Collapsing regions and black hole formation

TL;DR

This paper proves that rapidly collapsing compact regions in initial data surfaces necessarily contain future trapped regions, implying black hole formation under certain geometric and physical conditions, without requiring asymptotic or energy assumptions.

Contribution

It strengthens previous theorems by removing energy conditions and introduces new geometric criteria for black hole formation from collapsing regions.

Findings

Rapid collapse implies trapped regions in initial data surfaces.

Existence of trapped regions indicates black hole formation.

New geometric methods do not rely on asymptotic or energy conditions.

Abstract

Up to a conjecture in Riemannian geometry, we significantly strengthen a recent theorem of Eardley by proving that a compact region in an initial data surface that is collapsing sufficiently fast in comparison to its surface-to-volume ratio must contain a future trapped region. In addition to establishing this stronger result, the geometrical argument used does not require any asymptotic or energy conditions on the initial data. It follows that if such a region can be found in an asymptotically flat Cauchy surface of a spacetime satisfying the null-convergence condition, the spacetime must contain a black hole with the future trapped region therein. Further, up to another conjecture, we prove a strengthened version of our theorem by arguing that if a certain function (defined on the collection of compact subsets of the initial data surface that are themselves three-dimensional manifolds with boundary) is not strictly positive, then the initial data surface must contain a future trapped region. As a byproduct of this work, we offer a slightly generalized notion of a future trapped region as well as a new proof that future trapped regions lie within the black hole region.