Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics, with zero cosmological constant

TL;DR

This paper presents a simplified and complete invariant classification of conformally flat pure radiation metrics using the generalised invariant formalism, confirming previous results and clarifying subtle differences in classification approaches.

Contribution

It introduces an intrinsic coordinate approach for invariant classification, simplifying analysis and resolving discrepancies in earlier classifications of these metrics.

Findings

Complete classification of conformally flat pure radiation metrics (except plane waves)

Identification of genuinely non-redundant functions of one variable

Clarification of the orders of Cartan invariants for metric classification

Abstract

Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates; in particular we confirm that the three apparently non-redundant functions of one variable are genuinely non-redundant, and easily identify the subclasses which admit a Killing and/or a homothetic Killing vector. Most of our results agree with the earlier classification carried out by Skea in the different Koutras-McIntosh coordinates, which required much more involved calculations; but there are some subtle differences. Therefore, we also rework the classification in the Koutras-McIntosh coordinates, and by paying attention to some of the subtleties involving arbitrary functions, we are able to obtain complete agreement with the results obtained in intrinsic coordinates. In particular, we have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.