On the Goldberg–Sachs theorem in six dimensions

Abstract

We study the “shear-free” part of the Goldberg–Sachs theorem for type II and D six-dimensional Einstein spacetimes. A spacetime is of the type II or D if it admits a double Weyl aligned null direction (WAND). The Goldberg–Sachs theorem then restricts geometric properties of double WANDs. This restriction can be expressed in terms of constraints on the form of the “optical matrix” defining the expansion, rotation, and shear of the double WAND. We show that a rank-four and rank-three optical matrix is orthogonally similar to one of three canonical forms, reducing the number of free parameters of the optical matrix from ten to five.