Quantum Analysis of BTZ Black Hole Formation Due to the Collapse of a Dust Shell

TL;DR

This paper investigates the quantum properties of BTZ black hole formation via dust shell collapse in 2+1 dimensions, revealing a continuous-to-discrete spectrum transition and quantum tunneling effects near the horizon.

Contribution

It introduces a Hamiltonian reduction approach for a 2+1D gravity model with dust, analyzing the quantum spectrum and tunneling phenomena in black hole formation.

Findings

Spectrum is continuous outside the horizon.

Spectrum becomes discrete inside the horizon, with eigenvalues spaced by the square root of mass.

Quantum tunneling into classically forbidden regions is possible, with exponential suppression.

Abstract

We perform Hamiltonian reduction of a model in which 2+1 dimensional gravity with negative cosmological constant is coupled to a cylindrically symmetric dust shell. The resulting action contains only a finite number of degrees of freedom. The phase space consists of two copies of $ADS^2$ -- both coordinate and momentum space are curved. Different regions in the Penrose diagram can be identified with different patches of $ADS^2$ momentum space. Quantization in the momentum representation becomes particularly simple in the vicinity of the horizon, where one can neglect momentum non-commutativity. In this region, we calculate the spectrum of the shell radius. This spectrum turns out to be continuous outside the horizon and becomes discrete inside the horizon with eigenvalue spacing proportional to the square root of the black hole mass. We also calculate numerically quantum transition amplitudes between different regions of the Penrose diagram in the vicinity of the horizon. This calculation shows a possibility of quantum tunneling of the shell into classically forbidden regions of the Penrose diagram, although with an exponentially damped rate away from the horizon.