IDEAL characterization of vacuum pp-waves

TL;DR

This paper develops an extended IDEAL classification method for vacuum pp-wave spacetimes, enabling the identification of these solutions through tensorial equations despite their vanishing scalar invariants.

Contribution

It introduces a modified IDEAL approach that extends the Stewart-Walker lemma, allowing classification of vacuum pp-waves with isometry groups.

Findings

Successfully classifies vacuum pp-waves with higher-dimensional isometry groups.

Extends the IDEAL framework to handle spacetimes with vanishing scalar invariants.

Identifies all but one vacuum pp-wave solution within the classification scheme.

Abstract

An IDEAL characterization of a particular spacetime metric, $g_0$, consists of a set of tensorial equations $T[g] = 0$ arising from expressions constructed from the metric, $g$, its curvature tensor and its covariant derivatives and which are satisfied if and only if $g$ is locally isometric to the original metric $g_0$. Earlier applications of the IDEAL classification of spacetimes relied on the construction of particular scalar polynomial curvature invariants as an important step in the procedure. In this paper we investigate the well-known class of vacuum pp-wave spacetimes, where all scalar polynomial curvature invariants vanish, and determine the applicability of an IDEAL classification for these spacetimes. We consider a modification of the IDEAL approach which permits a corresponding extension of the Stewart-Walker lemma. With this change, we are able to construct invariants and IDEAL-ly classify all of the vacuum pp-wave solutions which admit a two- or higher-dimensional isometry group, with the exception of one case.