Refinements of the Weyl tensor classification in five dimensions

TL;DR

This paper refines the classification of the Weyl tensor in five-dimensional spacetimes by introducing spin types and analyzing the Weyl operator's Segre types, enhancing understanding of algebraically special cases and their physical examples.

Contribution

It introduces a new refinement of the Weyl tensor classification in five dimensions using spin types and Segre types, including analysis of nilpotence and differences from four-dimensional cases.

Findings

Refined classification for types N, III, II, D

Analysis of nilpotence in higher dimensions

Application to specific spacetimes like black holes

Abstract

We refine the null alignment classification of the Weyl tensor of a five-dimensional spacetime. The paper focusses on the algebraically special alignment types {\bf {N}}, {\bf {III}}, {\bf {II}} and {\bf {D}}, while types {\bf {I}} and {\bf {G}} are briefly discussed. A first refinement is provided by the notion of spin type of the components of highest boost weight. Second, we analyze the Segre types of the Weyl operator acting on bivector space and examine the intersection with the spin type classification. We present a full treatment for types {\bf {N}} and {\bf {III}}, and illustrate the classification from different viewpoints (Segre type, rank, spin type) for types {\bf {II}} and {\bf {D}}, paying particular attention to possible nilpotence, which is a new feature of higher dimensions. We also point out other essential differences with the four-dimensional case. In passing, we exemplify the refined classification by mentioning the special subtypes associated to certain important spacetimes, such as Myers-Perry black holes, black strings, Robinson-Trautman spacetimes, and purely electric/magnetic type {\bf {D}} spacetimes.