Noncommutative BTZ Black Hole in Polar Coordinates

TL;DR

This paper derives a noncommutative BTZ black hole solution using Chern-Simons theory and the Seiberg-Witten map in polar coordinates, revealing how noncommutativity affects horizon properties.

Contribution

It presents the first noncommutative BTZ black hole solution formulated in polar coordinates via the Seiberg-Witten map, highlighting noncommutative effects on horizons.

Findings

Apparent and Killing horizons coincide only in the non-rotating limit.

Noncommutativity influences the relation between horizons.

Solution is obtained as a first-order approximation in the noncommutativity parameter.

Abstract

Based on the equivalence between the three dimensional gravity and the Chern-Simons theory, we obtain a noncommutative BTZ black hole solution as a solution of $U(1,1)\times U(1,1)$ noncommutative Chern-Simons theory using the Seiberg-Witten map. The Seiberg-Witten map is carried out in a noncommutative polar coordinates whose commutation relation is equivalent to the usual canonical commutation relation in the rectangular coordinates up to first order in the noncommutativity parameter $\theta$. The solution exhibits a characteristic of noncommutative polar coordinates in such a way that the apparent horizon and the Killing horizon coincide only in the non-rotating limit showing the effect of noncommutativity between the radial and angular coordinates.