Greybody Radiation and Quasinormal Modes of Kerr-like Black Hole in Bumblebee Gravity Model
HHEP/123-qed
Greybody Radiation and Quasinormal Modes of Kerr-like BlackHole in Bumblebee Gravity Model
Sara Kanzi and ˙Izzet Sakallı Physics Department, Eastern Mediterranean University,Famagusta, North Cyprus via Mersin 10, Turkey. (Dated: February 15, 2021; Received)
Abstract
In the framework of the Lorentz symmetry breaking (LSB), we investigate the quasinormal modes(QNMs) and the greybody factors (GFs) of the Kerr-like black hole spacetime obtained from thebumblebee gravity model. In particular, we analyze the scalar and fermionic perturbations ofthe black hole within the framework of both semi-analytic WKB method and the time domainapproach. The impacts of the LSB on the bosonic/fermionic QNMs and GFs of the Kerr-like blackhole are investigated in detail. The obtained results are graphically depicted and discussed.
PACS numbers:Keywords: Hawking Radiation, Lorentz Symmetry Breaking, Bumblebee Gravity Model, Greybody factor,Quasinormal Modes,Exact Solution, Klein-Gordon Equation a r X i v : . [ h e p - t h ] F e b ontents I. Introduction II. Kerr-like Black Hole Spacetime of BGM III. Scalar Perturbations IV. Fermionic Perturbations V. Greybody radiation in Kerr spacetime of BGM
VI. Effect of LSB Parameter on QNMs
VII. Conclusion References I. INTRODUCTION
One of the cornerstones of modern physics is the Lorentz invariance, which is a funda-mental part of both general relativity (GR) and the standard model of particle physics.However, today there are several theories that Lorentz invariance can not be valid at allenergies [1]. Lorentz invariance violation (LIV) may yield a glimpse of quantum gravity(QG). Although from the theoretical point of view the exploration of this possibility hasbeen active for many years [see for instance [2], and references therein], the phenomenologyof LIV has been developed only within the last decade [3–5]. Before the mid-1990s, therewere only few works about the experimental consequences of LIV [6–8], because the neweffects were expected only in the particle interactions, which occur at energies of Planckmass: M Pl ≡ (cid:112) (cid:126) c/G N (cid:39) . × GeV /c . Then, it was realized that there exists specialcases in which the new effects could appear also at lower energies. Those particular cases2ave opened ”windows on the QG”. Nowadays, this subject has been investigating in variousfields of QG: string theory [9], loop QG theory [10], and even in the non-gravity theory [11].Standard model extension (SEM ) [12–14] is an effective field theory which describingthe standard model coupled to GR with the Lorentz and CPT invariance violations. It isworth noting that CPT invariance requires the physics to be unchanged under the com-bination of charge conjugation (C), parity inversion (P), and time reversal (T). In SEM,other important consequences can arise, like the appearance of Nambu-Goldstone and Higgsmodes. Unlike the effective framework provided by the SME, the properties of these modesare, in general, model dependent and cannot be completely discussed without knowledgeof the underlying fundamental theory. On the other hand, the most studied LIV modelsthat contemplate the role of the extra modes arising from the LSB mechanism involve thevacuum condensation of a vector field. These