BTZ black holes inspired by noncommutative ... - APS Link Manager

23 downloads 0 Views 640KB Size Report
Apr 4, 2013 - Department of Mathematics, Jadavpur University, Kolkata 700 032, West ... Saha Institute of Technology, Kolkata 700 150, West Bengal, India.
PHYSICAL REVIEW D 87, 084014 (2013)

BTZ black holes inspired by noncommutative geometry Farook Rahaman* Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India

P. K. F. Kuhfittig† Department of Mathematics, Milwaukee School of Engineering, Milwaukee, Wisconsin 53202-3109, USA

B. C. Bhui‡ and Mosiur Rahaman§ Department of Mathematics, Meghnad Saha Institute of Technology, Kolkata 700 150, West Bengal, India

Saibal Rayk Department of Physics, Government College of Engineering and Ceramic Technology, Kolkata 700 010, West Bengal, India

U. F. Mondal{ Department of Mathematics, Behala College, Parnasree, Kolkata 700 060, West Bengal, India (Received 17 January 2013; published 4 April 2013) In this paper, a Ban˜ados-Teitelboim-Zanelli (BTZ) black hole [Phys. Rev. Lett. 69, 1849 (1992)] is constructed from an exact solution of the Einstein field equations in a (2 þ 1)—dimensional anti—de Sitter spacetime in the context of noncommutative geometry. The BTZ black hole turns out to have either two horizons, no horizon, or a single horizon corresponding to a minimal mass. Certain thermodynamical properties are investigated, including Hawking temperature, entropy, and heat capacity. Also discussed is the geodesic structure of BTZ black holes for both massless and massive particles. In particular, it is shown that bound orbits for test particles are possible. DOI: 10.1103/PhysRevD.87.084014

PACS numbers: 04.40.Nr, 04.20.Dw, 04.20.Jb

I. INTRODUCTION Recent years have seen rapid advances in the applications of noncommutative geometry, an outgrowth of string theory. The approach is based on the realization that coordinates may become noncommuting operators on a D brane [1–4]. The result is a discretization of spacetime due to the commutator ½x ; x  ¼ i , where  is an antisymmetric matrix. It is similar to the way that the Planck constant ℏ discretizes phase space [1]. The noncommutativity eliminates pointlike structures and replaces them with smeared objects. The noncommutative geometry is an intrinsic property of spacetime and does not depend on particular features such as curvature. A number of studies inspired by noncommutative geometry can be found in the literature. In one of the earlier studies, Garattini and Lobo [5] obtained exact wormhole solutions that were then analyzed in semiclassical gravity. In a subsequent study [6], they found an exact gravastar solution and explored its physical characteristics. Rahaman et al. [7], discussing galactic rotation curves, concluded that a noncommutative-geometry background is sufficient

*[email protected][email protected][email protected] § [email protected] k [email protected] { [email protected]

1550-7998= 2013=87(8)=084014(8)

for producing stable circular orbits without the need for dark matter. Kuhfittig [8] found that a special class of thinshell wormholes that are unstable in classical general relativity exhibit small regions of stability in noncommutative geometry. In another study by Rahaman et al. [9] on higher-dimensional wormholes, it is shown that wormhole solutions exist in the usual four, as well as in five dimensions, but they do not exist in higher-dimensional spacetimes. In a more recent study, Radinschi et al. [10] calculated the energy momentum for a noncommuting radiating Schwarzschild black hole in order to obtain the expression for energy. Common to all these studies is that the effect of the smearing is mathematically implemented pffiffiffi by using a Gaussian distribution of minimal length  instead of the Dirac delta function. Interest in ð2 þ 1Þ—dimensional gravity has increased in recent years due to the discovery of various aspects of black hole solutions. Some general works in this line are quasinormal modes of charged dilaton black holes in ð2 þ 1Þ—dimensional solutions in low-energy string theory with asymptotically anti—de Sitter spacetimes [11]; branes with naked singularities, analogous to linear or planar defects in crystals and showing that zero-branes in AdS spacetimes are ‘‘negative-mass black holes’’ [12]; Hawking radiation from covariant anomalies in ð2 þ 1Þ— dimensional black holes [13]; and so on. On the other hand, specialized investigations have been carried out by Rahaman et al. [14] on gravastars in ð2 þ 1Þ anti—de

084014-1

Ó 2013 American Physical Society

RAHAMAN et al.

