What is the Shape of a Black Hole?

TL;DR

This paper surveys recent theoretical results and conjectures about the geometry and physical properties of black holes in higher dimensions, focusing on horizon shape, size measures, and related geometric questions.

Contribution

It provides an overview of recent advances and open questions regarding the geometric characterization of black holes in four and higher dimensions.

Findings

Discussion of the Hoop Conjecture and its implications

Analysis of measures like Birkhoff's invariant and geodesic lengths

Exploration of whether one can determine black hole shape from spectral data

Abstract

A brief, and certainly not exhaustive, survey is provided of some recent results and conjectures in four and higher spacetime dimensions, such as the Hoop Conjecture, relating the geometry of event horizons to dynamical quantities such as the total energy or mass of the spacetime containng the black hole. As a measure of the size of a hoop one may take Birkhoff's invariant based on sweepouts by circles or higher dimensional analogues or one may take the length of the shortest non-trivial closed geodesic. Also discussed are whether one can hear the shape of a black hole and to what extent one may associate with it a volume.