models are called “bumblebee models” [15–17],which were first introduced by Kostelecky and Samuel in 1989 [18, 19]. This model was thenextended to the various fields including the gravity [20–22]. Remarkably, in 2018, an exactSchwarzschild-like solution in this bumblebee gravity model (BGM) was found by Casanaet al [23]. Following this study, the accretion onto that black hole was studied by Rong-JiaYang et al [24] who showed that the LSB parameter slows down the mass accretion rate.Moreover, the problems of gravitational lensing and Hawking radiation of the black holehave been recently studied in [25] and [26], respectively. On the other hand, it is a factaccepted by researchers that rotating black hole spacetimes are the most relevant sub-casesfor astrophysics. Such solutions might also describe exterior metric for the rotating stars.The Kerr-like black solution in the BGM has been recently obtained by Ding et al [27]. Thisstationary, axisymmetric, and asymptotically flat 4-dimensional Kerr-like black hole solutionprescribed in the bumblebee gravity theory is obtained with a bumblebee vector field, whichis coupled to the spacetime curvature and acquires a vacuum expectation value that inducesthe LSB. When the rotation a →
0, then one can recover the Schwarzschild-like black solu-tion [23]. Furthermore, if LIV or LSB constant vanishes, (cid:96) → , then the well-known Kerrblack hole solution is recovered. Since a perturbed black hole emits gravitational radiationfrom its horizon, which reveals information about its inner properties [28], one can study thequantum structure of the black holes. Outside the horizon, potential barrier acts as a filterwhich depends on the frequencies of the propagating waves. Some waves are reflected bythe barrier and some transmitted to infinity [29]. However, the observers at infinity receives3nly a fraction of the emitted radiation. Therefore, there is a deviation between the radia-tion emanating from the black hole’s horizon and the observed radiation. This phenomenonalso manifests itself in the Hawking radiation [30, 31]. Namely, Hawking radiation whichis modified from the perfect black body spectrum is known as the GF [32, 33]. There aredifferent methods to compute the GF such as the WKB approximation [34, 35], matchingmethod [36, 37], rigorous bound method [38], and analytical method for the various of spinfields [39–44]. Furthermore, in the framework of GR for the radiation of gravitational waves,the most important phase described in function of the proper oscillation frequencies of theblack hole is called QNMs, which depend on the black hole parameters [46, 47]. The prop-agation of QNMs is different from normal modes. Namely, they possess a unique complexfrequency spectrum in which its real part represents the frequency of the oscillation and theimaginary part shows the damping [48, 49]. One of the appropriate methods to compute theQNM is the WKB approximation [50, 51], which is a semi-analytic technique. This methodwas also studied within the different contexts such as the AdS/CFT correspondence [52–54],the black hole spectroscopy, the black hole quantization [55–57], and quantum singularityof black holes, in particular for the rotating ones such as the Kerr black hole [58–60].This paper is organized as follows: In Sec. (II), we review the Kerr-like spacetime ofthe