On higher dimensional Einstein spacetimes with a non-degenerate double Weyl aligned null direction

TL;DR

This paper proves that higher-dimensional Einstein spacetimes with a specific null direction property are necessarily of type D under certain conditions, extending Goldberg-Sachs theorem insights to six dimensions.

Contribution

It establishes conditions under which a single double Weyl aligned null direction implies the existence of a second, leading to classification of these spacetimes as type D.

Findings

Spacetimes with a geodesic, non-degenerate double WAND are of type D if Weyl tensor satisfies certain conditions.

Twistfree null directions also imply the presence of a second double WAND.

Extension of Goldberg-Sachs theorem to six dimensions is discussed.

Abstract

We prove that higher dimensional Einstein spacetimes which possess a geodesic, non-degenerate double Weyl aligned null direction (WAND) $\ell$ must additionally possess a second double WAND (thus being of type D) if either: (a) the Weyl tensor obeys $C_{abc[d}\ell_{e]}\ell^c=0$ ($\Leftrightarrow\Phi_{ij}=0$, i.e., the Weyl type is II(abd)); (b) $\ell$ is twistfree. Some comments about an extension of the Goldberg-Sachs theorem to six dimensions are also made.