Isoperimetric inequality for higher-dimensional black holes

Abstract

The initial data sets for the five-dimensional Einstein equation have been examined. The system is designed such that the black hole $(\ensuremath{\simeq}{S}^{3})$ or the black ring $(\ensuremath{\simeq}{S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1})$ 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 ${V}_{n\ensuremath{-}2}\ensuremath{\lesssim}GM,$ where ${V}_{n\ensuremath{-}2}$ denotes the volume of a typical closed (n-2)-section of the horizon and M is typical mass scale, rather than $C\ensuremath{\lesssim}{(\mathrm{GM})}^{1/(n\ensuremath{-}2)}$ in terms of the hoop length C, which holds only when $n=3.$