The noncommutative BTZ black hole in polar coordinates

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)×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 θ. 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.

Non-commutativity in polar coordinates

“…It is worth recalling that the apparent small-r divergence here causes no problems for the analysis of black holes in [14] since attention is limited to coordinates r > 0 and besides, this expansion is appropriate only for r 2 θ . At the other extreme we can investigate the large θ or small r limit in which z → ∞.…”

“…One facet of non-commutative geometry that has not often been discussed is its application in non-Cartesian coordinates. However, in [14], as a refinement of [15], the authors considered black holes in a non-commutative version of Ad S 3 , using the polar variables,r ,φ, to describe the spatial coordinates -see also [16,17] for similar applications and [18] for a discussion of the use of these coordinates. One outstanding issue was that the transition to the basic commutator in polar coordinates, [r ,φ], was not justified, which in turn led Iskauskas to investigate how it might be deduced [19].…”

Quasinormal modes and quantization of area/entropy for noncommutative BTZ black hole

“…[18,32] as follows. Noncommutative field theories are constructed from commutative field by replacing in the lagrangian the usual multiplication of fields with the -product of fields, which is defined in terms of a real antisymmetric matrix θ μν that parameterizes the noncommutativity of Minkowski space-time [18].…”

“…The commutation relation for noncommutative space-time (we adopt the formula in Ref. [32]) takes the form of…”

“…Based on the three dimensional equivalence between gravity and Chern-Simons theory, the authors in Ref. [32] have studied the noncommutative BTZ black hole solutions in polar coordinates (r,φ, t) with the following commutation relation…”