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Nuclear Physics B 946 (2019) 114703 www.elsevier.com/locate/nuclphysb

GUP modified Hawking radiation in bumblebee

∗ Sara Kanzi, Izzet˙ Sakallı

Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey Received 3 May 2019; received in revised form 12 July 2019; accepted 18 July 2019 Available online 23 July 2019 Editor: Hubert Saleur

Abstract The effect of Lorentz symmetry breaking (LSB) on the Hawking radiation of Schwarzschild-like black hole found in the bumblebee gravity model (SBHBGM) is studied in the framework of quantum gravity. To 1 this end, we consider Hawking radiation spin-0 (bosons) and spin- 2 particles (fermions), which go in and out through the event horizon of the SBHBGM. We use the modified Klein-Gordon and Dirac equations, which are obtained from the generalized uncertainty principle (GUP) to show how Hawking radiation is affected by the GUP and LSB. In particular, we reveal that, independent of the spin of the emitted par- ticle, GUP causes a change in the Hawking temperature of the SBHBGM. Furthermore, we compute the semi-analytic greybody factors (for both bosons and fermions) of the SBHBGM. Thus, we reveal that LSB is effective on the greybody factor of the SBHBGM such that its redundancy decreases the value of the greybody factor. Our findings are graphically depicted. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

In spite of their overwhelming successes in describing nature, (GR) (i.e., detection of the gravitational waves [1,2] and observation of the shadow of the M87 supermassive black hole (BH) [3]) and Standard Model (SM) (i.e., detection of the Higgs boson [4]) of particle physics are incomplete theories. While Einstein’s theory of GR successfully describes gravity at a classical level, SM explains particles and the other three fundamental forces (electromagnetic,

* Corresponding author. E-mail address: [email protected] (I.˙ Sakallı). https://doi.org/10.1016/j.nuclphysb.2019.114703 0550-3213/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 and the strong and weak nuclear forces) at a quantum level. The unification of GR and SM is a fundamental quest, and this success will necessarily lead us to a deeper understanding of nature. In the search for this unification, some quantum gravity theories (QGTs) have been proposed, but direct tests of their features are beyond the energy scale of the currently available experiments. Because, they will be observed on the Planck scale which is around 1019 (GeV). However, it is possible that some signals of the QGT appear at sufficiently low energy scales and their effects can be observed in experiments on existing energy scales. One of these signals could be related to the LSB [5]. The theory of LSB has been under intense research since the proposed SM Extension (SME) [6–14], which is an effective field theory that includes the SM, GR, and every possible operator that breaks the Lorentz symmetry. With the SME, further investigations of the LSB can be made in the context of high energy particle physics, nuclear physics, gravitational physics, and astro- physics. The simplest models that contain a vector field which dynamically breaks the Lorentz symmetry are called bumblebee models [15–18]. These models, although owning a simpler form, have interesting features such as rotations, boosts, and CPT violations. In a bumblebee gravity model (BGM), potential V is included in the action SBM, which evokes a vacuum expectation value (VEV) for the vector field. The potential V is formed as a function of a scalar combination ℵ of the vector Bμ and the metric gμν (plus the other matter fields, if there are any). The poten- dV = tial has a minimum at dℵ 0. At the Vmin, the bumblebee field Bμ incorporates a vacuum value shown by Bμ = bμ, which is the so-called vacuum vector. In fact, the vacuum vector is nothing but a background vector that gives rise to local (spontaneous) LSB [19]. The scalar of the BGM, μ 2 in general, reads as ℵ = B Bμ ± b in which b is a constant having dimensions of mass (M). dV = ℵ = Thus, the Vmin satisfies the condition of dℵ 0for 0. Here, bμ is spontaneously induced as μ 2 a timelike vector abiding by bμb =−b . For instance, the aether models [20–23]are based on a vector field, which is in the Lagrangian density of the system with a non-vanishing VEV. The vector field dynamically selects a preferred frame at each point in the considered and spontaneously breaks the Lorentz invariance. This is a mechanism reminiscent of the breaking of local gauge symmetry described by the . In general, the subclass of aether models obeys the following action [24]: 1 1 S = d4x R + /cBμBνR − BμνB − V ℵ , (1) BM 16πG μν 4 μν where the parameter /c, having dimensions of M−2, denotes the coupling between the Ricci tensor μ (Rμν ) and B . Bμν is the bumblebee field strength:

Bμν =∇μBν −∇νBμ. (2) As mentioned above, V is the potential of the bumblebee field that drives the breaking of the Lorentz symmetry of the Lagrangian by collapsing onto a non-zero minimum at ℵ = 0or μ 2 BμB =∓b . In fact, Bμ is one of the Lorentz breaking coefficients and it shows a preferred direction in which the equivalence-principle is locally broken for a certain Lorentz frame. Ob- servations of Lorentz violation can emerge if the particles or fields interact with the bumblebee field [24]. It is worth noting that when a smooth quadratic potential is chosen as

V = Aℵ2, (3) where A is a dimensionless constant, one gets the Nambu-Goldstone excitations (massless bosons) besides the massive excitations [24]. Besides, the linear Lagrange-multiplier potential S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 3 is given by V = λℵ. These potentials (1) and (2) present also the breaking of the U(1) gauge in- variance and other implications to the behavior of the matter sector, the , and the . For a topical review (from experimental proposals to the test results) of the BGMs, the reader is referred to [17] and references therein. Furthermore, the studies using the bumblebee models have gained momentum for the last two decades. The vacuum solutions for the bumblebee field for purely radial, temporal-radial, and temporal-axial Lorentz symmetry breaking were obtained in [25]. New spherically static black hole (BH) [ 26] and traversable wormhole [27] solutions in the BGM have been recently discovered. Bluhm [28] discussed the Higgs mechanism in the BGM. The electrodynamics of the bumblebee fields was studied by [29]in which the bumble- bee field was considered as a photon field. Propagation velocity of the photon field, along with its possible effects on the accelerator physics and cosmic ray observations, was also investi- gated. BGMs are also used to limit the likelihood of Lorentz violation in astrophysical objects such as the Sun [30]. For other studies demonstrating the physical effects (quasinormal modes, thermodynamics, etc.) of the bumblebee field, the reader may refer to [29,31–34,22,35–38] and references therein. Hawking’s ground-breaking studies [39,40] can be considered as the onset of QGT [41,42]. Since then there have been numerous research papers on the subject of Hawking radiation (HR) in the literature (see, for instance, [43–75]). Several methods have been developed to calculate the HR of BHs [76–79]. In this study, we mainly focus on the quantum gravity effects on the HR of SBHBGM [26]in the tunneling paradigm. Although a number of QGTs have been proposed, however, physics literature does not as yet have a complete and consistent QGT. In the absence of a complete quantum description of the HR, we use effective models to describe the quantum gravitational behavior of the BH evaporation. In particular, , loop quantum gravity, and quantum geometry predict the minimal observable length on the Planck scale [80,81], which leads to the GUP [82–118]: h¯ xp ≥ 1 + β(p)2 , (4) 2 = α0 = hc¯ where β 2 in which Mp denotes the Planck mass and α0 is the dimensionless param- Mp G eter, which encodes the quantum gravity effects on the particle dynamics. The upper bound for 21 α0 was obtained as α0 < 10 [119]. Today, the effects of GUP on BHs have been extensively studied in the literature [120–122]. To amalgamate the GUP with the considered wave equation, the Wentzel-Kramers-Brillouin approximation [123]is generally used. Thus, one can obtain the quantum corrections to the HR of the BH [124,125]. Since the GUP and LSB effects are high energy modifications of the QGT, it is interesting to investigate their combined effects. To this end, we study the GUP-assisted HR of bosons (spin-0) 1 and fermions’ (spin- 2 ) tunneling [126,127] from the SBHBGM. Although the SBHBGM looks like the Schwarzschild BH, the differences in the Kretschmann scalars confirm that both BHs are physically different. The effects of spin and Lorentz-violating parameter L [128,129]on the quantum corrected HR are analyzed. We also study the problem of low energy greybody factors [130,131]for the bosons and fermions emitted by the SBHBGM. For this purpose, we imple- ment a method developed by Unruh [132,133]. It is also worth noting that Lorentz invariant massive gravity can be obtained dynamically from spontaneous symmetry breaking in a topolog- ical Poincare [134]. Besides, BH radiation in massive gravity (selecting a preferred direction of time) naturally corresponds to violations of the Lorentz symmetry [135–137]. The outline of the paper is as follows: In Sec. 2, we briefly introduce the SBHBGM and discuss some of its basic features. Section 3 is devoted to the computation of GUP-corrected 4 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703

