Trapped surfaces in spherical expanding open universes

Abstract

Consider spherically symmetric initial data for a cosmology which, in the large, approximates an open $k = -1 ,Λ= 0$ Friedmann-Lema{\^ı}tre universe. Further assume that the data is chosen so that the trace of the extrinsic curvature is a constant and that the matter field is at rest at this instant of time. One expects that no trapped surfaces appear in the data if no significant clump of excess matter is to be found. This letter confirms this belief by displaying a necessary condition for the existence of trapped surfaces.This necessary condition, simply stated, says that a relatively large amount of excess matter must be concentrated in a small volume for trapped surfaces to appear.