Spacetimes characterized by their scalar curvature invariants

Abstract

In this paper we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an $\mathcal{I}$-non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is $\mathcal{I}$-non-degenerate. This enables us to prove our main theorem that a spacetime metric is either $\mathcal{I}$-non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of \emph{strong} and \emph{weak} non-degeneracy.

38pages; v2: some corrections + refs

References in corpus (8)

• Classification of the Weyl Tensor in Higher Dimensions and Applications
• Kundt Spacetimes
• The Invar tensor package: Differential invariants of Riemann
• Lorentzian spacetimes with constant curvature invariants in four dimensions
• Lorentzian spacetimes with constant curvature invariants in three dimensions
• The type N Karlhede bound is sharp
• The curvature homogeneity bound for Lorentzian four-manifolds
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Cited by in corpus (5)

• Kundt Spacetimes
• Lorentzian spacetimes with constant curvature invariants in four dimensions
• General Kundt spacetimes in higher dimensions
• Gyratons on direct-product spacetimes
• Properties of kinematic singularities