Black holes on cylinders are not algebraically special

TL;DR

This paper classifies five-dimensional Ricci-flat metrics using Petrov types and demonstrates that a black hole on a cylinder is not algebraically special, unlike black strings and black holes.

Contribution

It introduces a Petrov classification for five-dimensional metrics and applies it to show black holes on cylinders are not algebraically special.

Findings

Black holes on cylinders are not algebraically special.

Black strings and black holes are algebraically special.

Classification helps distinguish geometric properties of higher-dimensional black objects.

Abstract

We give a Petrov classification for five-dimensional metrics. We classify Ricci-flat metrics that are static, have an SO(3) isometry group and have Petrov type 22. We use this classification to look for the metric of a black hole on a cylinder, i.e. a black hole with asymptotic geometry four-dimensional Minkowski space times a circle. Although a black string wrapped around the circle and the five-dimensional black hole are both algebraically special, it turns out that the black hole on a cylinder is not.