PHYSICAL REVIEW D 87, 084014 (2013)

Sitter spacetimes with charge as an alternative to charged black holes. Also, nonstatic charged BTZ-like black holes in ðN þ 1Þ dimensions have been considered by Ghosh [15], which in the static limit, for N ¼ 2, reduces to ð2þ1Þ BTZ black hole solutions. There are also charged, regular extensions of the BTZ black hole solutions available in the literature by employing nonlinear Born-Infeld electrodynamics to eliminate the inner singularity [16]. There are several proposals in the literature for constructing noncommutative black holes [17–20]. However, following Nicolini et al. [2], we construct a BTZ black hole solution from the Einstein field equations in ð2 þ 1Þ—dimensional anti—de Sitter spacetime, given a noncommutative-geometry background. This is followed by a discussion of the black hole’s thermodynamical properties such as Hawking temperature, entropy and heat capacity, as well as the geodesic structure for both massless and massive particles. Some of the thermodynamical properties are similar to those obtained by Liang and Liu [21], who used a Lorentzian smeared mass distribution instead of a Gaussian one in AdSp3ffiffiffispacetime. One consequence of this is that in the limit r=  ! 1, the solution reduces to a rotating BTZ black hole. Returning to a Gaussian framework, rotating black holes in ð2 þ 1Þ dimensions are also discussed in Ref. [22] by Tejeiro and Larranaga. They also describe charged black holes and compare them to charged BTZ black holes [23]. II. THE INTERIOR SPACETIME Let us write the line element describing the interior spacetime of a static, spherically symmetric distribution of matter in ð2 þ 1Þ dimensions in the following form: ds2 ¼ fðrÞdt2 þ ½fðrÞ1 dr2 þ r2 d2 ;

(1)

where fðrÞ is denoted by e2ðrÞ and ½fðrÞ1 by e2ðrÞ . We take the matter distribution to be anisotropic in nature and therefore choose the most general energy-momentum tensor in the form Tij ¼ ð þ pt Þui uj þ pt gij þ ðpr  pt Þi j ;

(2)

where , pr and pt represent the energy density, radial pressure and tangential pressure, respectively. Also, i ¼ eðrÞ ir is a unit 4-vector along the radial direction, and ui is the 4-velocity of the fluid. The Einstein field equations with cosmological constant  < 0, together with the general energy-momentum tensor given in Eq. (2), yield (letting G ¼ c ¼ 1) 2 þ  ¼ 2pr   ¼

We have, in addition, the conservation equation in ð2 þ 1Þ dimensions: 1 ð þ pr Þ0 þ p0r þ ðpr  pt Þ ¼ 0: r

(6)

In ð2þ1Þ dimensions, the maximally localized source of energy of the static and spherically symmetric distributions having a minimal-spread Gaussian profile is taken as [3]   M r2 ¼ : (7) exp  4 4 Here M is the total mass of the source. Due to the uncertainty,pit ffiffiffi is diffused throughout a region of linear dimension . The vacuum Einstein field equations in ð2 þ 1Þ spacetime dimensions, with a negative cosmological constant, admit a black hole solution known as a Ban˜adosTeitelboim-Zanelli (BTZ) solution [24]. For a BTZ black hole, we have grr ¼ g1 tt . So to retain the structure, we require that pr ¼ :

(8)

This ansatz is known in the literature as a ‘‘ vacuum’’ or ‘‘vacuum equation of state’’ in connection with the ‘‘zero-point energy of quantum fluctuation,’’ [25–28] where pressure is of a repulsive nature. With this equation, one can solve Eq. (6) to yield   r2 Mð2  1Þ r2 pt ¼ : (9) exp  4 4 Using the field equations, we get the following solution for the metric coefficients:   r2 2 2 e  r2 ; ¼ e ¼ A þ 2M exp  (10) 4 where A is an integration constant. In the limit, prffiffi ! 1, so that Eq. (10) reduces to a BTZ black hole, where the integration constant A plays the role of the mass of the BTZ black hole, i.e., A ¼ M. Observe that asymptotically far away,  ¼ pr ¼ pt ¼ 0. To determine the mass distribution from Eq. (7), we use an approach similar to that in Refs. [3,5]:    ~ 2 r 2 m M mðrÞ ¼ ðm2Þ=2 ; (11) ;

