Event horizons and apparent horizons in spherically symmetric geometries

TL;DR

This paper demonstrates that in spherically symmetric geometries, apparent horizons coincide with event horizons, and their propagation speed matches that of outgoing photons, without relying on Birkhoff's theorem.

Contribution

It provides a proof within (1+3) formalism that apparent horizons and event horizons are identical in spherical symmetry, independent of Birkhoff's theorem.

Findings

Apparent horizons propagate at the speed of outgoing photons.

Apparent and event horizons coincide in spherical geometries.

The proof is achieved without using Birkhoff's theorem.

Abstract

Spherical configurations that are very massive must be surrounded by apparent horizons. These in turn, when placed outside a collapsing body, must propagate outward with a velocity equal to the velocity of radially outgoing photons. That proves, within the framework of (1+3) formalism and without resorting to the Birkhoff theorem, that apparent horizons coincide with event horizons.