On a five-dimensional version of the Goldberg-Sachs theorem

TL;DR

This paper extends the Goldberg-Sachs theorem to five-dimensional Einstein spacetimes, characterizing the optical matrix of multiple Weyl Aligned Null Directions and exploring their algebraic and geometric properties.

Contribution

It provides necessary algebraic conditions for the optical matrix of multiple WANDs in five dimensions and classifies their canonical forms, advancing higher-dimensional algebraic classification.

Findings

Optical matrix can be reduced to three canonical forms with two parameters each.

Necessary conditions for multiple WANDs are identified but are not sufficient in 5D.

Examples of solutions for each canonical form are provided.

Abstract

Previous work has found a higher-dimensional generalization of the "geodesic part" of the Goldberg-Sachs theorem. We investigate the generalization of the "shear-free part" of the theorem. A spacetime is defined to be algebraically special if it admits a multiple Weyl Aligned Null Direction (WAND). The algebraically special property restricts the form of the "optical matrix" that defines the expansion, rotation and shear of the multiple WAND. After working out some general constraints that hold in arbitrary dimensions, we determine necessary algebraic conditions on the optical matrix of a multiple WAND in a five-dimensional Einstein spacetime. We prove that one can choose an orthonormal basis to bring the 3 x 3 optical matrix to one of three canonical forms, each involving two parameters, and we discuss the existence of an "optical structure" within these classes. Examples of solutions corresponding to each form are given. We give an example which demonstrates that our necessary algebraic conditions are not sufficient for a null vector field to be a multiple WAND, in contrast with the 4d result.