Gravitational Aharonov-Bohm effect due to noncommutative BTZ black hole

In this paper we consider the scattering of massless planar scalar waves by a noncommutative BTZ black hole. We compute the differential cross section via the partial wave approach, and we mainly show that the scattering of planar waves leads to a modified Aharonov-Bohm effect due to spacetime noncommutativity

associated with the circulation go to zero (non-rotating black-holes). In this limit, the noncommutative background forms a conical defect , which is also responsible for the appearance of the analogue AB effect [23].
The paper is organized as follows. In Sec. II we briefly introduce the noncommutative BTZ black hole background. In Sec. III we shall compute the differential cross section due to the scattering of planar waves that leads to a modified AB effect in the noncommutative spacetime. Finally in Sec. IV we present our final considerations.

II. NONCOMMUTATIVE BTZ BLACK HOLES
The metric of noncommutative BTZ black hole is given by [8] ds 2 = −F 2 dt 2 + N −2 dr 2 + 2r 2 N φ dtdφ + r 2 − θB 2 where here B is the magnitude of the magnetic field, θ is the noncommutative parameter, r + and r − are the outer and inner horizons of the commutative BTZ black hole given by The apparent horizons, for the noncommutative black hole, denoted byr ± can be determined by N 2 = 0 note that the event horizons in the noncommutative case, are shifted through constant θB/2. In the limit θ → 0, these reduce to the event horizons of the commutative case. The metric of noncommutative BTZ black hole can be rewritten as where The metric can be now written in the form with inverse g µν where −g = θB In order to study the Aharonov-Bohm effect, we shall now consider the Klein-Gordon equation in the noncommutative BTZ metric given by the background (7): As usual, one can make a separation of variables into the equation (12) as in the following Let us now concentrate on the radial function R(r) that satisfies the following linear second-order differential equation The equation (14) can be rewritten as where F (r) = √ −gQ(r). Now introducing the tortoise coordinate r * through the use of the following equation and considering a new radial function, G(r * ) = r 1/2 R(r) we get to a new radial equation obtained from (15) that reads where V (r) is the potential given by a form that resembles that given in Refs. [18,23].

III. GRAVITATIONAL AHARONOV-BOHM EFFECT
To address the issues concerning the gravitational Aharonov-Bohm effect we shall now consider the scattering of a monochromatic planar wave of frequency ω written in the form such a way that far from the vortex, the function ψ can be given in terms of the sum of a plane wave and a scattered wave: where e iωx = ∞ m=−∞ i m J m (ωr)e imφ and J m (ωr) is a Bessel function of the first kind. The scattering amplitude f ω (φ) has the following partial-wave representation To compute the phase shift δ m at some level of approximation we first rewrite the equation (17) in terms of the new function X(r) = F (r) 1/2 G(r * ), that is where dF (r) and The equation (22) can still be written in terms of a power series in 1/r as follows where where m 2 = m 2 (1 + 3ω 2 Θ Mb 2 ) + am M − 3a 2 8Mb 2 , Θ = θB/2, a = ωJ and b = lω. Now we apply the folowing approximation formula and using |m| ≫ √ a 2 + b 2 , we obtain [18,23] Therefore, we can compute the differential scattering cross section restricted to small angles φ and to lower orders in Θ that is given by Note that if Θ = 0 in (29), we simply have This phase shift leads to a gravitational AB effect and the differential scattering cross section (30) at small angles φ is given by whose leading term of the differential cross section reads where,ã = a/M . This is the differential scattering cross section for the gravitational AB effect.
On the other hand, for a = 0 and restricting ourselves to lower orders in Θ, the differential scattering cross section (30) now becomes Since we are taking φ → 0, the differential cross section is dominated by Note that the noncommutative gravitational AB scattering result through the use of partial waves approach is successfully obtained and contrarily to the usual Aharonov-Bohm effect, in the noncommutative case the differential scattering cross section is different from zero even if a = 0.

IV. CONCLUSIONS
In summary, in this paper we have considered the gravitational Aharonov-Bohm effect in the background of the noncommutative BTZ black hole. To address the issues concerning the noncommutative gravitational Aharonov-Bohm effect we considered the scattering of a monochromatic planar wave. Our results are qualitatively in agreement with that obtained in [24] for the AB effect in the context of noncommutative quantum mechanics and in [23] for an analogue Aharonov-Bohm effect due to an idealized draining bathtub vortex. The noncommutative correction vanishes in the limit Θ → 0 so that no singularities are generated. The leading correction (∼ Θ 2 ) due to effect of spacetime noncommutativity may be relevant at a scale where the spacetime noncommutativity takes place as expected in high energy physics. Our result shows that pattern fringes can appear even if a, which depends on the angular momentum, is equal to zero, unlike the commutative case. One can make some estimative of the aforementioned effect by estimating θ following similar calculations already known in the literature [29,30].