Spacetimes characterized by their scalar curvature invariants

Understand

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.