2 2M ~

0 e2 ; r

(3)

0 e2 ; r

(4)

where 2 ¼ M2 =, and is the lower incomplete gamma function:  Zx  a du (12) ;x ¼ ua=b eu : b u 0

(5)

~ ¼ 2, and we obtain from For a BTZ black hole, m

2 r2 =4M2 ¼ r2 =4 the expression for mass as

2pt   ¼ e2 ð02 þ 00  0 0 Þ:

084014-2

BTZ BLACK HOLES INSPIRED BY NONCOMMUTATIVE . . .



PHYSICAL REVIEW D 87, 084014 (2013)



 Z r2 =4 r2 mðrÞ ¼ M et dt ¼ M 1  exp  : 4 0

(13)

The parameter plays a critical role in determining the horizons, as we will see later on. At the origin, mð0Þ ¼ 0, which is consistent with Eq. (11). Near the origin, the geometry is given by   2M þ  r2 þ Oðr4 Þ: e2 ¼ e2 ¼ A þ 2M  4 One can identify this result with a BTZ black hole spacetime, where the total mass M and the noncommutative parameter  combine to modify the cosmological constant, a point also made in Ref. [3]. This indicates that different mass particles experience different cosmological constants. We therefore conclude that our line element describes the geometry of a noncommutative-geometry inspired BTZ black hole. III. FEATURES OF THE BLACK HOLE In this section, we study some of the effects of the noncommutative geometry on BTZ black holes. Let A ¼ M in Eq. (10). Then the equation gtt ðrh Þ ¼ 0 gives the event horizon(s):     M r2 2 exp  h  1 : (14) r2h ¼  4 Even though we cannot obtain a closed-form solution for rh in Eq. (14), we can readily write the mass M as a function of rh : M¼

r2h r2

2 exp ð 4h Þ  1

:

(15)

The existence of horizons and their radii can be found at the points pffiffiffiwhere gtt cuts the r axis, as shown in Fig. 1, using   ¼ 0:02. Here, three possibilities present themselves graphically in terms of the approximate value of : (i) Two phorizons when > 0:214, or M > M0 ¼ ffiffiffi 0:214 . (ii) One horizon corresponding to the extremal black hole with M ¼ M0 , i.e., ¼ 0:214; (iii) No horizon for < 0:214. Fig. 1 shows that a noncommutative-geometry-inspired BTZ black hole has two horizons, and that the distance between the horizons will increase with an increasing black hole mass. Figure 1 also indicates that there is a minimal mass M0 below which no black hole exists. Moreover, at the minimal mass M ¼ M0 , the two horizons coincide at the minimal horizon radius r0 , which lies between the horizons. This r0 is therefore the horizon radius of the extremal black hole. It can also be

FIG. 1 (color online). Singularities occur where gpttffiffiffi cuts the r axis. Representation: pffiffiffithe solid curve for M ¼ 1:2  and dotted curve for M ¼ 0:8 pffiffiffi  indicate two horizons. The dashed curve forpM ¼ 0:214  represents one degenerate horizon ffiffiffi pffiffiffi r0  2:59 —i.e., an extremal black hole. For M ¼ 0:1 , no horizon exists (chain curve). The intercepts on the r axis give the radii of the event horizons.

determined from the conditions f ¼ 0 and ing to the equation

df dr

¼ 0, lead-

 r20 r20 = ¼ 0: þ r2 4 1  2 exp ð 0 Þ

(16)

 2 exp

4

pffiffiffi Using the pcondition   ¼ 0:02 in Fig. 1, we can ffiffiffi obtain r0 =  ¼ 2:59, showing that the two approaches are consistent. The minimal mass of the extremal black hole can be written in terms of the minimal radius r0 , so that pffiffiffi r20 M0  ð Þ pffiffiffi ¼ ¼ 0:214: r2  2 exp ð 40 Þ  1

(17)

The variation of this factor with respect to the horizon radius usffiffiffi also plot pffiffiffi is shown in Fig. 2. At this point, let p   from Eq. (7) for various values of M= p (Fig. pffiffiffi ffiffiffi 3). Similarly, Fig. 4 shows pt  plotted against r=  from Eq. (9). It is worth noting that the radius of the extremal black hole is always less than the radius of the outer horizon. The result is significant: if the initial mass of the black hole is M > M0 , then it can radiate until the value M0 is reached. It follows that evaporation of the black hole may indeed be occurring. Next, consider the Hawking temperature, which is given by

084014-3

RAHAMAN et al.