HR of the bosons’ tunneling from the SBHBGM. In Sec. 4, we compute the quantum tunneling rate for the fermions of the SBHBGM using the GUP-modified Dirac equation and derive the modified HR. In the following section, we derive the greybody factor of the SBHBGM. In Sec. 6, we summarize our results. (Throughout the paper, we use geometrized units: c = G = 1.)

2. SBHBGM spacetime

The Lagrangian density of the BGM [138,139] yields the following extended vacuum Einstein equations 1 G = R − Rg = κT B ,(5) μν μν 2 μν μν B where Gμν and Tμν are the Einstein and bumblebee energy-momentum tensors, respectively. = B κ 8πGN is the gravitational coupling and Tμν is given by 1  ξ 1 T B =−B Bα − B Bαβ g − Vg + 2V B B + BαBβ R g − B BαR μν μα ν 4 αβ μν μν μ ν κ 2 αβ μν μ αν α 1 α 1 α 1 2 −BνB Rαμ + ∇α∇μ B Bν + ∇α∇ν B Bμ − ∇ BμBν 2 2 2 1 − g ∇ ∇ BαBβ , (6) 2 μν α β where ξ is the real coupling constant (having dimension M−1) that controls the non-minimal gravity-bumblebee interaction. From now on, the prime symbol shall denote the differentiation with respect to its argument. Meanwhile, there are other generic bumblebee models having non- zero torsion in the literature (see for instance [138]). In Eq. (6), the potential V ≡ V (ℵ) provides a non-vanishing VEV for Bμ. As it was stated above (see also [140,141]), the VEV of the bum- blebee field is determined when V = V  = 0. Taking the covariant divergence of the bumblebee Einstein equations (5) and using the contracted Bianchi identities, one gets ∇μ B = Tμν 0, (7) which gives the covariant conservation law for the bumblebee total energy-momentum tensor Tμν . Thus, Eq. (5) reduces to ξ ξ R = κT B + g ∇2 B Bα + g ∇ ∇ (BαBβ ).(8) μν μν 4 μν α 2 μν α β One can immediately see that when the bumblebee field Bμ vanishes, we recover the ordinary Einstein equations. Recently, the vacuum solution in the BGM induced by the LSB has been derived by Casana et al. [26]. The solution is obtained when the bumblebee field Bμremains frozen in its VEV bμ [25,27]. Namely, we have

Bμ = bμ ⇒ bμν ≡ ∂μbν − ∂νbμ.(9) Thus, the extended Einstein equations are found to be κ ξ R + κb bα + b bαβ g + ξb bαR + ξb bαR − bαbβ R g − μν μα ν 4 αβ μν μ αν ν αμ 2 αβ μν ξ ξ ξ ∇ ∇ bαb − ∇ ∇ bαb + ∇2 b b = 0. (10) 2 α μ ν 2 α ν μ 2 μ ν S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 5

Assuming a spacelike background for bμ as

bμ =[0,br (r), 0, 0], (11) μ 2 2 and using the condition b bμ = b =constant, LSB parameter (L) is defined as L = ξb ≥ 0 [138]. A spherically symmetric static vacuum solution to Eq. (10)is obtained as follows [26] − 2M 2M 1 ds2 =− 1 − dt2 + (1 + L) 1 − dr2 + r2 dθ2 + sin2 θdϕ2 , (12) r r which we call it SBHBGM solution. This BH solution represents a purely radial Lorentz- violation outside a spherical body characterizing a modified BH solution. In the limit L → 0 (b2 → 0), one can immediately see that the usual Schwarzschild metric is recovered. For the metric (12), the Kretschmann scalar becomes 4 12M2 + 4LMr + L2r2 K = , (13) r6 (1 + L)2 which is different than the Kretschmann scalar of the Schwarzschild BH. It means that none of the coordinate transformations link the metric (12)to the usual Schwarzschild BH. When r = 2M, Eq. (12) becomes finite: the coordinate singularity can be removed by applying a proper coordi- nate transformation. However, in the case of r = 0, physical singularity cannot be removed. So, we see that the behaviors of the physical (r = 0) and coordinate (r = rh = 2M : event horizon) singularities do not change in the BGM. The Hawking temperature of the metric (12) can be computed from Eq. (1), in which the surface gravity is given by [41]

μ ν κ =∇μχ ∇νχ , (14) where χμ is the timelike Killing vector field. Thus, the Hawking temperature of the SBHBGM (12) reads 1 dgtt 1 M 1 TH = √ = √ = √ . (15) 4π −g g dr + r2 + tt rr r=rh 2π 1 L r=rh 8πM 1 L One can easily see from Eq. (15) that the non-zero LSB parameter has the effect of reducing the Hawking temperature of a Schwarzschild BH.

