Bounded Area Theorems for Higher Genus Black Holes

TL;DR

This paper establishes lower bounds on the areas of higher genus black hole horizons in four-dimensional spacetimes, revealing geometric constraints and energy conditions, with implications for anti-de Sitter spaces.

Contribution

It introduces a modified topology theorem providing lower area bounds for higher genus black holes, extending Gibbons' recent results to more general settings.

Findings

Lower bounds for black hole horizon areas based on genus and cosmological constant

Higher genus black holes require negative mean energy density on the horizon

Bounds are saturated by specific extremal anti-de Sitter black hole solutions

Abstract

By a simple modification of Hawking's well-known topology theorems for black hole horizons, we find lower bounds for the areas of smooth apparent horizons and smooth cross-sections of stationary black hole event horizons of genus $g>1$ in four dimensions. For a negatively curved Einstein space, the bound is ${{4\pi (g-1)}\over {-\ell}}$ where $\ell$ is the cosmological constant of the spacetime. This is complementary to the known upper bound on the area of $g=0$ black holes in de Sitter spacetime. It also emerges that $g>1$ quite generally requires a mean negative energy density on the horizon. The bound is sharp; we show that it is saturated by certain extreme, asymptotically locally anti-de Sitter spacetimes. Our results generalize a recent result of Gibbons.