Noncommutative geometry inspired 3-dimensional charged black hole solution in an anti-de Sitter background space-time

We report a 3D charged black hole solution in an anti desetter space inspired by noncommutative geometry.In this construction,the black hole exhibits two horizon which turn into a single horizon in the extreme case.We investigate the impacts of the electromagnetic field on the location of the event horizon,mass and thermodynamic properties such as Hawking temperature,entropy and heat capacity of the black hole.The geodesics of the charged black hole are also analyzed.


I. INTRODUCTION
In recent years, noncommutative geometry has received a considerable interest to get insight into many conceptual issues in the quantum domain of high energy physics. It is argued that at sufficiently high energy scales the target-space structure coordinates may become noncommuting operators on a D-brane [1,2]. Consequently, the noncommutativity of the space-time can be encoded in a commutator [x µ , x ν ] = iθ µν , where, θ µν is an antisymmetric matrix of dimension [length] 2 which determines the fundamental cell discretization of the spacetime. It is similar to the way the Plank's constant discretizes phase-space [3]. The noncommutative approach provides the construction of a black hole with a minimum scale √ θ, known as noncommutative black hole [4,5], whose commutative limit is the Schwarzschild metric. It is to be noted that noncommutative geometry is an intrinsic property of the space-time and it does not depend on the curvature of space-time.
A large number of studies inspired by noncommutative geometry are available in the literature. Myung et al [6] have discussed thermodynamic behaviour and evaporation processes involved in a noncommutative black hole. In their investigations, they have considered a noncommutative 3-dimensional black hole whose commutative limit is the non-rotating Bañados, Teitelboim and Zanelli [7] (henceforth BTZ) black hole. Nozari and Mehdipour [8] have investigated the Hawking radiation from a noncommutative black hole. Rahaman et al [9], in the context of galactic rotation curves, have shown that a noncommutative geometrical background is sufficient for the existence of a stable circular orbit and one need not consider dark matter in this formalism. In a separate paper, Rahaman et al [10] have shown that wormhole solutions exist up to 5-dimensional space-times; however, * Electronic address: rahaman@iucaa.ernet.in † Electronic address: piyalibhar90@gmail.com ‡ Electronic address: rsharma@iucaa.ernet.in § Electronic address: rishitiwari59@rediffmail.com they do not exist in dimensions > 5. Banerjee et al [11] have investigated the thermodynamic properties (Hawking temperature, entropy and area law, in particular) of a Schwarzschild black hole in noncommutative spacetimes. By invoking a graphical analysis, the authors have shown the non-trivial behavior of the noncommutative Schwarzschild metric at different length scales. It has been shown that in the limit of large r (r >> √ θ) outside the horizon, the metric behaves like the standard Schwarzschild solution, whereas in the limit of small r (r << √ θ) inside the horizon, the space-time is de-Sitter type with a constant positive curvature.
The motivation of the present work is to construct a 3-dimensional charged black hole solution based on noncommutative geometry in an anti-de Sitter background space-time. It is to be stressed here that gravitational analyses in (2 + 1)-dimensions have become extremely fascinating with the discovery of the black hole solution by Bañados, Teitelboim and Zanelli [7]. The BTZ metric describes the exterior gravitational field of a circularly symmetric Einstein-Maxwell system in the presence of a negative cosmological constant and is characterized by its mass, angular momentum and charge. In standard (3 + 1)-dimensions, Einstein-Maxwell systems for static spherically symmetric charged distributions of matter have investigated by many (see for example, Tikekar and Singh [12], Sharma et al [13], Thirukkanesh and Maharaj [14] and also references therein). Analytic models of static charged fluid spheres has been reviewed by Ivanov [16]. In (2 + 1)-dimensions, Rahaman et al [15] have developed a model for a charged gravastar in an anti-de Sitter background space-time. Corresponding to the static BTZ exterior, Cataldo et al [17] have investigated the properties of a circularly symmetric static charged fluid distribution. Larrañaga and Tejeiro [18] have obtained a charged black hole solution in 3-D anti-de Sitter space inspired by noncommutative geometry where a Gaussian distribution of charge has been considered. In an earlier work, Rahaman et al [19] have studied BTZ black holes inspired by noncommutative geometry. The current investigation is an extension of Ref. [19] where we have incorporated a distribution of charge into the sys-tem. The resultant charged black hole appears to have two horizons which, however, get reduced to a single horizon in the extreme case. Thermodynamic properties such as Hawking temperature, entropy and heat capacity of the charged black hole have been analyzed. Effective potential and behaviour of test particles around the black hole have also been outlined.

