Report-no: NSF-ITP-94-47, gr-qc/9411024, J. Math. Phys. 36, 3004-3011 (1995).
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.