Spherically Symmetric Gravitational Collapse

TL;DR

This paper develops a formalism for analyzing the gravitational collapse of spherically symmetric spacetimes using Israel's junction conditions, deriving surface energy-momentum relations and applying them to identify collapse and expansion regions.

Contribution

It introduces a new formalism based on junction conditions for spherically symmetric spacetimes, providing explicit formulas for surface energy-momentum tensors and applying them to specific cases.

Findings

Identified regions of collapse and expansion in spherical shells.

Derived explicit formulas for surface energy density and pressure.

Validated the formalism on known spherically symmetric solutions.

Abstract

In this paper, we discuss gravitational collapse of spherically symmetric spacetimes. We derive a general formalism by taking two arbitrary spherically symmetric spacetimes with $g_{00}=1$. Israel's junction conditions are used to develop this formalism. The formulae for extrinsic curvature tensor are obtained. The general form of the surface energy-momentum tensor depending on extrinsic curvature tensor components is derived. This leads us to the surface energy density and the tangential pressure. The formalism is applied to two known spherically symmetric spacetimes. The results obtained show the regions for the collapse and expansion of the shell.