Collapsing Shells and the Isoperimetric Inequality for Black Holes

TL;DR

This paper applies recent isoperimetric inequalities to gravitational collapse, demonstrating that the apparent horizon area in black hole formation is bounded by mass, supporting the Cosmic Censorship Hypothesis.

Contribution

It introduces a novel application of Trudinger's inequalities to black hole physics, providing bounds on horizon area during collapse in higher dimensions.

Findings

Apparent horizon area is bounded above by 16πG^2 M^2.

Results support the Cosmic Censorship Hypothesis.

Applicable in four and higher spacetime dimensions.

Abstract

Recent results of Trudinger on Isoperimetric Inequalities for non-convex bodies are applied to the gravitational collapse of a lightlike shell of matter to form a black hole. Using some integral identities for co-dimension two surfaces in Minkowski spacetime, the area $A$ of the apparent horizon is shown to be bounded above in terms of the mass $M$ by the $16 \pi G^2 M^2$, which is consistent with the Cosmic Censorship Hypothesis. The results hold in four spacetime dimensions and above.