Black holes on cylinders are not algebraically special

TL;DR

This work extends the Petrov classification to five dimensions using Weyl spinors to define a Weyl polynomial and Petrov types. Focusing on Ricci-flat, static metrics with an $SO(3)$ isometry, the authors derive necessary and sufficient conditions for Petrov type $22$ and classify all such metrics, exposing a finite set of homogeneous solutions (including the 5D black hole and wrapped string) and their relations to known geometries. They then prove that a black hole on a cylinder (localized in the compact direction) cannot be algebraically special, i.e., does not belong to the type-$22$ sector, by showing it does not arise among the static, $SO(3)$-invariant type-$22$ solutions. This result constrains the possible higher-dimensional black-hole geometries and informs the search for localized horizon solutions in spacetimes with compact extra dimensions, with potential implications for string theory and braneworld scenarios.

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.

Figures (1)

• Figure 1: The 12 different Petrov types in 5 dimensions.