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Noncommutative correction to the entropy of BTZ black hole with GUP

We investigate the effect of noncommutativity and quantum corrections to the temperature and entropy of a BTZ black hole based on a Lorentzian distribution with the generalized uncertainty principle (GUP). To determine the Hawking radiation in the tunneling formalism we apply the Hamilton-Jacobi method by using the Wentzel-Kramers-Brillouin (WKB) approach. In the present study we have obtained logarithmic corrections to entropy due to the effect of noncommutativity and GUP. We also address the issue concerning stability of the non-commutative BTZ black hole by investigating its modified specific heat capacity.


I. INTRODUCTION
The study of three-dimensional gravity has been extensively explored in the literature [1]. It has become an excellent laboratory for a better understanding of the fundamentals of classical and quantum gravity and also to explore some ideas behind the AdS/CFT correspondence [2]. This special attention in three-dimensional gravity has been mainly due to the discovery of the black hole solution in 2 + 1 dimensions [3]. In addition, generalizations of the Bañados-Teitelboim-Zanelli (BTZ) black hole solution have also been constructed considering coupling with a dilaton/scalar field [4][5][6]. In recent years, the implementation of noncommutativity in black hole physics has been extensively explored (for a review see [7]). In [8] the authors have introduced a noncommutative Schwarzschild black hole solution in four dimensions. As shown in [8], one way to incorporate noncommutativity into General Relativity is to modify the source of matter. Thus, noncommutativity is introduced by replacing the point-like source term with a Gaussian distribution -or otherwise by a Lorentzian distribution [9]. In addition, noncommutativity in BTZ black hole has also been introduced in [10][11][12]. In [13] the gravitational Aharonov-Bohm effect due to BTZ black hole in a noncommutative background has been analyzed. The process of massless scalar wave scattering by a noncommutative black hole via Lorentzian smeared mass distribution has been explored in [14]. The thermodynamic properties of BTZ black holes in noncommutative spaces have been studied in [15][16][17].
It is well known that string theories, loop quantum gravity and noncommutative geometry presents important elements for the construction of a compatible theory of quantum gravity. Furthermore, these theories have a common feature, which is the appearance of a minimum length on the order of the Planck scale. This therefore leads to a modification of the Heisenberg uncertainty principle, which is called the generalized uncertainty principle (GUP) [18][19][20]. In recent years, several works have been devoted to investigating the effect of GUP on computing Hawking radiation from black holes in 2 + 1 dimensions. In this sense, the Hamilton-Jacobi method via the WKB approach to calculate the imaginary part of the action is an effective way of investigating Hawking radiation as a process of tunneling particles from a black hole [21][22][23][24][25][26][27]. In [28] the effect of GUP on Hawking radiation from the BTZ black hole has been investigated using the modified Dirac equation. Hawking radiation has been analyzed in [29], by considering the Martinez-Zanelli black hole in 2 + 1 dimensions [5] and using the Dirac equation modified by the GUP. By applying the quantum tunneling formalism, Hawking radiation from a new type of black hole in 2 + 1 dimensions has also been studied in [30], and in [31] was explored the Hawking radiation of charged rotating BTZ black hole with GUP. Moreover, in [32,33] the entropy of the BTZ black hole with GUP has been determined, and in [34], by adopting a new principle of extended uncertainty, its effect on the thermodynamics of the black hole has been examined.
The purpose of this paper is to investigate the effect of noncommutative and quantum corrections coming from the GUP for the calculation of the temperature and entropy of a BTZ black hole based on a Lorentzian distribution, by considering the tunneling formalism framework through the Hamilton-Jacobi method. Thus, Hawking radiation will be computed using the WKB approach. Therefore, we show that the entropy of the BTZ black hole presents logarithmic corrections due to the both aforementioned effects.
The paper is organized as follows. In Sec. II we consider noncommutative corrections for the BTZ black hole metric implemented via Lorentzian mass distribution. We also have applied the Hamilton-Jacobi approach to determine noncommutative corrections for Hawking temperature and entropy. In Sec. III we consider the GUP to compute quantum corrections to Hawking temperature and entropy and also briefly comment on the correction of the specific heat capacity at constant volume. In Sec. IV we make our final considerations.

