Isoperimetric inequality for higher-dimensional black holes

TL;DR

This paper investigates geometric inequalities for higher-dimensional black holes, proposing a new isoperimetric inequality relating horizon volume to mass, extending known results from four-dimensional cases.

Contribution

It introduces a conjecture that the isoperimetric inequality for n-dimensional black holes involves horizon volume and mass, generalizing the three-dimensional hoop conjecture.

Findings

Horizon length can become arbitrarily large.

Area of characteristic 2D submanifold is bounded by mass.

Proposed isoperimetric inequality for higher dimensions.

Abstract

The initial data sets for the five-dimensional Einstein equation have been examined. The system is designed such that the black hole ($\simeq S^3$) or the black ring ($\simeq S^2\times S^1$) can be found. We have found that the typical length of the horizon can become arbitrarily large but the area of characteristic closed two-dimensional submanifold of the horizon is bounded above by the typical mass scale. We conjecture that the isoperimetric inequality for black holes in $n$-dimensional space is given by $V_{n-2} \lesssim GM$, where $V_{n-2}$ denotes the volume of typical closed $(n-2)$-section of the horizon and $M$ is typical mass scale, rather than $C\lesssim (GM)^{1/(n-2)}$ in terms of the hoop length $C$, which holds only when $n=3$.