TRAPPED SURFACES AND THE PENROSE INEQUALITY IN SPHERICALLY SYMMETRICAL GEOMETRIES

We demonstrate that the Penrose inequality is valid for spherically sy mmetric geometries even when the horizon is immersed in matter. The ma tter field need not be at rest. The only restrict.ion is that the sour ce satisfies an energy condition outside the horizon. No restrictions are placed on the matter inside the horizon. The proof of the Penrose inequality gives a new necessary condition for the formation of trappe d surfaces. This formulation may also be adapted to give a sufficient condition. We show that a modification of the Penrose inequality propo sed by Gibbons for charged black holes can be broken in early stages o f gravitational collapse. This investigation is based exclusively on t he initial data formulation of general relativity.