Spacetimes characterized by their scalar curvature invariants

TL;DR

This paper classifies four-dimensional Lorentzian manifolds based on their scalar curvature invariants, establishing conditions under which the spacetime metric is uniquely determined by these invariants, and identifying degenerate cases as Kundt metrics.

Contribution

It introduces the concept of $ ext{I}$-non-degenerate spacetimes and proves that such metrics are uniquely characterized by their scalar curvature invariants, except for degenerate Kundt metrics.

Findings

Spacetimes are either $ ext{I}$-non-degenerate or Kundt.

A metric not characterized by invariants must be a degenerate Kundt.

Partial results on reconstructing spacetime properties from 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.