Higher dimensional Kerr-Schild spacetimes

TL;DR

This paper explores the properties of higher-dimensional Kerr-Schild spacetimes, classifying their Weyl tensor types, optical characteristics, and potential extensions of the Goldberg-Sachs theorem, with implications for black hole solutions.

Contribution

It provides a detailed classification of higher-dimensional Kerr-Schild solutions, including conditions for their Weyl types and optical properties, and proposes a weak extension of the Goldberg-Sachs theorem.

Findings

Non-expanding solutions are equivalent to vacuum Kundt solutions of type N.

Expanding solutions are of Weyl type II or D and have shearing, caustics, and curvature singularities.

A possible weak extension of the Goldberg-Sachs theorem for Kerr-Schild spacetimes is proposed.

Abstract

We investigate general properties of Kerr-Schild (KS) metrics in n>4 spacetime dimensions. First, we show that the Weyl tensor is of type II or more special if the null KS vector k is geodetic (or, equivalently, if T_{ab}k^ak^b=0). We subsequently specialize to vacuum KS solutions, which naturally split into two families of non-expanding and expanding metrics. After demonstrating that non-expanding solutions are equivalent to the known class of vacuum Kundt solutions of type N, we analyze expanding solutions in detail. We show that they can only be of the type II or D, and we characterize optical properties of the multiple Weyl aligned null direction (WAND) k. In general, k has caustics corresponding to curvature singularities. In addition, it is generically shearing. Nevertheless, we arrive at a possible "weak" n>4 extension of the Goldberg-Sachs theorem, limited to the KS class, which matches previous conclusions for general type III/N solutions. In passing, properties of Myers-Perry black holes and black rings related to our results are also briefly discussed.