3. GUP assisted HR of SBHBGM: bosons’ tunneling

The generic Klein-Gordon equation within the framework of GUP is given by [140] 2 t 2 μ 2 2 μ 2 −(ih)¯ ∂ ∂t  = (ih)¯ ∂ ∂μ + m 1 − 2β (ih)¯ ∂ ∂μ + m , (16) where β and m are the GUP parameter and mass of the scalar particle, respectively. Introducing the following ansatz for the wave function  i  = exp I(t,r,θ,ϕ) , (17) h¯ where I(t, r, θ, ϕ) is the classically forbidden action for quantum tunneling. Substituting Eq. (17), together with the metric functions of line-element (12), into Eq. (16), we get 6 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 −1 2 f 2 1 2 1 2 2 (f ) (∂t I) = (∂r I) + (∂θ I) + (∂ϕI) + m 1 + L r2 r2 sin2 θ h h f 2 1 2 1 2 2 × 1 − 2β (∂r I) + (∂θ I) + (∂ϕI) + m , (18) 1 + L r2 r2 sin2 θ where 2M f = 1 − . (19) r It is easy to see that SBHBGM (12) admits two Killing vectors <∂t , ∂ϕ >. The existence of these symmetries implies that we can assume a following separable solution for the action I =−ωt + R(r) + S(θ) + Jϕ, (20) where ω and J denote the energy and angular momentum of the radiated particle, respectively. Substituting Eq. (20)in Eq. (18), we obtain ω2 f 1 J 2 = (∂ R)2 + (∂ S)2 + + m2 + r 2 θ 2 f 1 L r sin θ 2 f 2 1 2 J 2 × 1 − 2β (∂r R) + (∂θ S) + + m . (21) 1 + L r2 sin2 θ We focus only on the radial trajectories in which only the (r − t) sector is considered. Thus, one can set 2 1 2 J (∂θ S) + = e, (22) r2 sin2 θ where e is a constant. So, Eq. (21) becomes f f ω2 (∂ R)2 + e + m2 1 − 2β (∂ R)2 + e + m2 = , (23) 1 + L r 1 + L r f which can be rewritten as a bi-quadratic equation as follows 4 2 a(∂r R) + b(∂r R) + c = 0, (24) where f 2 a =−2β , (25) (1 + L)2 f b = 1 − 4β m2 + e , (26) 1 + L ω2 c = e − 2βe2 − 4βem2 + m2 − − 2βm4. (27) f Eq. (24) has four roots if b2 − 4ac > 0. We deduced from our analytical computations that only two roots (R±) have physical meaning at the event horizon of the SBHBGM. These roots are 2 − 2 + 4 ω m f 2βm f 2 R± =± dr (1 + L) 1 + 2βm f 2 √ = iπωM 1 + L 1 + 2βm2 . (28) S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 7

It is worth noting that a +/− sign represents an outgoing/ingoing wave. On the other hand, the integrand of the integral (28) has a pole at r = rh. Evaluating the integral by using the Cauchy’s integral formula, we obtain the imaginary part of the action as √ 2 Im R± ≡ Im I± =±πωM 1 + L 1 + 2βm . (29)

Thus, the tunneling rate of the scalar particles becomes

P(emission) exp(−2ImI+)  = = = exp(−4ImI+) P(absorbtion) exp(−2ImI−) √ = exp −4ωMπ 1 + L 1 + 2βm2 . (30)

Recalling the expression of the Boltzmann factor ω  = exp(− ), (31) T one can read the modified Hawking temperature (TH ) as follows

1 TH TH = √ = . (32) 8πM 1 + L 1 + 2βm2 (1 + 2βm2) As can be seen above, after terminating the GUP parameter i.e., β = 0, one can recover the standard Hawking temperature (15).

4. GUP-assisted HR of SBHBGM: fermions’ tunneling

In this section, we aim to derive the modified Hawking temperature in the case of radiating fermions. To this end, we consider the Dirac equation, which is given by [141] 0 μ 2 2 j k i hγ¯ ∂0 + m + i hγ¯ μ + hβ∂¯ μ 1 − βm + β h¯ gjk ∂ ∂ ψ = 0, (33) where ψ denotes the test spinor field. The γ μ matrices for the metric (12)are given by i 0 0 σ 3 γ t = √ 1 γ r = f(r) f(r) 0 −i 1+L σ 3 0 , (34) 0 σ 1 0 σ 2 γ θ = 1 γ φ = 1 r σ 1 0 r sin θ σ 2 0 in which σ i ’s represent the well-known Pauli matrices [142]. One can easily ignore the terms having β2 since β is the effect of quantum gravity and it is a relatively very small quantity. For spin-up particles, the wave function can be expressed as [141] ⎛ ⎞ 0 ⎜ ⎟ ⎜ X ⎟ i  = ⎝ ⎠ exp I , (35) 0 h¯ Y where X, Y , and I are functions of coordinates (t,r,θ,φ). I is the action of the emitted fermion. It is worth noting that here we only consider the spin-up case since it is physically same with the spin-down case; the only difference is the sign. Substitution of the wave function in the generalized Dirac equation (33) results in the following coupled equations 8 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 ! 1 2 f − iX√ ∂t I − Y(1 − βm ) ∂r I f 1 + L f − Xmβ (∂ I)2 + gθθ (∂ I)2 + gϕϕ ∂ I 2 1 + L r θ ϕ ! f f + Yβ ∂ I (∂ I)2 + gθθ (∂ I)2 + gφφ ∂ I 2 + Xm 1 − βm2 = 0, 1 + L r 1 + L r θ ϕ (36) and ! 1 2 f iY √ ∂t I − X 1 − βm ∂r I f 1 + L f − Ymβ (∂ I)2 + gθθ (∂ I)2 + gϕϕ ∂ I 2 1 + L r θ ϕ ! f f + Xβ ∂ I (∂ I)2 + gθθ (∂ I)2 + gϕϕ ∂ I 2 + Ym 1 − βm2 = 0. 1 + L r 1 + L r θ ϕ (37) Then, one can get the following decoupled equations f X − 1 − βm2 gθθ∂ I + β gθθ∂ I (∂ I)2 + gθθ (∂ I)2 + gϕϕ ∂ I 2 θ θ 1 + L r θ ϕ " " f −i 1 − βm2 gϕϕ∂ I + iβ gϕϕ∂ I (∂ I)2 + gθθ (∂ I)2 + gϕϕ ∂ I 2 = 0, ϕ ϕ 1 + L r θ ϕ (38) and f Y − 1 − βm2 gθθ∂ I + β gθθ∂ I (∂ I)2 + gθθ(∂ I)2 + gφφ(∂ I)2 θ θ 1 + L r θ φ f −i 1 − βm2 gφφ∂ I + iβ gφφ∂ I (∂ I)2 + gθθ (∂ I)2 + gϕϕ ∂ I 2 = 0. φ φ 1 + L r θ φ (39)

∂ By using the fact that SBHBGM spacetime has a timelike Killing vector ∂t , one can obtain the radial action by performing the separation of variables technique:

