$\mathcal{I}$-degenerate pseudo-Riemannian 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).

23 pages; v2: changed title+notation and cleaned up a bit

References in corpus (7)

• Spacetimes characterized by their scalar curvature invariants
• Higher dimensional VSI spacetimes
• Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension
• A spacetime not characterised by its invariants is of aligned type II
• A simple test for spacetime symmetry
• Pseudo-Riemannian VSI spaces II
• 4D neutral signature VSI and CSI spaces

Cited by in corpus (7)

• A frame based approach to computing symmetries with non-trivial isotropy groups
• Spacetime invariants and their uses
• Universal spacetimes in four dimensions
• $\mathcal{I}$-Preserving Diffeomorphisms of Lorentzian Manifolds
• Teleparallel geometries not characterized by their scalar polynomial torsion invariants
• Locally Homogeneous Kundt Triples and CSI Metrics
• A Class of Integrable Metrics II