Twisting algebraically special solutions in five dimensions

TL;DR

This paper classifies five-dimensional vacuum Einstein solutions with a cosmological constant that are algebraically special and have a specific null direction structure, revealing both known warped products and new solutions.

Contribution

It provides the general form of such solutions, explicitly determines their coordinate dependence, and identifies new solution families beyond warped products.

Findings

Solutions include warped products of 4D algebraically special solutions.

Several new solution families are identified.

The Einstein equations reduce to PDEs in three coordinates.

Abstract

We determine the general form of the solutions of the five-dimensional vacuum Einstein equations with cosmological constant for which (i) the Weyl tensor is everywhere type II or more special in the null alignment classification of Coley et al., and (ii) the $3 \times 3$ matrix encoding the expansion, shear and twist of the aligned null direction has rank 2. The dependence of the solution on 2 coordinates is determined explicitly, so the Einstein equation reduces to PDEs in the 3 remaining coordinates, just as for four-dimensional algebraically special solutions. The solutions fall into several families. One of these consists of warped products of four-dimensional algebraically special solutions. The others are new.