Noncommutative BTZ Black Hole in Different Coordinates

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 $\theta$.