Lorentzian manifolds and scalar curvature invariants

TL;DR

This paper explores Lorentzian manifolds and scalar curvature invariants, extending previous four-dimensional results to higher dimensions, and discusses their implications for spacetime characterization and physical theories.

Contribution

It generalizes the classification of Lorentzian manifolds based on scalar invariants from four to higher dimensions, revealing new insights into spacetime structure.

Findings

Higher-dimensional Lorentzian manifolds are either characterized by scalar invariants or are degenerate Kundt spacetimes.

The results have implications for understanding spacetime geometry in theories of gravity.

Potential applications in identifying physically relevant solutions in higher-dimensional gravity theories.

Abstract

We discuss (arbitrary-dimensional) Lorentzian manifolds and the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. Recently, we have shown that in four dimensions a Lorentzian spacetime metric is either $\mathcal{I}$-non-degenerate, and hence locally characterized by its scalar polynomial curvature invariants, or is a degenerate Kundt spacetime. We present a number of results that generalize these results to higher dimensions and discuss their consequences and potential physical applications.