findings

Broader class of conformally flat, pure radiation metrics identified

Systematic derivation confirms Wils' metric as a special case

Demonstrates effectiveness of GHP formalism in classification tasks

Abstract

Held has proposed an integration procedure within the GHP formalism built around four real, functionally independent, zero-weighted scalars. He suggests that such a procedure would be particularly simple for the `optimal situation', when the formalism directly supplies the full quota of four scalars of this type; a spacetime without any Killing vectors would be such a situation. Wils has recently obtained a metric which he claims is the only conformally flat, pure radiation metric which is not a plane wave; this metric has been shown by Koutras to admit no Killing vectors, in general. Therefore, as a simple illustration of the GHP integration procedure, we obtain systematically the complete class of conformally flat, pure radiation metrics. Our result shows that the conformally flat, pure radiation metrics, which are not plane waves, are a larger class than Wils has obtained.