Quantum Collapse of a Small Dust Shell

TL;DR

This paper develops a quantum mechanical model for a small relativistic dust shell's collapse, revealing a low probability of total collapse and a concentration of solutions near the classical Schwarzschild radius.

Contribution

It introduces a novel quantum framework for shell collapse, addressing multivalued Hamiltonians and half-line momentum operators with analytical and numerical methods.

Findings

Total collapse probability is very small.

Solutions concentrate around the classical Schwarzschild radius.

Numerical results are supported by detailed WKB analysis.

Abstract

The full quantum mechanical collapse of a small relativistic dust shell is studied analytically, asymptotically and numerically starting from the exact finite dimensional classical reduced Hamiltonian recently derived by H\'aj{\'\i}\v{c}ek and Kucha\v{r}. The formulation of the quantum mechanics encounters two problems. The first is the multivalued nature of the Hamiltonian and the second is the construction of an appropriate self adjoint momentum operator in the space of the shell motion which is confined to a half line. The first problem is solved by identifying and neglecting orbits of small action in order to obtain a single valued Hamiltonian. The second problem is solved by introducing an appropriate lapse function. The resulting quantum mechanics is then studied by means of analytical and numerical techniques. We find that the region of total collapse has very small probability. We also find that the solution concentrates around the classical Schwarzschild radius. The present work obtains from first principles a quantum mechanics for the shell and provides numerical solutions, whose behavior is explained by a detailed WKB analysis for a wide class of collapsing shells.