Quantum gravitational collapse as a Dirac particle on the half-line

TL;DR

This paper models quantum gravitational collapse of a spherical shell using the Dirac equation on a half-line, revealing a discrete energy spectrum, potential negative ground states, and implications for singularity avoidance and black hole radiation.

Contribution

It establishes an equivalence between quantum shell dynamics and the Coulomb-Dirac equation, introducing a family of self-adjoint extensions and analyzing their spectral properties.

Findings

Discrete energy spectrum with bound states for |E|<m

Existence of scattering states for |E|>m

Negative ground state energy for large shell mass

Abstract

We show that the quantum dynamics of a thin spherical shell in general relativity is equivalent to the Coulomb-Dirac equation on the half line. The Hamiltonian has a one-parameter family of self-adjoint extensions with a discrete energy spectrum $|E| < m$, and a continuum of scattering states for $|E|>m$, where $m$ is the rest mass of the shell and $E$ is the Arnowitt-Deser-Misner mass. For sufficiently large $m$, the ground state energy level is negative. This suggests that classical positivity of energy does not survive quantization. The scattering states provide a realization of singularity avoidance. We speculate on the consequences of these results for black hole radiation.