I =−ωt + W(r)+ (θ, φ), (40) where ω is the fermion energy. Substituting Eq. (40)into Eqs. (38) and (39), we find out the identical equations for X and Y equations. Thus, we have f " β (∂ W)2 + gθθ(∂ )2 + gφφ(∂ )2 gθθ∂  + i gϕϕ∂  + 1 + L r θ φ θ φ " 2 θθ ϕϕ 1 − βm g ∂θ  + i g ∂ϕ = 0, (41) or S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 9 " θθ ϕϕ g ∂θ  + i g ∂ϕ f × β (∂ W)2 + gθθ(∂ )2 + gφφ(∂ )2 + m2 − 1 = 0. (42) 1 + L r θ φ It is obvious that the expression inside the square brackets can not vanish; thus, one should have " θθ ϕϕ g ∂θ  + i g ∂ϕ = 0, (43) and the solution of , therefore, does not contribute to the tunneling rate. The above result helps us to simplify Eqs. (36) and (37) [with ansatz (40)] as follows iω f X √ − mβ ∂ W 2 + m − βm2 + + ( r ) 1 f# 1 L $ ! ! f f f Y − 1 − βm2 ∂ W + β ∂ W (∂ W)2 = 0, (44) 1 + L r 1 + L r 1 + L r and iω f Y √ − mβ ∂ W 2 + m − βm2 + + ( r ) 1 f# 1 L $ ! ! f f f X − 1 − βm2 ∂ W + β ∂ W (∂ W)2 = 0. (45) 1 + L r 1 + L r 1 + L r In the simple way, one can set XA + YB = 0, (46) YA+ XB = 0, (47) where iω fβ 2 2 A = √ − m (∂r W) + 1 − βm , (48) f 1 + L and ! ! f f f B =− 1 − βm2 ∂ W + β ∂ W (∂ W)2 . (49) 1 + L r 1 + L r 1 + L r After making some manipulations, we see that A2 − B2 = 0 2 iω f 2 2 √ − mβ (∂r W) + m(1 − βm ) − f 1 + L % & ! ! 2 f f f − 1 − βm2 ∂ W + β ∂ W ∂2W = 0, (50) 1 + L r 1 + L r 1 + L r which yields 6 4 2 L6 (∂r W) + L4 (∂r W) + L2 (∂r W) + L0 = 0, (51) where 10 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703

f L = β2f( )3, (52) 6 1 + L f L = β( )2f m2β − 2 , (53) 4 1 + L 2 f 2iωm 2 2 L2 = √ + 1 − βm 1 + 2m β , (54) 1 + L f 2 " 2 2 2 2 L0 = ω − m f 1 − βm − 2iωm f 1 − βm , (55) ignoring O(β2) terms, Eq. (51) reduces to

4 2 L4 (∂r W) + L2 (∂r W) + L0 = 0. (56) Therefore, we have (1 + L) ω2 + m2f 2 2 ω W± =± dr 1 + β m + f f ∼ 2 = ±iπωr+ 1 + 2βm (57) √ =±i2πMω 1 + L 1 + 2βm2 , in another form √ 2 Im W± = 2πMω 1 + L 1 + 2βm . (58)

Recalling Eq. (30), we find the tunneling rate of fermions as follows √ 2  exp (−4ImW+) = exp 8πMω 1 + L 1 + 2βm . (59)

Thus, with the help of the Boltzmann factor (31), we get the GUP-consolidated temperature of the SBHBGM via the emission of the fermions: 1 T T = √ = 0 , (60) 8πM 1 + L 1 + 2βm2 1 + 2βm2 in which T0 represents the original Hawking temperature (15) 1 T0 = √ . (61) 8πM 1 + L The above result (60)shows that GUP corrected temperature deviates from the standard Hawking temperature.

5. Greybody factors of SBHBGM

In this section, we shall first derive the effective potentials of the scalar and fermion perturba- tions in the geometry of the SBHBGM. Then, the obtained effective potentials will be used for computing the greybody factors of the SBHBGM. The results will be depicted with some plots and discussed. S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 11

5.1. Scalar perturbations of SBHBGM

The massless Klein- Gordon equation is given by

 = 0, (62) √  = √1 − μν where the D’Alembert operator is denoted by the box symbol and −g ∂μ( gg ∂ν). For the SBHBGM (12), we have √ √ −g = r2 sin θ 1 + L, (63) and therefore Eq. (62) reads 1 1  = ∂2 − 2rf ∂  + r2∂ f∂r+ r2f∂2 + f t r2(1 + L) r r r

1 2 1 2 (− cos θ∂θ  − sin θ∂ ) − ∂ . r2 sin θ θ r2 sin2 θ φ We invoke the following ansatz for the scalar field  in the above equation: −  = p(r)A(θ)e iωteimφ, (64) so that we have 2 ω 1 2f      =− − p + f p + fp − f p(1 + L) r 2 1   m cos θA + sin θA − A = 0. r2A sin θ sin θ When one changes the independent variable θ to cos−1 z, the angular equation is found to be   λ 1 − z2 A + 2zA − m2 + s 1 − z2 A = 0, (65) 1 + L where λs denotes the eigenvalue. The above equation is nothing but the Legendre differential equation when one sets

λs =−l(l + 1)(1 + L). (66) The radial equation then becomes    f 2 1 + L λ p + p + + ω2 + s p = 0. (67) f r f 2 r2f = u Introducing a new variable p r , we get a Schrödinger-like wave equation 2 du + 2 − = 2 ω Veff u 0, (68) dr∗ where r∗ is the tortoise coordinate defined by √ dr r∗ = 1 + L . (69) f 12 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703

r∗ Fig. 1. Veff versus M graph. The plots are governed by Eq. (70).

The effective potential felt by the scalar field then becomes f  l(l + 1) V = f + . (70) eff (1 + L) r r2 It is obvious from Fig. (1) that the effective potential vanishes both at the event horizon of the SBHBGT and at spatial infinity. This behavior will help us to analytically derive the greybody factor of the scalar field emission from the SBHBGT .