FIG. 2. The variation of mass radius prhffiffi .

PHYSICAL REVIEW D 87, 084014 (2013)

Mffiffi p 

with respect to horizon

  1 dgtt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gtt grr jr¼rh TH ¼ 4 dr    r M r2 ¼  h 2 þ exp  h :  4 4

(18)

The plot, Fig. 5, is obtained from pffiffiffi    pffiffiffi M rh =  r2h 2  þ pffiffiffi exp  TH ¼  : 4 4 

(19)

pffiffiffi FIG. 3 (color online). Plot for   vs. prffiffi for different pffiffiffi values of pMffiffi . Representation: the solid curve for M ¼ 1:2 , dotted pffiffiffi pffiffiffi curve for M ¼ 0:8 , dashed curve for M ¼ 0:214 , and pffiffiffi chain curve for M ¼ 0:1 .

pffiffiffi FIG. 4 (color online). Plot for pt  vs. prffiffi for different pffiffiffi values of pMffiffi . Representation: the solid curve for M ¼ 1:2 , dotted pffiffiffi pffiffiffi curve for M ¼ 0:8 , dashed curve for M ¼ 0:214 , and pffiffiffi chain curve for Mpffiffiffi ¼ 0:1 . Note that up to the degenerate horizon r0  2:59  for an extremal black hole, pt assumes negative values, and beyond that, positive values. For large values of r, i.e., for prffiffi  1, it dies out.

Equation (19) and Fig. 5 show that the noncommutative geometry leads to the minimal horizon radius r0 , since TH tt cannot be negative. This is exactly where dg dr ¼ 0 in Eq. (10). Observe that the temperature sinks to absolute zero at r0 . For prhffiffi  1, the Hawking temperature assumes the value

FIG. 5 (color online). Plot for Hawking temperature vs. prffiffi for different values of pMffiffi . Representation: the solid curve for M ¼ pffiffiffi pffiffiffi 1:2 p, ffiffiffidotted curve for M ¼ 0:8 , pdashed curve for M ¼ ffiffiffi 0:214 , and chain curve for M ¼ 0:1 .

084014-4

BTZ BLACK HOLES INSPIRED BY NONCOMMUTATIVE . . .

TH  

rh  : 2

      1 dgtt  rh M r2h   ¼ 2 þ exp  ; ¼  r¼rh 2 dr   2 4

S ¼ 2L ¼ 4rh :

(22)

As a final comment, the noncommutative-geometryinspired BTZ black hole is stable if the heat capacity C is positive [21], where 1 @Tðrh @rh

For small  and written as

the above equation can be

  pffiffiffiffiffi pffiffiffiffiffi Ml2 S ¼ 4rh  4 Ml  4 Ml exp  : 4

The nature of the heat capacity is shown in Fig. 6. The plot shows that C vanishes at the extremal event horizon r0 and becomes negative for prhffiffi < pr0ffiffi , just as in the case of the Hawking temperature. So this region is definitely unphysical. On the other hand, for prhffiffi > pr0ffiffi , C is positive, which implies that the BTZ black hole is stable. At this point, we would like to comment on how the physical quantities such as temperature, entropy, etc., could be affected by the noncommutativity for small . We calculate the different physical quantities of the standard BTZ black hole plus perturbative terms. The BTZ black hole has the horizon located at pffiffiffiffiffi rh ¼ Ml; (23)

(25)

Here, the first term is the BTZ black hole horizon radius, while the second term is the  correction. It now becomes apparent that the effect of noncommutativity is small, as expected, because spacetime should be a smooth classical manifold at large distances. The next step is to find the  corrections of the Bekenstein-Hawking entropy (S) of the BTZ black hole:

