# Multipole Hair of Schwarzschild-Tangherlini Black Holes

## TL;DR

This paper investigates how electric charges influence higher-dimensional black holes, revealing that multiple multipole moments persist and lead to complex final states, challenging traditional black hole characterizations.

## Contribution

It demonstrates that for dimensions greater than three, black holes develop infinite multipole moments and exhibit novel final states after charge infall, expanding understanding of higher-dimensional black hole physics.

## Key findings

- Infinite multipole moments appear for n>3
- Final states differ from classical black hole solutions
- Odd and even dimensions exhibit distinct topological behaviors

## Abstract

We study the field of an electric point charge that is slowly lowered into an $n + 1$ dimensional Schwarzschild-Tangherlini black hole. We find that if $n > 3$, then countably infinite nonzero multipole moments manifest to observers outside the event horizon as the charge falls in. This suggests the final state of the black hole is not characterized by a Reissner-Nordstr\"om-Tangherlini geometry. Instead, for odd $n$, the final state either possesses a degenerate horizon, undergoes a discontinuous topological transformation during the infall of the charge, or both. For even $n$, the final state is not guaranteed to be asymptotically-flat.