Sufficient Conditions for Apparent Horizons in Spherically Symmetric Initial Data

TL;DR

This paper derives sufficient conditions involving energy and current for the formation of apparent horizons in spherically symmetric initial data, extending previous results to more general gauge conditions.

Contribution

It provides new inequalities involving energy and current that guarantee apparent horizons in spherically symmetric spacetimes under extrinsic foliation.

Findings

If M - P ≥ ℓ, a future apparent horizon exists.

The inequality holds under various gauge conditions with different constants.

Extends previous work on time-symmetric and maximal slices.

Abstract

We establish sufficient conditions for the appearance of apparent horizons 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 $\ell$. Suppose that the dominant energy condition is satisfied. We show that if $M- P \ge \ell$ then the region must possess a future apparent horizon for some non -trivial 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 \'O Murchadha on a maximal slice.