: Þ

where  ¼  l12 to emphasize that  is negative.

rhffiffi p 1, 

  pffiffiffiffiffi  Ml2 : rh  Ml 1  exp  4

(21)

as well as the Bekenstein-Hawking entropy (S) of the black hole. It is twice the perimeter L of the event horizon:

@Mðrh Þ @Mðrh Þ ¼ @Tðrh Þ @rh

Equation (14) can be solved by iteration. The result is   pffiffiffiffiffi  Ml2 12 rh ¼ Ml 1  2 exp  : (24) 4

(20)

For completeness, let us also state the closely related surface gravity



PHYSICAL REVIEW D 87, 084014 (2013)

(26)

Here, the first term is the Bekenstein-Hawking entropy (S) of the BTZ black hole, while the second term is the  correction. Our final task is to find the  corrections for the Hawking temperature and surface gravity of the BTZ black hole. From Eq. (18), we have     1 pffiffiffiffiffi Ml2 TH ¼  Ml 1  exp  4 4      2 2 M Ml Ml2 1  2exp    2 þ exp  : (27)  4 4 l 2

Keeping the first order of exp ð Ml 4 Þ, the above expression yields pffiffiffiffiffi  pffiffiffiffiffi   pffiffiffiffiffi   M Ml Ml2 Ml2 M M exp  þ :  exp  TH  4 4 4 2l 2l (28) Note that the first term is the Hawking temperature of the BTZ black hole, while the second term is the  correction. The correction term of the surface gravity is given by pffiffiffiffiffi  pffiffiffiffiffi   pffiffiffiffiffi   M M M Ml Ml2 Ml2  exp  þ :  exp  2 4 4 l l (29) IV. THE GEODESICS From the geodesics equation 

d2 x  dx dx ¼ 0; þ   d d d 2

FIG. 6. Plot for C vs. prhffiffi .

we obtain [29]

084014-5

(30)

RAHAMAN et al.

PHYSICAL REVIEW D 87, 084014 (2013)



1 dr fðrÞ d

2

¼

E2 p2  2 þ L; fðrÞ r

(31)

  d ¼ p; d

(32)

E dt ¼ ; fðrÞ d

(33)

r2 and

r where fðrÞ ¼ M þ 2M exp ð 4 Þ  r2 , and the constants E and p are identified as the energy per unit mass and angular momentum, respectively. Here is the affine parameter, and L is the Lagrangian having values 0 and 1, respectively, for massless and massive particles. From the geodesic [Eq. (31)], we can write      p2 1 dr 2 1 2 ¼ E þ fðrÞ L  2 : (34) 2 r 2 d 2

Now, comparing Eq. (34) with r2_ þ Veff ¼ 0, the effective potential can be written    1 2 p2 Veff ¼  E þ fðrÞ L  2 : (35) 2 r 2

A. Null geodesics For massless particles, i.e., for photons, we have L ¼ 0, and the corresponding effective potential is    2 E2 p2 r Veff ¼  þ 2 M þ 2M exp   r2 : (36) 2 r 4

pffiffiffi FIG. 7 (color online). Plot for Veff  vs. prffiffi for different pffiffiffi values of pMffiffi . Representation: the solid curve for M ¼ 1:2 , dotted pffiffiffi pffiffiffi curve for M ¼ 0:8 , dashed curve for M ¼ 0:214 pffiffiffi , and pffiffiffi chain curve for M ¼ 0:1 . The shape of the Veff  indicates that a photon will fall into black hole.

indicates that the particle can move only inside the black hole. Since the effective potential assumes negative values between the horizons, the particle is confined to the region between the two horizons, and as a result, cannot hit the singularity. Finally, observe that the minimum of Veff ðrÞ occurs between the horizons, which means that stable circular orbits are going to exist.

