TRAPPED SURFACES IN NONSPHERICAL INITIAL DATA SETS AND THE HOOP CONJECTURE

The existence of outer trapped surfaces in conformally flat, axisymmetric, momentarily static initial data sets for Einstein's equations is investigated. It is shown that none of the level surfaces of the conformal factor can be outer trapped, whenever the minimum value of the circumferences (or of the square roots of the areas) of all the surfaces surrounding the source region is greater than a constant times the Arnowitt-Deser-Misner mass. This result is along the lines of the hoop conjecture. It also provides evidence in favor of the conclusion of Shapiro and Teukolsky, drawn from recent numerical relativity calculations, that the gravitational field on a spacelike hypersurface can become arbitrarily singular without the appearance of an apparent horizon.