II. NONCOMMUTATIVE CORRECTIONS TO THE BTZ BLACK HOLES
In this section we introduce the noncomutativity by considering a Lorentzian mass distribution, given by [8,9,12,14] where θ is the noncommutative parameter with dimension of length 2 and M is the total mass diffused throughout the region of linear size √ θ. In this case the smeared mass distribution function becomes [12] By considering the above modified mass, the metric of noncommutative BTZ black hole is given by where We shall now analyze the non-rotating case (J = 0), so the metric (4) becomes where The horizons are found by solving the equation which is equivalent to solving a cubic equation The roots of this cubic equation are given by [35] The three roots for ǫ = 1, 0, −1, up to first order in √ θ, are given respectively bỹ where r h = √ l 2 M ,r h is the event horizon, r c the cosmological horizon and r v the virtual (unphysical) horizon. From Eq. (8) we obtain the mass of the noncommutative black hole, up to first order in √ θ, that is given by In order to compute the Hawking temperature we use the Klein-Gordon equation for a scalar field Φ in the curved space given by where m is the mass of a scalar particle. In the sequel we apply the WKB approximation such that we obtain By applying the metric (6) in the above equation we have Now we can write the solution of equation (18) as follows where being J φ a constant. By substituting (19) into equation (18) and solving for W (r) the classical action is written as follows: Next, in the regime near the event horizon of the noncommutative BTZ black hole, r →r h , we can write f (r) ≈ κ(r −r h ) and so the spatial part of the action function reads where κ is the surface gravity of the noncommutative BTZ black hole given by The next step is to determine the probability of tunneling for a particle with energy E and for this we use the following expression In order to calculate the Hawking temperature of the noncommutative BTZ black hole we can compare equation (24) with the Boltzmann factor exp(−E/T H ), so we can find Moreover, the above result can be rewritten in terms of r h = √ l 2 M as follows Therefore, the result above shows that the Hawking temperature is modified due to the presence of the noncommutative parameter θ. Note that when we take θ = 0, we recover the temperature of the commutative BTZ black hole, which is T h = r h /(2πl 2 ). At this point, we are prepared to go further. Let us now consider the noncommutative BTZ black hole in the rotating regime (J = 0). Now the line element of equation (4) can be written in the form where Thus, to find the horizons we have to solve which is equivalent to solving a quartic equation We can now rewrite this equation as follows [35] ( that for θ = 0 we have where r + is the outer event horizon and r − is the inner event horizon of the commutative BTZ black hole. Now rearranging the equation (32) in the form where r h = √ l 2 M , we can solve it perturbatively. So, in the first approximation we get the event horizoñ or by keeping terms up to first order in √ θ, we obtaiñ for the outer horizon. For the internal horizon we havẽ so that forr − , we findr In order to determine the Hawking temperature for the case of the rotating black hole, we can follow the same steps as presented above and so for the tunneling probability we have where the surface gravity is given byκ Again, by comparing Γ with the Boltzmann factor exp(−E/T H ), we obtain the Hawking temperature of the noncommutative rotating BTZ black hole (42) For θ = 0 we recover the result for the Hawking temperature of the rotating BTZ black hole which is given by From Eq. (30) we obtain the mass of the noncommutative black hole, up to first order in √ θ, that is given by In order to analyze the entropy we consider the following equation: where The next step is to perform an expansion in T −1 H up to first order in √ θ, so we can find Now, by replacing (46) and (47) in (45), we obtain or rewriting in terms of r + , we havê For θ = 0 in (50) we have S = 4πr + , which is the entropy of the commutative rotating BTZ black hole. On the other hand, for the case J = 0, we have r + = r h and the entropy becomeŝ Note that we have obtained a logarithmic correction for the noncommutative BTZ black hole.

III. QUANTUM CORRECTION TO THE ENTROPY
In this section in order to derive quantum corrections to the Hawking temperature and entropy of the noncommutative BTZ black hole, we will apply tunneling formalism using the Hamilton-Jacobi method. So, we will adopt the following GUP [36][37][38][39][40] where α is a dimensionless positive parameter and l p is the Planck length.
In sequence, without loss of generality, we will adopt the natural units G = c = k B = = l p = 1 and by assuming that ∆p ∼ E and following the steps performed in [21] we can obtain the following relation for the corrected energy of the black hole Thus, performing the same procedure as previously described, we have the following result for the probability of tunneling with corrected energy E gup given by where a is the surface gravity. Again, we compare with the Boltzmann factor and we obtain the corrected Hawking temperature of the noncommutative BTZ black hole Here for simplicity we will consider the case J = 0. The temperatureT H is given by equation (25). Furthermore, since near the event horizon of the BTZ black hole the minimum uncertainty in our model is of the order of the radius of the horizon, so the corrected temperature due to the GUP is given by We can also write the result above in terms of r h = l √ M as follows Next, we will compute the entropy of the noncommutative BTZ black hole by using the following formula: where from Eq. (14) we have So, now we can obtain the corrected entropy or by expressing the result above in terms of the r h we have (63) Therefore, by analyzing the result we have obtained corrections to the entropy due to the effects of GUP and also noncommutative correction. Note that due to the effect of noncommutativity and GUP we have found logarithmic corrections for the entropy of the BTZ black hole. For α = 0, we have precisely the noncommutative correction to the entropy given by (51).
The correction of the specific heat capacity at constant volume Now if α = 0 in Eq. (64) we have (66) Note that the specific heat vanishes at the point r h = √ θ/2. In this case, we have a minimum radius r min = √ l 2 M min = √ θ/2 and then the noncommutative black hole reaches a minimum mass given by Thus, this result indicates that the black hole ceases to evaporate completely and becomes a remnant.

IV. CONCLUSIONS
In summary, we have considered the metric of a noncommutative BTZ black hole implemented via Lorentzian mass distribution. Thus, applying the Hamilton-Jacobi approach and the WKB approximation we have obtained noncommutative corrections to Hawking temperature and entropy. In addition, we have found a logarithmic correction to the entropy of the BTZ black hole due to the effect of noncommutativity. We also have verified the stability of the BTZ black hole by calculating the specific heat capacity and have shown that the noncommutative BTZ black hole becomes a remnant with a minimum mass M min = θ/4l 2 . Therefore, the contribution of the noncommutative corrections introduces a GUP effect. We also investigated the effect of GUP by calculating Hawking temperature and entropy of the noncommutative BTZ black hole. Due to the effect of noncommutativity and GUP we have found a logarithmic corrections for the entropy of the BTZ black hole, in the form S gup ∼ S + (c 1 + c 2 ) ln S + ..., where the 'species' c i = (−α, θ) are essentially related to each corresponding parameter of correction.