5.2. Fermion perturbations of SBHBGM

In this subsection, we shall employ the Newman-Penrose formalism [143]to find the effective potential of the fermion fields propagating in the geometry of the SBHBGM. Chandrasekar-Dirac equations (CDEs) are given by [144] ∗ (D + ε − ρ) F1 + (δ + π − α)F2 = iμ G1, ∗ (δ + β − τ) F1 + ( + μ − γ ) F2 = iμ G2, ∗ (D + ε − ρ) G − (δ + π − α) = iμ F , 2 2 ∗ ( + μ − γ ) G1 − δ + β − τ G2 = iμ F1, (71) where F1, F2, G1, and G2 represent the components of the wave functions or the so-called Dirac spinors. ε, ρ, π, α, β, τ , μ, and γ are the spin coefficients, and a bar over a quantity denotes complex conjugation. The non-zero spin coefficients are found to be √ √ √ 2f  2f 2 cot θ ε = γ = √ √ ,μ= ρ =− √ ,β=−α =− . (72) 8 f 1 + L 2r 1 + L 4r To have separable solutions for the CDEs (71), we introduce the following ansatzes

F1 = f1(z)A1(θ) exp [i(ωt + mφ)] ,

G1 = g1(z)A2(θ) exp [(ωt + mφ)] , S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 13

F2 = f2(z)A3(θ) exp [i(ωt + mφ)] ,

G2 = g2(z)A4(θ) exp [(ωt + mφ)] , (73) where m denotes the azimuthal number and ω is the frequency of the spinor fields. Since the a a a directional derivatives [144]are defined by D =  ∂a,  = n ∂a, and δ = m ∂a, we have 1 f D = √ ∂t + ∂r , 2f 2 (1 + L) 1 f  = √ ∂t − ∂r , 2f 2(1 + L) 1 i δ = √ ∂θ + √ ∂ϕ, r 2 r 2sinθ 1 i δ = √ ∂θ − √ ∂ϕ. (74) r 2 r 2sinθ After substituting Eqs. (72-74) into the CDEs (71), one can obtain the following set of equa- tions:  √  √ iω r f rf f f1 LA3 g1A2 √ + √ ∂r + √ + √ + − iμr = 0, f 1 + L 4 f (1 + L) 1 + L f2 A1 f2A1 √ √   † iω r f rf f f2 L A1 g2A4 √ − √ ∂r − √ − √ + − iμr = 0, f 1 + L 4 f (1 + L) 1 + L f1 A3 f1A3 √ √   † iω r f rf f g2 L A2 f2A3 √ + √ ∂r + √ + √ − − iμr = 0, f 1 + L 4 f (1 + L) 1 + L g1 A4 g1A4  √  √ iω r f rf f g1 LA4 f1A1 √ − √ ∂r − √ − √ − − iμr = 0, (75) f 1 + L 4 f (1 + L) 1 + L g2 A2 g2A2 √   where μ = 2μ∗. L and L† are the angular operators, which are known as the laddering opera- tors:   m cot θ m cot θ L = ∂ + + , L† = ∂ − + , (76) θ sin θ 2 θ sin θ 2 which lead to the spin-weighted spheroidal harmonics [145,146] with the following eigenvalue [147,148]: 1 λ =− l + . (77) f 2

By considering g1 = f2, g2 = f1, A2 = A1, and A4 = A3, then we reduce the CDEs (75)to two coupled differential equations: √ % √ & r f d iω 1 + L f  1 √ + + + g2 = −λf + iμr g1, (78) 1 + L dr f 4f r √ % √ & r f d iω 1 + L f  1 √ − + + g1 = −λf − iμr g2. (79) 1 + L dr f 4f r 14 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703

r∗ Fig. 2. Plots of V± versus M . The plots are governed by Eq. (89).

Moreover, if one sets   g (r) = 1 and g (r) = 2 , (80) 1 r 2 r and substitute them into Eqs. (78) and (79), after some manipulations, we get: % √ & √  r f d iω 1 + L f λf √ + + 2 = − + iμ 1, (81) 1 + L dr f 4f r % √ & √  r f d iω 1 + L f λf √ − + 1 = − − iμ 2. (82) 1 + L dr f 4f r

− 1 − 1 By defining 1 = f 4 R1(r) and 2 = f 4 R2(r) and introducing the tortoise coordinate f d d (r∗) as √ = , we obtain 1+L dr dr∗ " d λf + iω R2(r) = f − + iμ R1, (83) dr∗ r " d λf − iω R1(r) = f − − iμ R2. (84) dr∗ r One can combine the above equations by letting

Z+ = R1 + R2, (85)

Z− = R2 − R1. (86) Thus, we end up with the following pair of one dimensional Schrödinger-like wave equations: 2 d 2 + ± = ± ± 2 ω Z V Z , (87) dr∗ where the effective potentials for the Dirac field read S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 15 √ √ 2 λf 1 d f iμ f V± = f − ± iμ ± λf √ − ± , (88) r 1 + L dr r λf √ √ 2l + 1 2 1 1 d f 2iμ f = f ± iμ ∓ l + √ − ∓ . (89) 2r 2 1 + L dr r 2l + 1

Behaviors of V± (89)are depicted in Fig. 2.

5.3. Greybody factor computations

In general relativity, the greybody factor is one of the most important physical quantities related to the quantum nature of a BH. A high value of the greybody factor indicates a high probability that HR can reach to spatial infinity. Among the many methods (see for example [42] and references therein) for obtaining the greybody factor, here we employ the method of [149], which formulates the general semi-analytic bounds for greybody factors: ⎛ ⎞ +∞ 2 ⎝ ⎠ σ(ω) ≥ sec h ℘dr , (90) −∞ where σ(ω) are the dimensionless greybody factors that depend on the angular momentum quan- tum number  and frequency ω of the emitted particles, and  2 2 2 2 (h ) + (ω − Veff − h ) ℘ = , (91) 2h in which prime denotes the derivation with respect to r. We have two conditions for the certain positive function h: 1) h(r) > 0 and 2) h(−∞) = h(∞) = ω [149]. Without loss of generality, we simply set h = ω, which reduces the integration of Eq. (90)to +∞ √ +∞ 1 + L Veff ℘dr = dr. (92)  2ω f(r) −∞ rh For a massless scalar field φ, considering the effective potential given in Eq. (70), Eq. (90) becomes ⎛ ⎞ √ +∞ 1 + L f (r) l(l + 1) σ s(ω) ≥ sec h2 ⎝ + dr⎠ . (93)  2ω (1 + L) r r2 rh Taking cognizance of the integral part of Eq. (93): +∞ 1 l(l + 1) f (r) f(r)dr + f(r)dr , 2ω r2  r (1 + L)  −∞ ⎡ ⎤ √ +∞ ∞ 1 + L dr dr 2M = ⎣l(l + 1) + ⎦ , 2ω r2 r (1 + L) r2 r r √ h h 1 + L 1 = l(l + 1) + , (94) 2ωrh 2 (1 + L) 16 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 the greybody factor of the SBHBGM due to scalar field radiation yields #√ $ + s ≥ 2 1 L + + 1 σ (ω) sec h l(l 1) . (95) 2ωrh 2 (1 + L) When one considers the effective potential (88)of the massless Dirac fields: 2 √ λf 1 d f V±| = = f ± λf √ − , (96) μ 0 r2 1 + L dr r the integral seen in Eq. (90) can be easily computed. Thus, we find the greybody factor expression of the SBHBGM arising from the fermion radiation: ⎛ ⎞ +∞ √ λ2 f 2 ⎝ 1 f 1 d f ⎠ σ (ω) ≥ sec h f ± λf √ − dr . (97)  2ω r2 1 + L dr r −∞

