Bel-Debever criteria for the classification of the Weyl tensors in higher dimensions

TL;DR

This paper extends the Bel-Debever criteria to higher-dimensional spacetimes, providing algebraic, frame-independent conditions to classify Weyl tensors and their principal types, with implications for special geometries.

Contribution

It introduces a higher-dimensional Bel-Debever characterization of the Weyl tensor, enabling algebraic classification of Weyl aligned null directions in a frame-independent manner.

Findings

Provides algebraic conditions for Weyl tensor classification in higher dimensions.

Encompasses subtypes of Weyl tensor algebraic types.

Discusses geometric interpretation of WANDs and restrictions in special spacetimes.

Abstract

An extension to higher dimensions of the Bel-Debever characterization of the Weyl tensor is considered. This provides algebraic conditions that uniquely determine the multiplicity of a Weyl aligned null direction (WAND), and thus the principal Weyl type, in a frame independent way. The specification of several "subtypes" is also encompassed by the criteria. We further comment on a Cartan-like geometrical interpretation of WANDs in terms of their invariance properties under parallel transport around infinitesimal loops. As a result, restrictions on the algebraic types permitted in spacetimes that admit a recurrent/covariantly constant vector field are outlined.