II. INTERIOR SPACE-TIME
We write the line element for the interior space-time of a static circularly symmetric charged distribution of matter in (2 + 1)-dimensions in the form The energy-momentum tensor corresponding to the charged fluid distribution is assumed as where, ρ is the energy-density, p r is the radial pressure and p t is the tangential pressure. In (2), u i is the velocity of the fluid and η i is a radial unit vector satisfying the relation u i u i = −η i η j = 1. The electromagnetic field tensor is governed by the Maxwell's equations where, J µ is the current 3-vector defined as In (4), σ(r) is the proper charged density of the distribution. The electromagnetic field tensor may be expressed as where, E(r) is the electric field intensity. In the presence of a negative cosmological constant (Λ < 0), the Einstein-Maxwell equations for the metric (1) yield the following set of four independent equations (in geometrized units G = c = 1) where a 'prime' denotes differentiation with respect to the radial parameter r. Eq. (10) can equivalently be expressed as where q(r) is the total charge contained within a sphere of radius r. We assume that the volume charge density has the form where n is an arbitrary constant and σ 0 is the central charged density. This particular choice is different from the Gaussian distribution of charge assumed in Ref. [18]. Consequently, from Eq. (11) we get Further, we consider the maximally localized source of energy of the static spherically symmetric charge distribution as a Gaussian distribution with minimal width √ θ expressed as [19], where M is the total mass of the source diffused throughout a region of linear dimension √ θ. Substituting (13) and (15) in (7), we obtain which, on integration, yields where, A is a constant of integration. In Ref. [9] it has been shown that the constant of integration A plays the role of the mass of the black hole, i.e., A = M . In our construction, if we set σ 0 = 0 and consider the limit r √ θ → ∞, then it follows that A = M . Accordingly, we rewrite Eq. (17) as Note that the vacuum Einstein field equations in (2 + 1)dimensional space-time in the presence of a negative cosmological constant (Λ < 0) admit a black hole solution known as BTZ [7] solution for which we have g rr = (g tt ) −1 . To retain the structure of BTZ black hole, we must have e ν = e −λ . From Eqs. (7)- (8), it then follows that which is the 'ρ vacuum' EOS in reference to 'zero-point energy of quantum fluctuations' [19]. Combining Eqs. (7)-(9) together with (19), we obtain From Eq. (16), we have Combining Eqs. (20) and (21), we get (22) We, thus, have have a noncommutative geometry inspired analytic model of a 3-D charged black hole. In this construction, the spacetime near the origin behaves as Note that for n = 0, by adjusting the parameters suitably, one can regain the (2+1)-dimensional static charged distribution of Liang and Liu [20] In the following sections, we shall analyze some features of our charged BTZ black hole.

A. Formation of event horizons
The condition for the formation of event horizon is given by g tt (r h ) = 0, which implies where r h is the horizon distance. Though a closed-form solution for r h cannot be obtained from the above equation, one can express the mass parameter in terms of r h as In Fig. 1, we have shown the existence of horizons and their corresponding radii by identifying the intersections of gtt √ θ with r √ θ . We note that, for a given value of Λ √ θ = −0.02, there exists a minimal mass M 0 = 0.36 √ θ below which no black hole exists though two distinct horizons exist for M > 0.36 √ θ. If mass of the black hole increases the distance between the two horizons increases. The two horizons coincide at a minimal mass M = M 0 . In our illustrative example, r = r 0 = 3.7 √ θ is the distance where the two horizons coincide, i.e., r 0 is the radius of the extremal black hole. The results may be summarized as follows:  26), it is interesting to note that the lower limit mass M 0 increases when charge is incorporated into the system. Moreover, the distance of the outer event horizon increases in the presence of electromagnetic field as shown in Fig. 2.
In Fig. 3-4, we have shown behaviour of the density parameter (ρ)/radial pressure (p r √ θ) and tangential pressure (p t √ θ), respectively, between the inner and outer horizons and outside the outer horizon for different choices of the mass parameter. Variation of mass M √ θ with respect to the horizon radius r h √ θ has been shown in Fig. 5. We note a decrease in mass M √ θ within the horizon distance r h √ θ when charge is incorporated into system. This has been shown in Fig. 6.

B. Thermodynamical properties
The Hawking temperature in our model turns out to be The surface gravity takes the form and the Bakenstein-Hawking entropy can be obtained from the equation In Fig. 7, we have shown variation of the Hawking temperature T H with respect to the radial distance r √ θ . Interestingly, the Hawking temperature T H decreases with the inclusion of the charge as shown in Fig. 8.
The heat capacity of the charged BTZ black hole can be obtained from the relation For stability of the black hole, C has to be positive [20]. In Fig. 9, the nature of the heat capacity in our model has been shown. We note that C vanishes at the extremal event horizon r 0 . Since C is negative in the re- √ θ , C is positive which indicates that the noncommutative geometry inspired charged BTZ black hole is stable in this region. We notice an appreciable change in the heat capacity due to the inclusion of charge as shown in Fig. 10.

