Cardy-Verlinde Formula of Noncommutative Schwarzschild Black Hole

Abstract

Few years ago, Setare (2006) has investigated the Cardy-Verlinde formula of noncommutative black hole obtained by noncommutativity of coordinates. In this paper, we apply the same procedure to a noncommutative black hole obtained by the coordinate coherent approach. The Cardy-Verlinde formula is entropy formula of conformal field theory in an arbitrary dimension. It relates the entropy of conformal field theory to its total energy and Casimir energy. In this paper, we have calculated the total energy and Casimir energy of noncommutative Schwarzschild black hole and have shown that entropy of noncommutative Schwarzschild black hole horizon can be expressed in terms of Cardy-Verlinde formula.

1. Introduction

Verlinde [1] proved that the entropy of conformal field theory in arbitrary dimension is related to its total energy and Casimir energy; this is known as generalized Verlinde formula (commonly termed as Cardy-Verlinde formula). Recently, it has been investigated that this formula holds well for Reissner-Nordström de-Sitter black hole (BH) [2] and charged Kerr BH [3]. Birmingham and Mokhtari proved the validity of Birmingham and Mokhtari [4] proved the Verlinde formula for Taub-Bolt-Anti-de-Sitter BH. Setare and Jamil [5] discussed the Cardy-Verlinde formula for charged BTZ BH. Many authors [6–11] have proved the validity of Cardy-Verlinde for different BHs. The purpose of this paper is to investigate the validity of Cardy-Verlinde entropy formula for NC Schwarzschild BH.

In classical general relativity (GR), the curvature singularity is such a point where physical description of the gravitational field is impossible. This problem can be removed in GR by taking into account the quantum mechanical treatment to the standard formulation of GR. Motivated by such reasoning, some BH solutions in noncommutative (NC) field theory have been derived. In these solutions, curvature singularity at origin is removed by de-Sitter core which is introduced due to NC nature of spacetime [12]. Ansoldi et al. [13] formulated the NC charged BHs solutions; this was extended to rotating noncommutative BHs case by Modesto and Nicolini [14]. Mann and Nicoloni [15] have discussed the cosmological production of NC BHs. The first NC version of wormholes solution was investigated by Nicolini and Spallucci [16]. Farook et al. [17] have investigated the higher dimensional wormhole solutions in NC theory of gravity. Motivated by such NC correction to BHs, Sharif and Abbas [18] studied the thin shell collapse in NC Reissner-Nordström geometry. Banerjee and Gangopadhyay [19] derived the Komar energy and Sammar formula for NC Schwarzschild BH.

Motivated by the recent development in NC theory of gravity, we have proved that the entropy of NC Schwarzschild BH horizon can be expressed in terms of Cardy-Verlinde formula. For this purpose, we have used the Setare and Jamil method [5]. The plan of the paper is as follows. In Section 2, we briefly discussed the the thermodynamical relations of NC Schwarzschild BH and Cardy-Verlinde formula and proved that entropy of noncommutative Schwarzschild BH horizon can be expressed in terms of Cardy-Verlinde formula. Section 3 is devoted to the concluded remarks of the work done.

2. Noncommutative Schwarzschild Black Hole and Cardy-Verlinde Formula

According to GR, singularity is such a region of spacetime at which the usual laws of physics break down. This problem can be removed by applying the formulation of NC field theory to GR. For example, the NC BHs are one of the outcomes of string theory. These have such geometric structure in which curvature singularity is recovered by the minimal length introduced by the NC nature of coordinates. Further, all types of NC BHs expose the de-Sitter core due to quantum fluctuations at the center of the manifold.

The NC formulation of GR is one of the long standing problems which hava no solution yet. An extensive literature survey [20–22] implies that the application of Moyal -product among the tetrad fields in the gravitational action is a mathematically correct approach but not physically. It is due to the fact that the expansion of -product in NC parameter is truncated up to a desirable order which causes destruction of the nonlocal nature of NC theory. This result is to face the BH geometry with the same curvature singularities as in GR. Instead of using -product, one can formulate NC form of GR using the coordinate coherent state approach.

In this approach, the density of point like source in NC spacetime can be governed by a Gaussian distribution by using the relation [12] where is constant gravitational mass and is constant having the dimension of length squared. The line element for NC Schwarzschild BH is [12] where and is lower incomplete gamma function which is defined by In the commutative limit , that is, , (2) reduces to conventional Schwarzschild metric. The event horizons of BH can be found by setting , which yields We take the large radius regime where we can expand the incomplete gamma function to solve by iteration. By keeping the terms up to order , we find Now the Hawking temperature for NC Schwarzschild BH up to order is given by The entropy of the NC Schwarzschild BH () up to order is given by The generalized form of Cardy formula (also known as Cardy-Verlinde formula) is given by [5] where , is radius of -dimensional FRW universe, is the Casimir energy, and is the total energy of underlying field. The definition of Casimir energy is derived by the violation of Euler’s relation as [6] where the pressure for CFT is , and are zero for NC Schwarzschild BH, and is the volume of the system bounded by the apparent horizon. The total energy may be written as sum of extensive part and Casimir energy as The Casimir energy as well as purely extensive part of energy can be expressed in terms of entropy and radius

After the work of Witten [23] on the / correspondence, Savonije and Verlinde [24] proposed that Cardy-Verlinde formula can be derived using the thermodynamical relations of arbitrary BHs in arbitrary dimensions. In this point of view, we shall prove the validity of Cardy-Verlinde formula for NC Schwarzschild BH.

From (8) and (10), we get By using (11) and (12) in the above equation, The Casimir energy given by (9) for with (6) and (7) takes the following: By using the above equation in (10), we get the purely extensive part of total energy as Further,

From the comparison of (12) and (16), we get

Also, the comparison of (11) and (18) gives By taking the product of (20) and (22), we get By using (16), (18), and (24) in (8), we get This result shows that the entropy of the NC Schwarzschild BH can be expressed in terms of Cardy-Verlinde formula. As the BH geometric and thermodynamic quantities are evaluated by assuming large-radius approximations, the Cardy-Verlinde formula is valid only for large BHs.

3. Outlook

As a prolongation of the research on BH and gravitational collapse [12–31] in this paper, we derive the entropy formula in conformal field theory of a 4D static spherically symmetric NC Schwarzschild BH. This NC BH solution is obtained by introducing the NC effect through a coordinate coherent state approach, which is in fact the substitution of the point distributions by smeared source throughout a regular region of linear size. We perform the analysis by obtaining entropy and temperature, which show a deviation from their usual relations depending on the NC parameter . We have proved that the entropy of the NC Schwarzschild BH can be expressed in terms of Cardy-Verlinde formula. For this purpose, we have used the approximate of values of incomplete gamma functions up to the term . With the same order of approximation the entropy and temperature of NC BH horizons has been calculated. The procedure adopted in this paper has been already used by Setare and Jamil [5, 6]. It would be interesting to generalize this work for charged and charged rotating NC BHs. The Cardy-Verlinde formula of charged NC BH [32] is in progress.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.