Twist of stationary black hole/ring in five dimensions

TL;DR

This paper investigates whether higher multipole moments can uniquely classify stationary black hole solutions in five-dimensional spacetimes, focusing on potential functions related to rotational symmetries and their dependence on mass and angular momentum.

Contribution

It explores the role of higher multipole moments in classifying vacuum solutions and examines the potential $\sigma$ as evidence for uniqueness in five-dimensional black holes.

Findings

Potential $\sigma$ cannot be expressed explicitly in terms of M and J.

Higher multipole moments may help classify black hole solutions.

Evidence suggests multipole moments could determine uniqueness in higher dimensions.

Abstract

It is unlikely that uniqueness theorem holds for stationary black holes in higher dimensional spacetimes. However, we will examine the possibility that the higher multipole moments classify vacuum solutions uniquely. Especially, we compute the potentials associated with rotational Killing vectors and look at the dependence on the total mass M and angular momentum J. Consequently, there is a potential $\sigma$ which we cannot write down in terms of integer power of M and J explicitly. This may be regarded as an evidence for the uniqueness using multipole moments generated by $\sigma$.