Noncommutative BTZ Black Hole in Different Coordinates

TL;DR

This paper explores noncommutative BTZ black hole solutions in polar and rectangular coordinates using Chern-Simons theory and the Seiberg-Witten map, revealing differences at first order in noncommutativity.

Contribution

It provides the first analysis of noncommutative BTZ black holes in different coordinate systems and demonstrates coordinate-dependent solutions at first order in noncommutativity.

Findings

Solutions differ in polar and rectangular coordinates at first order in noncommutativity

Noncommutative extension maintains the classical equivalence between BTZ and Chern-Simons solutions

Coordinate choice affects the form of noncommutative black hole solutions

Abstract

We consider noncommutative BTZ black hole solutions in two different coordinate systems, the polar and rectangular coordinates. The analysis is carried out by obtaining noncommutative solutions of $U(1,1)\times U(1,1)$ Chern-Simons theory on $AdS_3$ in the two coordinate systems via the Seiberg-Witten map. This is based on the noncommutative extension of the equivalence between the classical BTZ solution and the solution of ordinary $SU(1,1)\times SU(1,1)$ Chern-Simons theory on $AdS_3$. The obtained solutions in these noncommutative coordinate systems become different in the first order of the noncommutativity parameter $\theta$.