Noncommutative BTZ black hole and discrete time

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 of the angular momentum we obtain two two-parameter families of contact structures. We obtain the symplectic leaves, which characterize the irreducible representations of the corresponding noncommutative theory. The requirement that they be invariant under the action of the isometry group restricts to symplectic leaves which are topologically , where is associated with the Schwarzschild time. Quantization may then lead to a discrete spectrum for the time operator.