BGM. Sections (III) and (IV) are devoted to the scalar and fermionic perturbations inthe Kerr-like black hole geometry, respectively. To this end , we first derive the associatedeffective potentials of bosons and fermions by using the Klein-Gordon and Dirac equations,respectively. Then, we study the GFs of both particles from the black hole in Sec. (V).Bosonic and fermionic QNMs of the Kerr-like black hole are numerically computed with theaid of the WKB method in Sec. (VI). Finally, we draw our conclusions in Sec. (VII).( Throughout the paper, we follow the metric convention ( − , + , + , +) and the geometrizedunits: G = c = (cid:126) = 1 . ) 4 I. KERR-LIKE BLACK HOLE SPACETIME OF BGM
In the bumblebee gravity theory, action of an electromagnetic field coupled to the bum-blebee vector field for the curved spacetime is given by [27] S = (cid:90) d x √− g (cid:20) πG N ( (cid:60) + (cid:37)B µ B υ (cid:60) µν ) − B µν B µν − V ( B µ ) (cid:21) , (2.1)where (cid:37) is coupling constant and B µ is bumblebee field with mass dimension-1, which requiresa non-zero vacuum expectation value as (cid:104) B µ (cid:105) = b µ . The bumblebee field strength is definedas follows B µυ = ∂ µ B ν − ∂ ν B µ . (2.2)The function of potential V ( B µ ) is given by [14, 45] V = V (cid:0) B µ B µ ± b (cid:1) , (2.3)where b is a real constant. The Kerr-like rotating black hole metric in the BGM was recentlyfound by [27] as follows ds = − (cid:18) − M rρ (cid:19) dt − M ra √ L sin ( θ ) ρ dtdφ + ρ ∆ dr + ρ dθ + A sin ( θ ) ρ dϕ , (2.4)where ∆ = r − M r L + a , ρ = r + (1 + L ) a cos ( θ ) , (2.5) A = (cid:2) r + (1 + L ) a (cid:3) − ∆ (1 + L ) a sin ( θ ) . (2.6)One can immediately see that as L → a → ds = − (cid:18) − Mr (cid:19) dt + 1 + L − M/r dr + r dθ + r sin θ. (2.7)In short, metric (2.4) is nothing but a solution of LIV black hole with a rotation parameter,which is equal to the angular momentum per unit mass: a = JM . Its singularities appear at ρ = 0 and ∆ = 0. For ρ = 0, we have a ring-shape physical singularity at the equatorial5lane of the center of rotating black hole having radius a . The roots of Eq. (2.5) reveal thelocations of the event horizon and ergosphere: r ± = M ± (cid:112) M − a (1 + L ) , r ergo ± = M ± (cid:112) M − a (1 + L ) cos θ, (2.8)in which ± signs indicate the outer and inner horizon/ergosphere, respectively. For havinga black hole solution, it is conditional on a ≤ M √ L . (2.9)Now, we can write the metric tensor of the Kerr-like spacetime as g µυ = − (cid:16) − Mrρ (cid:17) − Mra √ L sin θρ ρ ∆ ρ − Mra √ L sin θρ A sin θρ , (2.10)from which one can compute the determinant of the metric tensor as follows g ≡ det ( g µν ) = − ρ (1 + L ) sin θ. (2.11)Thus, the contravariant form of g µν can be easily obtained as g µν = − Aρ ∆(1+ L ) − Mraρ ∆ √ L ∆ ρ ρ − Mraρ ∆ √ L ρ − Mrρ ∆(1+ L ) sin θ . (2.12)One can also get the Hawking temperature of this Kerr-like black hole which is derivedfrom its surface gravity ( κ ) [27] as follows T = κ π , κ = −
12 lim r → r + (cid:114) − Y dYdr , Y ≡ g tt − g tϕ g ϕϕ . (2.13)By using the relevant metric components given in Eq. (2.10) and substituting them intoEq. (2.13), the Hawking temperature is found to be T = (cid:112) M − a (1 + L )4 πM √ L (cid:16) M + (cid:112) M − a (1 + L ) (cid:17) . (2.14)6 II. SCALAR PERTURBATIONS