As r ! 0, the effective potential Veff ðrÞ becomes very 2 large, but it approaches  E2  p2 as r ! 1. At the 2 horizons, Veff ¼  E2 . The shape of Veff ðrÞ, shown in Fig. 7, indicates that a photon will fall into a black hole [29]. Taking various values for the masses does not alter the nature of the geodesics. B. Timelike geodesics For massive particles, L ¼ 1, and the corresponding effective potential is     2 E2 p2 r Veff ¼  þ 1 þ 2 M þ 2M exp   r2 ; 2 4 r (37) as shown in Fig. 8. The effective potential becomes very large as r ! 0, as well as when r ! 1. At the minimal 2 horizon r0 , it assumes the constant value Veff ¼  E2 , while Fig. 9 shows that the roots of the Veff coincide with the horizons. Also, the shape of the effective potential

pffiffiffi FIG. 8 (color online). Plot for Veff  vs. prffiffi for different pffiffiffi values of pMffiffi . Representation: the solid curve for M ¼ 1:2 , dotted pffiffiffi pffiffiffi curve for M ¼ 0:8 , p dashed curve for M ¼ 0:214 , and ffiffiffi chain curve for M ¼ 0:1 . The effective potential has a minimum between two horizons; i.e., stable circular orbits do exist.

084014-6

BTZ BLACK HOLES INSPIRED BY NONCOMMUTATIVE . . .

pffiffiffi FIG. 9 (color online). Plot for Veff  vs. prffiffi for different pffiffiffi values of pMffiffi . Representation: the solid curve for M ¼ 0:8 , dashed pffiffiffi pffiffiffi curve for M ¼ 0:214 , and chain curve for M ¼ 0:1 . The effective potential has a minimum between two horizons; i.e., stable circular orbits do exist.

PHYSICAL REVIEW D 87, 084014 (2013)

pffiffiffi pffiffiffi dV prffiffi FIG. 10. Plot for dV dr  vs.  . Note that dr  is zero for a r certain value of pffiffi . This implies that VðrÞ has at least one extremal.

   2 dV 2p2 r 2 ¼  r M þ 2M exp  dr 4 m0 r3   2   2 p Mr r exp   2r ¼ 0: þ 1þ   4 m0 r2

V. TEST PARTICLES Let us consider a test particle having mass m0 and moving in the gravitational field of the BTZ black hole inspired by noncommutative geometry and described by the metric ansatz [Eq. (1)]. The Hamilton-Jacobi (HJ) equation for the test particle is [30,31]    2   1 @S 2 @S 1 @S 2  þf þ 2 þ m20 ¼ 0: (38) f @t @r r @ As there is no explicit dependence on t and , let us choose the HJ function S as [30,31]

While difficult to tell from the equation, the plot of dV dr , given in Fig. 10, shows that real positive solutions exist prffiffi wherever p1ffiffi dV dr cuts the  axis. Hence, bound orbits for the test particles exist. In other words, the test particles can be trapped by BTZ black holes inspired by noncommutative geometry.

Sðt; r; ; Þ ¼ Et þ S1 ðrÞ þ p;

VI. CONCLUSION

where E and p are identified as the energy and angular momentum of the particle. The radial velocity of the particle is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3   f2 E2 p2 dr ¼  m20  2 : (39) E f r dt For detailed calculations, see Refs. [30,31]. The turning points of the trajectory are determined from dr dt ¼ 0 and, as a consequence, the potential curve is given by  pffiffiffi E p2 1=2 VðrÞ  ¼ f 1þ 2 2 : (40) m0 m0 r The extremals of the potential curve are the solutions of the equation dV dr ¼ 0, and are found to be

This paper investigates the properties of a BTZ black hole constructed from the exact solution of the Einstein field equations in a ð2 þ 1Þ—dimensional anti—de Sitter spacetime in the context of noncommutative geometry. It finds that a BTZ black hole has either two horizons, no horizons, or a single horizon r ¼ r0 corresponding to a minimal mass M ¼ M0 . In this connection, we note the comments by Mazharimousavi et al. [16]: ‘‘It is well known that unlike its chargeless version the charged Banados-Teitelboim-Zanelli (BTZ) black hole solution in (2 þ 1)—dimensional spacetime is singular.’’ Thus, they construct a charged, regular extension of the BTZ black hole solution by employing nonlinear Born-Infeld electrodynamics, supplemented with the Hoffmann term and gluing different spacetimes. However, our observation is that even the noncommutative-geometry-inspired BTZ black hole is not free from any singularity.

084014-7

RAHAMAN et al.

