Trapped surfaces and the Penrose inequality in spherically symmetric geometries

TL;DR

This paper proves the Penrose inequality holds in spherically symmetric spacetimes with matter outside the horizon, providing new conditions for trapped surface formation and examining modifications for charged black holes.

Contribution

It establishes the validity of the Penrose inequality under broader conditions and introduces new necessary and sufficient criteria for trapped surface formation in spherical symmetry.

Findings

Penrose inequality valid with matter outside horizon

New necessary condition for trapped surfaces

Modification of Gibbons' inequality can fail early in collapse

Abstract

We demonstrate that the Penrose inequality is valid for spherically symmetric geometries even when the horizon is immersed in matter. The matter field need not be at rest. The only restriction is that the source satisfies the weak 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 trapped surfaces. This formulation can also be adapted to give a sufficient condition. We show that a modification of the Penrose inequality proposed by Gibbons for charged black holes can be broken in early stages of gravitational collapse. This investigation is based exclusively on the initial data formulation of General Relativity.