In this section, we shall examine the scalar perturbations of the Kerr-like black hole andderive the effective potential to which scalar waves will be exposed in this geometry. To thisend, we employ the Klein-Gordon equation:1 √− g ∂ µ (cid:0) √− gg µν ∂ ν (cid:1) Ψ = µ Ψ , (3.1)where µ is the mass of the scalar particle. Using Eqs. (2.11) and (2.12) in Eq. (3.1), weget − Aρ ∆ (1 + L ) ∂ t Ψ − M ra ∆ ρ √ L ∂ t ∂ φ Ψ + 1 ρ sin θ ∂ θ (sin θ∂ θ ) Ψ+1 ρ ∂ r (∆ ∂ r ) − M raρ ∆ √ L ∂ φ ∂ t Ψ + ρ − M r ∆ ρ (1 + L ) sin θ ∂ φ Ψ = µ Ψ . (3.2)To apply the technique of separation of variables in Eq. (3.2), one can use the followingansatz: Ψ ( r, t ) = R ( r ) S ( θ ) e imφ e − iωt , (3.3)where m is azimuthal quantum number and ω represents the energy of the particles. There-fore, the radial equation becomes1 R ( r ) ddr (cid:18) ∆ dR ( r ) dr (cid:19) + ω ( r + (1 + L ) a ) ∆ (1 + L ) + m a ∆ − M ramω ∆ √ L − ω a (1 + L ) − µ r , (3.4)and the angular part reads1 S ( θ ) sin θ ddθ (cid:18) sin θ dS ( θ ) dθ (cid:19) − m sin θ + c cos θ. (3.5)As is known, angular and radial equations admit two same (in absolute) eigenvalueswith opposite signs. Angular differential equation (3.5) has solutions in terms of the oblatespherical harmonic functions S lm ( ic, cos θ ) having eigenvalue λ lm [62] in which l, m are in-tegers such that | m | ≤ l, and c = a (1 + L ) ( ω − µ ) [63]. For simplicity, we consider theseparation constant as λ = λ lm . Thus, the radial differential equation becomes∆ ddr (cid:18) ∆ dR ( r ) dr (cid:19) + (cid:26) m a + ω (1 + L ) (cid:0) r + (1 + L ) a (cid:1) − M ramω √ L − (cid:0) µ r + ω a (1 + L ) + λ (cid:1) ∆ (cid:27) R ( r ) = 0 . (3.6)7he radial solution is in general associated with a free oscillation mode of the propagatingfield. Stable modes have particular frequencies ω with complex negative imaginary values:so we have an exponentially subsidence in amplitude. Conversely, if the imaginary valuesof the frequencies are positive, then the amplitude of the oscillations exponentially increaseand the modes consequently become unstable. If we consider M ω (cid:28) µM (cid:28)
1, whichwas first noticed by Starobinskii [63, 64], then Eq. (3.5) is amenable to analytic methods.If we assume the inequalities to hold then the angular part can be thought as sphericalharmonics with λ (cid:39) l ( l + 1).For having one dimensional wave equation, we first use the following transformation R ( r ) = U ( r ) (cid:112) r + (1 + L ) a , (3.7)together with the tortoise coordinate: dr ∗ dr = r + (1 + L ) a √ L ∆ . (3.8)Thus, Eq. (3.6) can be expressed as a one-dimensional Schr¨odinger equation d Udr ∗ + (cid:0) ω − V eff (cid:1) U = 0 , (3.9)where the effective potential reads V eff = (1 + L ) ∆( r + (1 + L ) a ) × (cid:20) ∆´ r ( r + (1 + L ) a ) + ∆( r + (1 + L ) a ) − r ∆( r + (1 + L ) a ) + 4 M ramω ∆ √ L − m a ∆ + (cid:0) µ r + ω a (1 + L ) + λ (cid:1)(cid:21) . (3.10)From now on, the prime symbol denotes the derivation with respect to r . The behavior ofthe effective potential under the effect of LSB parameter for scalar particles is illustrated inFig. 1, which shows a significant deduction on the potential peak when the LSB parameteris increased. IV. FERMIONIC PERTURBATIONS
To proceed our analysis with the Dirac fields in the geometry of the Kerr-like black hole,we shall use the four-dimensional Dirac equation formulated in the Newman-Penrose (NP)8 eff