PHYSICAL REVIEW D 87, 084014 (2013)

Besides this, in the present paper we continue our investigation with a discussion of Hawking temperature, entropy and heat capacity. We observe that the noncommutativity leads to the same minimal radius r0 at which the black hole cools down to absolute zero. A discussion of the geodesic structure leads to the effective potential for both massless and massive particles. It is shown that photons will fall into the black hole, while massive particles are trapped between the two horizons. The use of the Hamilton-Jacobi equation confirms that bound orbits are possible for test particles.

F. R. and S. R. are thankful to the authority of the InterUniversity Centre for Astronomy and Astrophysics, Pune, India for providing them Visiting Associateship, under which a part of this work was carried out. F. R. is also thankful to UGC, Government of India under its research award scheme, for providing financial support. We are very grateful to an anonymous referee for his/her insightful comments that have led to significant improvements, particularly on the interpretational aspects.

[1] A. Smailagic and E. Spalluci, J. Phys. A 36, L467 (2003). [2] P. Nicolini, A. Smailagic, and E. Spalluci, Phys. Lett. B 632, 547 (2006). [3] E. Spallucci, A. Smailagic, and P. Nicolini, Phys. Lett. B 670, 449 (2009). [4] P. Nicolini and E. Spalluci, Classical Quantum Gravity 27, 015010 (2010). [5] R. Garattini and F. R. S. Lobo, Phys. Lett. B 671, 146 (2009). [6] F. S. N. Lobo and R. Garattini, arXiv:1004.2520. [7] F. Rahaman, P. K. F. Kuhfittig, K. Chakraborty, A. A. Usmani, and S. Ray, Gen. Relativ. Gravit. 44, 905 (2012). [8] P. K. F. Kuhfittig, Adv. High Energy Phys. 2012, 462493 (2012). [9] F. Rahaman, P. K. F. Kuhfittig, S. Ray, and S. Islam, Phys. Rev. D 86, 106010 (2012). [10] I. Radinschi, F. Rahaman, and U. F. Mondal, Int. J. Theor. Phys. 52, 96 (2013). [11] S. Fernanado, Gen. Relativ. Gravit. 36, 71 (2004). [12] J. Zanelli, J. Phys. Conf. Ser. 222, 012008 (2010). [13] S. Nam and J.-D. Park, Classical Quantum Gravity 26, 145015 (2009). [14] F. Rahaman, A. A. Usmani, S. Ray, and S. Islam, Phys. Lett. B 717, 1 (2012). [15] S. G. Ghosh, Int. J. Mod. Phys. D 21, 1250022 (2012).

[16] S. H. Mazharimousavi, M. Halilsoy, and T. Tahamtan, Phys. Lett. A 376, 893 (2012). [17] J. C. Lo´pez-Domı´nguez, O. Obrego´n, M. Sabido, and C. Ramı´rez, Phys. Rev. D 74, 084024 (2006). [18] O. Obrego´n, M. Sabido, and E. Mena, Mod. Phys. Lett. A 24, 1907 (2009). [19] M. Rinaldi, Classical Quantum Gravity 28, 105022 (2011). [20] Y. S. Myung and M. Yoon, Eur. Phys. J. C 62, 405 (2009). [21] J. Liang and B. Liu, Europhys. Lett. 100, 30001 (2012). [22] J. M. Tejeiro and A. Larranaga, Pramana J. Phys. 78, 1 (2012). [23] A. Larranaga and J. M. Tejeiro, Abraham Zelmanov Journal 4, 28 (2011). [24] M. Ban˜ados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992). [25] J. J. Blome and W. Priester, Naturwissenschaften 71, 528 (1984). [26] P. C. W. Davies, Phys. Rev. D 30, 737 (1984). [27] C. Hogan, Nature (London) 310, 365 (1984). [28] N. Kaiser and A. Stebbins, Nature (London) 310, 391 (1984). [29] S. Fernando, D. Krug, and C. Curry, Gen. Relativ. Gravit. 35, 1243 (2003). [30] F. Rahaman and S. Chakraborty, Pramana J. Phys. 51, 689 (1998). [31] F. Rahaman, Int. J. Mod. Phys. D 09, 627 (2000).

ACKNOWLEDGMENTS

084014-8