Spacetimes of Weyl and Ricci type N in higher dimensions

TL;DR

This paper investigates the geometric properties of higher-dimensional spacetimes with specific Weyl and Ricci tensor types, establishing alignment conditions and deriving canonical forms of optical matrices for null congruences.

Contribution

It proves alignment of Ricci and Weyl type N tensors and derives the canonical form of optical matrices in higher-dimensional spacetimes.

Findings

Type N Ricci and Weyl tensors must be aligned.

The multiple WAND is necessarily geodetic.

Canonical forms of optical matrices are obtained for twisting and non-twisting cases.

Abstract

We study the geometrical properties of null congruences generated by an aligned null direction of the Weyl tensor (WAND) in spacetimes of the Weyl and Ricci type N (possibly with a non-vanishing cosmological constant) in an arbitrary dimension. We prove that a type N Ricci tensor and a type III or N Weyl tensor have to be aligned. In such spacetimes, the multiple WAND has to be geodetic. For spacetimes with type N aligned Weyl and Ricci tensors, the canonical form of the optical matrix in the twisting and non-twisting case is derived and the dependence of the Weyl and the Ricci tensors and Ricci rotation coefficients on the affine parameter of the geodetic null congruence generated by the WAND is obtained.