Spinor classification of the Weyl tensor in five dimensions

TL;DR

This paper refines the spinor classification of the Weyl tensor in five dimensions, revealing a reality condition that limits possible types, and classifies vacuum solutions, including the black ring, within this framework.

Contribution

It introduces a revised classification scheme for the Weyl tensor in five dimensions accounting for a reality condition and links it to existing tensor classifications.

Findings

A reality condition reduces classification types.

All vacuum solutions of the most special type are classified.

The black ring is algebraically general in this classification.

Abstract

We investigate the spinor classification of the Weyl tensor in five dimensions due to De Smet. We show that a previously overlooked reality condition reduces the number of possible types in the classification. We classify all vacuum solutions belonging to the most special algebraic type. The connection between this spinor and the tensor classification due to Coley, Milson, Pravda and Pravdov\'a is investigated and the relation between most of the types in each of the classifications is given. We show that the black ring is algebraically general in the spinor classification.