Gravitational Collapse of Vacuum Gravitational Field Configurations

TL;DR

This paper establishes a theorem in general relativity that predicts apparent horizon formation, indicating black hole creation, based on the geometric and curvature properties of vacuum and matter configurations.

Contribution

It introduces a new criterion involving surface-to-volume ratio and extrinsic curvature to determine apparent horizon formation in vacuum gravitational fields.

Findings

Proves a condition for apparent horizon existence in vacuum configurations.

Links the product of region radius and curvature to black hole formation.

Provides a geometric measure for gravitational collapse in general relativity.

Abstract

This paper proves a theorem about the existence of an apparent horizon in general relativity, which applies equally well to vacuum configurations and matter configurations. The theorem uses the reciprocal of the surface-to-volume ratio of a region on a space slice to measure the radius of the region, and uses the minimum value $K_{\rm min}$ of certain components of the extrinsic curvature to measure the strengh of the gravitational field in the region. The theorem proves that, if the product of the radius times $K_{\rm min}$ is larger than unity, then an apparent horizon must form, signalling the formation of a black hole.