A higher-dimensional generalization of the geodesic part of the Goldberg-Sachs theorem

TL;DR

This paper proves that in higher-dimensional Einstein spacetimes, the existence of a non-geodesic multiple Weyl-aligned null direction implies the existence of a geodesic one, and classifies all five-dimensional cases.

Contribution

It extends the Goldberg-Sachs theorem to higher dimensions by showing the equivalence of geodesic and non-geodesic multiple WANDs in Einstein spacetimes and classifies five-dimensional solutions.

Findings

Any higher-dimensional Einstein spacetime with a non-geodesic multiple WAND also has a geodesic one.

All five-dimensional Einstein spacetimes with a non-geodesic multiple WAND are classified.

Abstract

In more than four spacetime dimensions, a multiple Weyl-aligned null direction (WAND) need not be geodesic. It is proved that any higher-dimensional Einstein spacetime admitting a non-geodesic multiple WAND also admits a geodesic multiple WAND. All five-dimensional Einstein spacetimes admitting a non-geodesic multiple WAND are determined.