Noncommutative BTZ Black Hole and Discrete Time

TL;DR

This paper explores noncommutative deformations of the BTZ black hole geometry, identifying Poisson structures compatible with the classical solution and analyzing their implications for quantum properties like discrete time spectra.

Contribution

It classifies Poisson brackets consistent with BTZ geometry and derives symplectic leaves, linking geometric structures to potential quantum discretization of time.

Findings

Identified two families of contact structures compatible with BTZ geometry.

Derived symplectic leaves that determine irreducible representations.

Suggested quantization could lead to a discrete spectrum for the time operator.

Abstract

We search for all Poisson brackets for the BTZ black hole which are consistent with the geometry of the commutative solution and are of lowest order in the embedding coordinates. For arbitrary values for the angular momentum we obtain two two-parameter families of contact structures. We obtain the symplectic leaves, which characterize the irreducible representations of the noncommutative theory. The requirement that they be invariant under the action of the isometry group restricts to $R\times S^1$ symplectic leaves, where $R$ is associated with the Schwarzschild time. Quantization may then lead to a discrete spectrum for the time operator.