Pseudo-Riemannian VSI spaces II

TL;DR

This paper classifies pseudo-Riemannian VSI spaces, showing their algebraic properties, and explicitly constructs new 4D neutral VSI metrics, expanding the understanding of their geometric structures.

Contribution

It introduces a classification framework for VSI spaces using boost-weight decomposition and constructs new 4D neutral VSI metrics not previously studied.

Findings

VSI spaces must have the ${f N}^G$-property.

All 4D neutral VSI spaces are either Kundt or Walker metrics.

New VSI metrics with invariant null planes are explicitly constructed.

Abstract

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). Using an algebraic classification of pseudo-Riemannian spaces in terms of the boost-weight decomposition we first show more generally that a space which is not characterised by its invariants must possess the ${\bf S}_1^G$-property. As a corollary, we then show that a VSI space must possess the ${\bf N}^G$-property (these results are the analogues of the alignment theorem, including corollaries, for Lorentzian spacetimes). As an application we classify all 4D neutral VSI spaces and show that these belong to one of two classes: (1) those that possess a geodesic, expansion-free, shear-free, and twist-free null-congruence (Kundt metrics), or (2) those that possess an invariant null plane (Walker metrics). By explicit construction we show that the latter class contains a set of VSI metrics which have not previously been considered in the literature.