From now on, without loss of generality, we consider only V+. After some manipulation, one can get ⎡ ⎛ ⎞⎤ √ ∞ ∞ 1 + L 1 λf M 1 3M σ f (ω) ≥ sec h2 ⎣ ⎝λ2 dr ± √ 1 + − dr⎠⎦ ,  2ω f r2 1 + L r r2 r3 rh rh (98) which recasts in √ % & 2 1 + L 1 λf 1 M M σ f (ω) ≥ sec h2 λ2 ± √ − − , (99)  f + 2 3 2ω rh 1 L rh rh rh or, in more compact form: √ l + 1 1 + L 1 1 σ f (ω) ≥ sec h2 2 l + ± √ . (100)  4Mω 2 4 1 + L We depict the greybody factors of the SBHBGM arising from the scalar (95) and fermion (100) fields in Figs. 3 and 4, respectively. As is well-known, the greybody factor of the HR must be < 1 since a BH does not perform a complete black body radiation with a 100% absorption coefficient. Our findings, as shown in Figs. 3 and 4, are in good agreement with the latter remark. Also, it can be seen from these figures that the peak values of the greybody factors decreases with increasing LSB parameter L. In summary, LSB has a greybody factor-reducing effect.

6. Conclusion

In this paper, we studied the quantum thermodynamics [150]of the SBHBGM. During this analysis, we had mainly two aims: 1) to obtain the modified Hawking temperature of the SB- HBGM, within the framework of GUP, arising from the emission of bosons and fermions; 2) to compute the greybody factors of the scalar and fermion fields from the SBHBGM. To this end, we first derived the effective potentials of the Klein-Gordon and Dirac equations. Next, we used the obtained effective potentials in the greybody expression (90). Then, we illustrated the S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 17

s = Fig. 3. σ (ω) versus ω graph. The plots are governed by Eq. (95) with M 1.

f Fig. 4. Plots of σ (ω) versus ω. The plots are governed by Eq. (100). obtained greybody factors in Figs. 3 and 4. It was clear from those figures that as LSB effect (L) increases, the greybody factor decreases: low values of the greybody factor indicate a low prob- ability that HR can reach spatial infinity. In the future, the latter observation might shed light on the LSB effects from both terrestrial experiments and astrophysical observations. Any discovery of the LSB would be an important signal beyond the SM physics [151]. In future work, we plan to extend the GUP and greybody factor analysis [152]to the various BHs in gravity’s rainbow, which is also a result of quantum gravity [153–161]. The deformation of a spacetime owing to the rainbow gravity effect leads to Lorentz violations [162,163]. In this 18 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 way, we hope to achieve new results that will help us to understand the QGT and its effect on the LSB.

Acknowledgements

The authors are grateful to the editor and anonymous referees for their valuable comments and suggestions to improve the paper.

References

[1] B.P. Abbott, et al., LIGO Scientific and Virgo Collaborations, Phys. Rev. Lett. 116 (2016) 061102. .[2] B.P Abbott, et al., LIGO Scientific and Virgo Collaborations, Phys. Rev. Lett. 116 (2016) 241103. [3] K. Akiyama, et al., Event Horizon Telescope Collaboration, Astrophys. J. 875 (2019) L1. [4] G. Aad, et al., ATLAS Collaboration, Science 338 (2012) 1576. [5] Y. Nomura, J. Varela, S.J. Weinberg, Phys. Lett. B 733 (2014) 126. [6] D. Colladay, V.A. Kostelecky, Phys. Rev. D 55 (1997) 6760. [7] D. Colladay, V.A. Kostelecky, Phys. Rev. D 58 (1998) 116002. [8] S.R. Coleman, S.L. Glashow, Phys. Rev. D 59 (1999) 116008. [9] V.A. Kostelecky, S. Samuel, Phys. Rev. Lett. 63 (1989) 224. [10] V.A. Kostelecky, S. Samuel, Phys. Rev. D 39 (1989) 683. [11] V.A. Kostelecky, S. Samuel, Phys. Rev. Lett. 66 (1991) 1811. [12] V.A. Kostelecky, R. Potting, Nucl. Phys. B 359 (1991) 545. [13] V.A. Kostelecky, R. Potting, Phys. Lett. B 381 (1996) 89. [14] V.A. Kostelecky, R. Potting, Phys. Rev. D 51 (1995) 3923. [15] V.A. Kostelecky, S. Samuel, Phys. Rev. D 40 (1989) 1886. [16] V.A. Kostelecky, R. Lehnert, Phys. Rev. D 63 (2001) 065008. [17] Q.G. Bailey, V.A. Kostelecky, Phys. Rev. D 74 (2006) 045001. [18] R. Bluhm, Int. J. Mod. Phys. D 16 (2008) 2357. [19] R. Lehnert, Hyperfine Interact. 193 (2009) 275. [20] T. Jacobson, D. Mattingly, Phys. Rev. D 64 (2001) 024028. [21] T. Jacobson, PoS QG PH (2007) 020. [22] D. Capelo, J. Páramos, Phys. Rev. D 91 (2015) 104007. [23] A. Coley, G. Leon, arXiv :1905 .02003. [24] V.A. Kostelecky, Phys. Rev. D 69 (2004) 105009. [25] O. Bertolami, J. Paramos, Phys. Rev. D 72 (2005) 044001. [26] R. Casana, A. Cavalcante, F.P. Poulis, E.B. Santos, Phys. Rev. D 97 (2018) 104001. [27] A. Övgün, K. Jusufi, I. Sakalli, Phys. Rev. D 99 (2019) 024042. [28] R. Bluhm, PoS QG PH (2007) 009, arXiv :0801 .0141. [29] M.D. Seifert, Phys. Rev. D 81 (2010) 065010. .[30] A.V Kostelecky, J.D. Tasson, Phys. Rev. D 83 (2011) 016013. [31] J. Paramos, G. Guiomar, Phys. Rev. D 90 (2014) 082002. [32] C.A. Hernaski, Phys. Rev. D 90 (2014) 124036. [33]. R.V Maluf, J.E.G. Silva, C.A.S. Almeida, Phys. Lett. B 749 (2015) 304. [34]. A.F Santos, A.Yu. Petrov, W.D.R. Jesus, J.R. Nascimento, Mod. Phys. Lett. A 30 (2015) 1550011. [35] C.A. Escobar, Phys. Rev. D 95 (2017) 0095006. [36] A. Övgün, K. Jusufi, I. Sakalli, Ann. Phys. 399 (2018) 193. [37] D.A. Gomes, R.V. Maluf, C.A.S. Almeida, arXiv :1811 .08503. [38] R. Oliveria, D.M. Dautas, et al., Class. Quantum Gravity 36 (2019) 105013. [39]. S.W Hawking, Nature 248 (1974) 30. .[40] S.W Hawking, Commun. Math. Phys. 43 (1975) 199, Erratum: Commun. Math. Phys. 46 (1976) 206. [41] R.M. Wald, General Relativity, The University of Chicago Press, Chicago and London, 1984. [42] R.M. Wald, The thermodynamics of black holes, in: A. Gomberoff, D. Marolf (Eds.), Lectures on Quantum Grav- ity. Series of the Centro de Estudios Científicos, Springer, Boston, MA, 2005. [43] M.K. Parikh, F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042. S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 19

