The spectrum of states of Ba~nados-Teitelboim-Zanelli black hole formed by a collapsing dust shell

TL;DR

This paper analyzes the quantum states of a BTZ black hole formed by a collapsing dust shell, revealing a finite-dimensional Hilbert space with discrete spectra inside the black hole, depending on the total energy.

Contribution

It introduces a canonical framework for 2+1D gravity coupled to a dust shell, showing the quantum algebra and finite spectra of black hole states.

Findings

Finite-dimensional phase space with SO(2,2) structure

Quantum algebra is a deformed SL(2) double, with deformation depending on energy

Discrete spectra of shell radius and time inside the black hole

Abstract

We perform canonical analysis of an action in which 2+1-dimensional gravity with negative cosmological constant is coupled to cylindrically symmetric dust shell. The resulting phase space is finite dimensional having geometry of SO(2; 2) group manifold. Representing the Poisson brackets by commutators results in the algebra of observables which is a quantum double D(SL(2)q). Deformation parameter q is real when the total energy of the system is below the threshold of a black hole formation, and a root of unity when it is above. Inside the black hole the spectra of the shell radius and time operator are discrete and take on a finite set of values. The Hilbert space of the black hole is thus finite-dimensional.