C. The geodesics
The geodesics equation in our construction yields the following results: where, The constants Σ and p represent the energy per unit mass and angular momentum, respectively. In Eqs. (33)-(35), τ is the affine parameter and L is the Lagrangian having values 0 and −1 for massless and massive particles, respectively. From Eq. (33), we have Combining Eqs. (35) and (36), we get Comparing Eq. (36) with we obtain the effective potential in the form (38)

Null geodesics
For a massless particle, i.e., photon with L = 0 and p = 0, we have (39) which in integral form can be expressed as (40) Eq. (40) is not integrable, however, one can obtain the values of t for different choices of r * √ θ by numerical procedures. This has been shown in Table 1 and a graphical representation of it is given in Fig. 11.  The effective potential of the system is obtained as which for p = 0 reduces to The nature of V ef f for massless particles has been shown in Fig. 12.

Time-like geodesics
For massive particles with L = −1 and p = 0, we have which in integral form can be written as For a time-like particle with p = 0, the effective potential V ef f gets the form The nature of effective potential for massive particles has been shown in Fig. 13. The shape of the effective potential indicates the possibility of stable circular orbits for massive particles around the black hole.

IV. TEST PARTICLES
Let us now analyze the motion of a test particle around noncommutative charged black hole using Hamilton-Jacobi(H-J) approach. Let m and e denote the respective mass and charge of the test particle. The Hamilton-Jacobi equation for the test particle has the standard form where, g ik is the background space and S(t, r, θ, φ) is the Hamilton's characteristic function. The gauge potential A i is given by To solve the above partial differential equation, we use the method of separation of variables and choose the H-J function S as where, E is the energy and J is the angular momentum of the particle. The radial velocity of the particle is then obtained as where, p is the momentum of the particle. Now, the turning points of the trajectory are determined from the requirement dr/dt = 0, which yields Consequently, the potential curve can be obtained from the relation For a stationary system, V (r) must have an extremal value, i.e., dV /dr = 0 which implies (50)

A. Test particle in static equilibrium
In static equilibrium (p = 0), from Eq. (50), the value of r at which potential curve will be an extremal can be obtained from the requirement A graphical representation of H(r), as shown in Fig. 14, indicates that there does not exist any r for which H(r) = 0. This implies that the test particle can not be trapped by the black hole. In other words, the black exerts no gravitation force on the test particle in static equilibrium situation.
B. Test particle in non-static equilibrium For a text particle in non-static equilibrium (p = 0), we consider two possibilities:

Uncharged test particle (e = 0)
In this case, the value of r at which potential curve will be an extremal can be obtained from the requirement In Fig. 15, we note that P (r) = 0 at some finite value of r. Therefore, the particle can be trapped by the black hole. In other words, the black hole exerts gravitation force on the uncharged test particle in non-static equilibrium.

Test particle with charge (e = 0)
For a charged particle, the value of r at which potential curve will be an extremal can be obtained from the requirement S(r) ≡ eQ mr 2 1 + Fig. 16 shows that S(r) = 0 for any finite value of r. This implies that the black exerts no gravitation force on the charged test particle in non-static equilibrium.
Our calculations are in agreement with the notion that a non-static charged test particle would behave differ-ently from its neutral counterpart in the gravitational field of a charged black hole. The motion of a neutral particle is solely governed by the gravitational field of the black hole whereas the motion of a charged particle in the gravitational field of a charged black hole also ex- periences electromagnetic force. Therefore, while a neutral particle moves along the geodesic and gets trapped, a charged test particle in motion may not be trapped by the black hole.

V. DISCUSSIONS
In this work, we have constructed a charged BTZ-like black hole solution by adopting the formalism of noncommutative geometry. The uncharged limit of this solution reported in Ref. [19] can be regained simply by putting σ = 0. The noncommutative effects on the physical quantities are found to be similar to the properties discussed in Ref. [19]. To investigate the impact of electromagnetic field we have considered two cases (σ = 0 and σ = 0) and compared the results graphically. Our results show that the limiting mass M 0 below which no black hole can exist depends on its associated charge. In fact, the value of the limiting mass increases as charge is incorporated into the system. The location of the event horizon, Hawking temperature and heat capacity also do get modified in the presence of the electromagnetic field. Particle trajectories around the black hole are also affected by the inclusion charge. In particular, we note that the motion of a charged particle around a charged black hole is quite different from the motion of a neutral test particle. We hope that our results will contribute towards getting a better understanding of the effects of noncommutative geometry on the structure and properties of black holes. In addition, the findings may be useful to understand how the electromagnetic field can change a number of astrophysical phenomena such as properties of accretion disks around charged black holes and Hawking radiation.