[44] P. Kanti, Int. J. Mod. Phys. A 19 (2004) 4018324. [45] C. Barcelo, S. Liberati, M. Visser, Living Rev. Relativ. 8 (2005) 12. [46] H. Pasaoglu, I. Sakalli, Int. J. Theor. Phys. 48 (2009) 3517. [47] R.G. Cai, L.M. Cao, Y.P. Hu, Class. Quantum Gravity 26 (2009) 155018. [48] S.H. Mazharimousavi, I. Sakalli, M. Halilsoy, Phys. Lett. B 672 (2009) 177. [49] I. Sakalli, K. Jusufi, A. Övgün, Gen. Relativ. Gravit. 50 (2018) 125. [50] P.A. Gonzalez, A. Övgün, J. Saavedra, Y. Vasquez, Gen. Relativ. Gravit. 50 (2018) 62. [51] K. Jusufi, I. Sakalli, A. Övgün, Gen. Relativ. Gravit. 50 (2018) 10. [52] K. Jusufi, A. Övgün, G. Apostolovska, Adv. High Energy Phys. 2017 (2017) 8798657. [53] G. Gecim, Y. Sucu, Adv. High Energy Phys. 2018 (2018) 8728564. [54] G. Gecim, Y. Sucu, Adv. High Energy Phys. 2018 (2018) 7031767. [55] A. Övgün, K. Jusufi, Eur. Phys. J. Plus 131 (2016) 177. [56] A. Övgün, Int. J. Theor. Phys. 55 (2016) 2919. [57] A. Övgün, Adv. High Energy Phys. 2017 (2017) 1573904. [58] X.M. Kuang, J. Saavedra, A. Övgün, Eur. Phys. J. C 77 (2017) 613. [59] A. Övgün, I. Sakalli, Int. J. Theor. Phys. 57 (2018) 322. [60] K. Jusufi, A. Övgün, Int. J. Theor. Phys. 56 (2017) 1725. [61] K. Jusufi, A. Övgün, Astrophys. Space Sci. 361 (2016) 207. [62] W. Javed, R. Babar, A. Övgün, Mod. Phys. Lett. A 34 (2019) 1950057. [63] I. Sakalli, A. Övgün, Europhys. Lett. 118 (2017) 60006. [64] I. Sakalli, A. Övgün, Eur. Phys. J. Plus 131 (2016) 184. [65] I. Sakalli, A. Övgün, J. Astrophys. Astron. 37 (2016) 21. [66] I. Sakalli, A. Övgün, Gen. Relativ. Gravit. 48 (2016) 1. [67] I. Sakalli, A. Övgün, Astrophys. Space Sci. 359 (2015) 32. [68] I. Sakalli, A. Övgün, Eur. Phys. J. Plus 130 (2015) 110. [69] I. Sakalli, A. Övgün, J. Exp. Theor. Phys. 121 (2015) 404. [70] I. Sakalli, A. Övgün, Europhys. Lett. 110 (2015) 10008. [71] I. Sakalli, A. Övgün, S.F. Mirekhtiary, Int. J. Geom. Methods Mod. Phys. 11 (2014) 1450074. [72] I. Sakalli, M. Halilsoy, H. Pasaoglu, Int. J. Theor. Phys. 50 (2011) 3212. [73] I. Sakalli, M. Halilsoy, H. Pasaoglu, Astrophys. Space Sci. 340 (2012) 155. [74] I. Sakalli, Astrophys. Space Sci. 340 (2012) 317. [75]. S.F Mirekhtiary, I. Sakalli, Theor. Mat. Fiz. 198 (2019) 523. [76] L. Vanzo, G. Acquaviva, R. Di Criscienzo, Class. Quantum Gravity 28 (2011) 183001. [77]. E.T Akhmedov, V. Akhmedova, D. Singleton, Phys. Lett. B 642 (2006) 124. [78] K. Srinivasan, T. Padmanabhan, Phys. Rev. D 60 (1999) 024007. [79] S. Hemming, E. Keski-Vakkuri, Phys. Rev. D 64 (2001) 044006. [80] G. Amelino-Camelia, M. Arzano, G. Mandanici, Class. Quantum Gravity 23 (2006) 0264. [81] K. Nouicer, Phys. Lett. B 646 (2007) 63. [82] R.J. Adler, P.S. Chen, D.I. Santiago, Gen. Relativ. Gravit. 33 (2001) 1430411. [83] G. Amelino-Camelia, M. Arzano, Y. Ling, G. Mandanici, Class. Quantum Gravity 23 (2006) 2585. [84] M. Cavaglia, S. Das, R. Maartens, Class. Quantum Gravity 20 (2003) L205. [85] M. Faizal, M.M. Khalil, Int. J. Mod. Phys. A 30 (2015) 1550144. [86] S. Gangopadhyay, A. Dutta, M. Faizal, Europhys. Lett. 112 (2015) 20006. [87] M. Salah, F. Hammad, M. Faizal, A.F. Ali, J. Cosmol. Astropart. Phys. 02 (2017) 035. [88] A. Iorio, P. Pais, I.A. Elmashad, A.F. Ali, M. Faizal, L.I. Abou-Salem, Int. J. Mod. Phys. D 27 (2018) 1850080. [89] Q. Zhao, M. Faizal, Z. Zaz, Phys. Lett. B 770 (2017) 564. [90] A. Bhat, S. Dey, M. Faizal, C. Hou, Q. Zhao, Phys. Lett. B 770 (2017) 325. [91] M. Khodadi, K. Nozari, S. Dey, A. Bhat, M. Faizal, Sci. Rep. 8 (2018) 1659. [92] M.C. Braidotti, et al., Europhys. Lett. 124 (2018) 44001. [93] M. Faizal, T.S. Tsun, Nucl. Phys. B 924 (2017) 588. [94] M. Faizal, A.F. Ali, A. Nassar, Phys. Lett. B 765 (2017) 238. [95] M. Faizal, B.P. Mandal, Gravit. Cosmol. 21 (4) (2015) 270. [96]. A.F Ali, M. Faizal, M.M. Khalil, J. Cosmol. Astropart. Phys. 09 (2015) 025. [97] A. Mohammadi, A.F. Ali, T. Golanbari, A. Aghamohammadi, K. Saaidi, M. Faizal, Ann. Phys. 385 (2017) 214. [98] M. Faizal, S.L. Kruglov, Int. J. Mod. Phys. D 25 (2016) 1650013. [99] Q. Zhao, M. Faizal, M.B. Shah, A. Bhat, P.A. Ganai, Z. Zaz, S. Masood, J. Raza, R.M. Irfan, Eur. Phys. J. C 77 (2017) 612. 20 S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703

