The fuzzy BTZ

TL;DR

This paper presents a noncommutative model of the BTZ black hole, incorporating quantum geometry, covariant calculus, and a detailed spectral analysis of the radial coordinate, advancing understanding of quantum black hole structures.

Contribution

It introduces a fuzzy BTZ black hole model derived from quantising Poincaré coordinates, with a covariant calculus and spectral properties analysis, extending noncommutative geometry in black hole physics.

Findings

The fuzzy BTZ satisfies Einstein's equations with negative curvature.

The radial coordinate spectrum includes scattering and bound states.

The model connects to fuzzy anti-de Sitter space through discrete identifications.

Abstract

We introduce a model of a noncommutative BTZ black hole, obtained by quantisation of Poincar\'e coordinates together with a moving frame. The fuzzy BTZ black hole carries a covariant differential calculus, satisfies Einstein's equations and has a constant negative curvature. The construction passes through a larger space, the fuzzy anti-de Sitter, and implements discrete BTZ identifications as conjugations by a unitary operator. We derive the spectrum of the suitably regularised radial coordinate: it consists of a continuum of scattering states outside the horizon $r_+$ and an infinite discrete set of bound states inside.