On algebraically special vacuum spacetimes in five dimensions

TL;DR

This paper classifies and analyzes five-dimensional vacuum spacetimes with special null directions, extending known four-dimensional solutions and introducing new classes, by simplifying Einstein equations and exploring their geometric structures.

Contribution

It extends the classification of algebraically special vacuum spacetimes to five dimensions, identifying new solutions and simplifying Einstein equations using coordinate transformations.

Findings

Rank 2 solutions include warped product and Kaluza-Klein versions of 4d Robinson-Trautman solutions.

Rank 1 solutions include product and analytically continued Schwarzschild spacetimes.

The paper provides a unified framework for understanding these 5d solutions.

Abstract

Vacuum solutions admitting a hypersurface-orthogonal repeated principal null direction are an important class of 4d algebraically special spacetimes. We investigate the 5d analogues of such solutions: vacuum spacetimes admitting a hypersurface-orthogonal multiple Weyl aligned null direction (WAND). Such spacetimes fall into 4 families determined by the rank of the 3X3 matrix that defines the expansion and shear of the multiple WAND. The rank 3 and rank 0 cases have been studied previously. We investigate the 2 remaining families. We show how to define coordinates which lead to a considerable simplification of the Einstein equation with cosmological constant. The rank 2 case gives warped product and Kaluza-Klein versions of the 4d Robinson-Trautman solutions as well as some new solutions. The rank 1 case gives product, or analytically continued Schwarzschild, spacetimes.