[100] S.R. Garattini, M. Faizal, Nucl. Phys. B 905 (2017) 313. [101] M. Faizal, B. Majumder, Ann. Phys. 357 (2015) 49. [102] S. Pramanik, M. Faizal, M. Moussa, A.F. Ali, Ann. Phys. 362 (2015) 24. [103] M. Faizal, Int. J. Mod. Phys. A 29 (2014) 1450106. [104] M. Faizal, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550022. [105] M. Faizal, Phys. Lett. B 757 (2016) 244. [106] M. Faizal, M.M. Khalil, S. Das, Eur. Phys. J. C 76 (2016) 30. [107] E.C. Vagenas, A.F. Ali, H. Alshal, Eur. Phys. J. C 79 (2019) 276. [108] H. Hinrichsen, A. Kempf, J. Math. Phys. 37 (1996) 2121. [109] J.R. Mureika, Phys. Lett. B 789 (2019) 88. [110] G. Gecim, Y. Sucu, Gen. Relativ. Gravit. 50 (2018) 152. [111] S. Deb, S. Das, E.C. Vagenas, Phys. Lett. B 755 (2016) 17. [112] P. Pedram, M. Amirfakhrian, H. Shababi, Int. J. Mod. Phys. D 24 (2015) 1550016. L.[113] T. Antonacci Oakes, R.O. Francisco, J.C. Fabris, J.A. Nogueira, Eur. Phys. J. C 73 (2013) 2495. [114] L. Menculini, O. Panella, P. Roy, Phys. Rev. D 87 (2013) 065017. [115] K. Nozari, M. Karami, Mod. Phys. Lett. A 20 (2005) 3095. [116] Y. Gim, H. Um, W. Kim, Phys. Lett. B 784 (2018) 206. [117] M. Khodadi, K. Nozari, A. Hajizadeh, Phys. Lett. B 770 (2017) 556. [118] L.B. Castro, A.E. Obispo, J. Phys. A, Math. Theor. 50 (2017) 285202. [119] S. Hossenfelder, Living Rev. Relativ. 16 (2013) 2. [120] A. Tawfik, A. Diab, Int. J. Mod. Phys. D 23 (2014) 1430025. [121] K. Nozari, T. Azizi, Gen. Relativ. Gravit. 38 (2006) 0262. [122] S. Gangopadhyay, A. Dutta, A. Saha, Gen. Relativ. Gravit. 46 (2014) 1661. [123] T. Padmanabhan, Class. Quantum Gravity 19 (2002) 5387. [124] R.A. Konoplya, Phys. Rev. D 68 (2003) 024018. .[125] A.F Ali, Phys. Rev. D 89 (2014) 104040. [126]. R.V Maluf, C.A.S. Almeida, R. Casana, M.M. Ferreira, Phys. Rev. D 90 (2014) 025007. [127] R. Bluhm, V.A. Kostelecky, Phys. Rev. D 71 (2005) 065008. [128] A. Zee, Phys. Rev. D 25 (1982) 1864. [129] S.M. Carroll, G.B. Field, R. Jackiw, Phys. Rev. D 41 (1990) 1231. [130] J. Maldacena, A. Strominger, Phys. Rev. D 55 (1997) 861. [131] I.R. Klebanov, M. Krasnitz, Phys. Rev. D 55 (1997) R3250. [132] L.C.B. Crispino, A. Higuchi, G.E.A. Matsas, Rev. Mod. Phys. 80 (2008) 787. [133] I. Fuentes-Schuller, R.B. Mann, Phys. Rev. Lett. 95 (2005) 120404. [134] M. Torabian, Phys. Lett. B 780 (2018) 81. [135] I. Arraut, Europhys. Lett. 109 (2015) 10002. [136] I. Arraut, Eur. Phys. J. C 77 (2017) 501. [137] I. Arraut, Universe 4 (2018) 27. [138] R. Bluhm, S.H. Fung, V.A. Kostelecky, Phys. Rev. D 77 (2008) 065020. [139] R. Bluhm, et al., Phys. Rev. D 77 (2008) 125007. [140] P. Wang, H. Yang, S. Ying, arXiv :1410 .5065. [141] M.A. Rahman, M.I. Hossain, Phys. Lett. B 712 (2012) 1. [142]. C.P Poolr, H.A. Farach, Found. Phys. 12 (1982) 719. [143] S. Jogia, J.B. Griffiths, Gen. Relativ. Gravit. 12 (1980) 597. [144] S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, Oxford, UK, 1983. [145] E. Newman, R. Penrose, J. Math. Phys. 7 (1966) 863. [146] J.N. Goldberg, A.J. Macfarlane, E. Newman, F. Rohrlich, E.C.G. Sudarshan, J. Math. Phys. 8 (1967) 2155. [147] S.A. Teukolsky, Astrophys. J. 185 (1973) 635. [148] C.H. Lee, J. Math. Phys. 17 (1976) 1226. [149] Y.G. Miao, Z.M. Xu, Phys. Lett. B 772 (2017) 542. [150] J. Anders, M. Esposito, New J. Phys. 19 (2017) 10201. [151] D. Mattingly, Living Rev. Relativ. 8 (2005) 5. [152] F. Gray, M. Visser, Universe 4 (2018) 93. [153]. A.F Ali, M. Faizal, M.M. Khalil, Phys. Lett. B 743 (2015) 295. [154] A. Ashour, M. Faizal, A.F. Ali, F. Hammad, Eur. Phys. J. C 76 (2016) 264. [155] Y. Heydarzade, P. Rudra, F. Darabi, A.F. Ali, M. Faizal, Phys. Lett. B 774 (2017) 46. S. Kanzi, I.˙ Sakallı / Nuclear Physics B 946 (2019) 114703 21

[156] S. Chougule, S. Dey, B. Pourhassan, M. Faizal, Eur. Phys. J. C 78 (2018) 685. [157]. A.F Ali, M. Faizal, B. Majumder, Europhys. Lett. 109 (2015) 20001. [158] S.H. Hendi, A. Dehghani, M. Faizal, Nucl. Phys. B 914 (2017) 117. [159]. A.F Ali, M. Faizal, M.M. Khalil, Nucl. Phys. B 894 (2015) 341. [160] S.H. Hendi, M. Faizal, Phys. Rev. D 92 (2015) 044027. [161] S.H. Hendi, M. Faizal, B.E. Panah, S. Panahiyan, Eur. Phys. J. C 76 (2016) 296. [162] N.A. Nilsson, M.P. Dabrowski, Phys. Dark Universe 18 (2017) 115. [163] S.H. Hendi, S. Panahiyan, B. Eslam Panah, M. Momennia, Eur. Phys. J. C 76 (2016) 150.