Algebraic classification of higher dimensional spacetimes based on null alignment

TL;DR

This paper reviews the algebraic classification of the Weyl tensor in higher-dimensional Lorentzian geometries, discussing methods, results, and applications including specific spacetime families and implications for quadratic gravity.

Contribution

It provides a comprehensive overview of classification techniques, recent results, and applications for higher-dimensional spacetimes, extending classical theorems and exploring invariant families.

Findings

Partial extension of Goldberg-Sachs theorem

Characterization of spacetimes with constant curvature invariants

Applications to quadratic gravity

Abstract

We review recent developments and applications of the classification of the Weyl tensor in higher dimensional Lorentzian geometries. First, we discuss the general setup, i.e. main definitions and methods for the classification, some refinements and the generalized Newman-Penrose and Geroch-Held-Penrose formalisms. Next, we summarize general results, such as a partial extension of the Goldberg-Sachs theorem, characterization of spacetimes with vanishing (or constant) curvature invariants and the peeling behaviour in asymptotically flat spacetimes. Finally, we discuss certain invariantly defined families of metrics and their relation with the Weyl tensor classification, including: Kundt and Robinson-Trautman spacetimes; the Kerr-Schild ansatz in a constant-curvature background; purely electric and purely magnetic spacetimes; direct and (some) warped products; and geometries with certain symmetries. To conclude, some applications to quadratic gravity are also overviewed.