Emergence of thin shell structure during collapse in isotropic coordinates

TL;DR

This paper numerically investigates gravitational collapse in isotropic coordinates, revealing the formation of a thin shell near the apparent horizon, and describes the interior as a collapsing isotropic universe approaching a singularity.

Contribution

It provides a detailed numerical analysis of the interior structure during collapse, highlighting the emergence of a thin shell and the nature of the interior spacetime in isotropic coordinates.

Findings

Formation of a thin shell inside the apparent horizon.

Interior described as a collapsing isotropic universe.

Exterior matches Schwarzschild solution in isotropic coordinates.

Abstract

Numerical studies of gravitational collapse in isotropic coordinates have recently shown an interesting connection between the gravitational Lagrangian and black hole thermodynamics. A study of the actual spacetime was not the main focus of this work and in particular, the rich and interesting structure of the interior has not been investigated in much detail and remains largely unknown. We elucidate its features by performing a numerical study of the spacetime in isotropic coordinates during gravitational collapse of a massless scalar field. The most salient feature to emerge is the formation of a thin shell of matter just inside the apparent horizon. The energy density and Ricci scalar peak at the shell and there is a jump discontinuity in the extrinsic curvature across the apparent horizon, the hallmark that a thin shell is present in its vicinity. At late stages of the collapse, the spacetime consists of two vacuum regions separated by the thin shell. The interior is described by an interesting collapsing isotropic universe. It tends towards a vacuum (never reaches a perfect vacuum) and there is a slight inhomogeneity in the interior that plays a crucial role in the collapse process as the areal radius tends to zero. The spacetime evolves towards a curvature (physical) singularity in the interior, both a Weyl and Ricci singularity. In the exterior, our numerical results match closely the analytical form of the Schwarzschild metric in isotropic coordinates, providing a strong test of our numerical code.