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

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• Local Invariants Vanishing on Stationary Horizons: A Diagnostic for Locating Black Holes
• The Curious Case of Null Warped Space
• Gyratons on direct-product spacetimes
• Weyl Tensor Classification in Four-dimensional Manifolds of All Signatures
• Pseudo-Riemannian VSI spaces II
• Gyratons on Melvin spacetime
• Pseudo-Riemannian VSI spaces
• Algebraic classification of five-dimensional spacetimes using scalar invariants
• Properties of kinematic singularities
• Geometric Properties of Stationary and Axisymmetric Killing Horizons
• Killing vectors in higher dimensional spacetimes with constant scalar curvature invariants
• All metrics have curvature tensors characterised by its invariants as a limit: the ε-property