Sufficient conditions for apparent horizons in spherically symmetric initial data

We establish sufficient conditions for the appearance of both apparent horizons and singularities in spherically symmetric initial data when spacetime is foliated extrinsically. Let $M$ and $P$ be, respectively, the total material energy and the total material current contained in some ball of radius $\mathcal{l}$. Suppose that the dominant energy condition is satisfied. We show that if $M-P>~\mathcal{l}$ then the region must possess a future apparent horizon for some nontrivial closed subset of such gauges. The same inequality holds on a larger subset of gauges but with a larger constant of proportionality which depends weakly on the gauge. This work extends substantially both our joint work on moment of time symmetry initial data as well as the work of Bizon, Malec, and Ó Murchadha on a maximal slice.