Optical structures, algebraically special spacetimes, and the Goldberg-Sachs theorem in five dimensions

TL;DR

This paper extends the Goldberg-Sachs theorem to five-dimensional spacetimes, establishing new algebraic conditions on the Weyl tensor that facilitate the analysis of optical structures and simplifying Einstein's field equations.

Contribution

It introduces a novel algebraic condition on the Weyl tensor in five dimensions, generalizing Petrov type II, and explores implications for optical structures and higher-dimensional generalizations.

Findings

The vacuum black ring has optical structures but is not algebraically special.

A new algebraic condition on the Weyl tensor ensures optical structures in 5D.

The ideas extend to six and seven dimensions, verified through examples.

Abstract

Optical (or Robinson) structures are one generalisation of four-dimensional shearfree congruences of null geodesics to higher dimensions. They are Lorentzian analogues of complex and CR structures. In this context, we extend the Goldberg-Sachs theorem to five dimensions. To be precise, we find a new algebraic condition on the Weyl tensor, which generalises the Petrov type II condition, in the sense that it ensures the existence of such congruences on a five-dimensional spacetime, vacuum or under weaker assumptions on the Ricci tensor. This results in a significant simplification of the field equations. We discuss possible degenerate cases, including a five-dimensional generalisation of the Petrov type D condition. We also show that the vacuum black ring solution is endowed with optical structures, yet fails to be algebraically special with respect to them. We finally explain the generalisation of these ideas to higher dimensions, which has been checked in six and seven dimensions.