$\mathcal{I}$-degenerate pseudo-Riemannian metrics

TL;DR

This paper investigates pseudo-Riemannian spaces with degenerate curvature invariants, introducing a new class of $ ext{I}$-degenerate metrics beyond known examples like Kundt and Walker metrics, using invariant theory and boost techniques.

Contribution

It develops a new approach based on invariant theory and boosts to identify and classify $ ext{I}$-degenerate metrics, expanding the known examples beyond Kundt and Walker metrics.

Findings

Identifies a broad class of $ ext{I}$-degenerate metrics

Provides a criterion for $ ext{I}$-degeneracy in metrics

Extends understanding of VSI and CSI metrics

Abstract

In this paper we study pseudo-Riemannian spaces with a degenerate curvature structure i.e. there exists a continuous family of metrics having identical polynomial curvature invariants. We approach this problem by utilising an idea coming from invariant theory. This involves the existence of a boost which is assumed to extend to a neighbourhood. This approach proves to be very fruitful: It produces a class of metrics containing all known examples of $\mathcal{I}$-degenerate metrics. To date, only Kundt and Walker metrics have been given, however, our study gives a plethora of examples showing that $\mathcal{I}$-degenerate metrics extend beyond the Kundt and Walker examples. The approach also gives a useful criterion for a metric to be $\mathcal{I}$-degenerate. Specifically, we use this to study the subclass of VSI and CSI metrics (i.e., spaces where polynomial curvature invariants are all vanishing or constants, respectively).