FIG. 1: Plots of V eff versus r for the spin-0 particles. The physical parameters are chosenas; M = m = 1 , ω = 15 , a = 0 . , and λ = 2.formalism [65]. By this way, we aim to derive the effective potentials for the fermionicfields propagating in this geometry. To achieve this goal, we use the orthogonal (dragging)coordinates [66, 67] for the metric (2.4) and get ds = − ∆ (1 + L )Σ d (cid:101) τ + Σ∆ dr + Σ dθ + sin θ Σ d (cid:101) ϕ , (4.1)where Σ = ρ and d (cid:101) τ = (cid:16) dt − a √ L sin θdϕ (cid:17) , (4.2) d (cid:101) ϕ = (cid:16)(cid:0) r + (1 + L ) a (cid:1) dϕ − a √ Ldt (cid:17) . (4.3)The NP tetrad of the Kerr-like black hole geometry can be given by l µ = 1∆ (cid:20) r + a (1 + L ) √ L , ∆ , , a (cid:21) ,n µ = 12Σ (cid:20) r + a (1 + L ) √ L , − ∆ , , a (cid:21) , µ = 1 (cid:0) r + ia √ L cos θ (cid:1) √ (cid:20) ia √ L sin θ, , , i sin θ (cid:21) ,m µ = 1 (cid:0) r − ia √ L cos θ (cid:1) √ (cid:20) − ia √ L sin θ, , , − i sin θ (cid:21) . (4.4)where a bar over a quantity denotes complex conjugation. Thus, the dual co-tetrad of Eq.(4.4) reads l µ = (cid:20) √ L, − Σ∆ , , − a (1 + L ) sin θ (cid:21) ,n µ = ∆2Σ (cid:20) √ L, Σ∆ , , − a (1 + L ) sin θ (cid:21) ,m µ = 1 (cid:0) r + ia √ L cos θ (cid:1) √ (cid:104) ia √ L sin θ, , − Σ , − i ( r + a (1 + L ) sin θ ) (cid:105) ,m µ = 1 (cid:0) r − ia √ L cos θ (cid:1) √ (cid:104) − ia √ L sin θ, , − Σ , i ( r + a (1 + L ) sin θ ) (cid:105) . (4.5)Before deriving the non-zero spin coefficients, one can re-normalize the NP tetrad byusing the spin boost Lorentz transformations: l → l = ζl, n → n = ζ − n, (4.6) m → m = e iφ m, m → m = e − iφ m, (4.7)where ζ = (cid:114) ∆2Σ and e iφ = √ Σ r − ia √ L cos θ . (4.8)Thus, we have l µ = 1 √ (cid:20) r + a (1 + L ) √ L , ∆ , , a (cid:21) , n µ = 1 √ (cid:20) r + a (1 + L ) √ L , − ∆ , , a (cid:21) , µ = 1 √ (cid:20) ia √ L sin θ, , , i sin θ (cid:21) , m ´ µ = 1 √ (cid:20) − ia √ L sin θ, , , − i sin θ (cid:21) , (4.9)and l µ = (cid:114) ∆2Σ (cid:20) √ L, − Σ∆ , , − a (1 + L ) sin θ (cid:21) , n µ = (cid:114) ∆2Σ (cid:20) √ L, Σ∆ , , − a (1 + L ) sin θ (cid:21) , m µ = 1 √ (cid:104) ia √ L sin θ, , − Σ , − i (cid:0) r + a (1 + L ) (cid:1) sin θ (cid:105) , m µ = 1 √ (cid:104) − ia √ L sin θ, , − Σ , i (cid:0) r + a (1 + L ) (cid:1) sin θ (cid:105) . (4.10)The non-zero spin coefficients [61] can then be computed as π = − τ = Σ θ (cid:112) L ) + i a sin θ Σ r √ ,β = − α = − Σ θ (cid:112) L ) + cot θ (cid:112) L ) − ia sin θ Σ r √ ,ρ = µ = − Σ r √ ∆2Σ √ − i a (cid:112) ∆(1 + L ) cos θ Σ √ ,ε = γ = ∆ r √ − ∆Σ r √ − i a (cid:112) ∆(1 + L ) cos θ √ . (4.11)After this step, we employ the Chandrasekar-Dirac equations (CDEs) [61] to find theequations governing the fermion fields. CDEs are given by( D + ε − ρ ) F + (cid:16)(cid:101) δ + π − α (cid:17) F = iµ ∗ G , (∆ + µ − γ ) F + ( δ + β − τ ) F = iµ ∗ G , D + ε − ρ ) G − ( δ + π − α ) G = iµ ∗ F , (∆ + µ − γ ) G − (cid:0) δ + β − τ (cid:1) G = iµ ∗ F , (4.12)where F , F , G , and G are the spinor fields and D, ∆ , δ, and δ are the directional covariantderivative operators, which are given by D = l µ ∂ µ , ∆ = n µ ∂ µ , δ = m µ ∂ µ , δ = m µ ∂ µ . (4.13)The form of the CDEs suggests that F i ( t, r, θ, ϕ ) = 1 (cid:113) ( r − ia √ L cos θ ) e − i ( ωt + mφ ) Ψ i ( r, θ ) ,G i ( t, r, θ, φ ) = 1 (cid:113) ( r + ia √ L cos θ ) e − i ( ωt + mφ ) Φ i ( r, θ ) . (4.14)Inserting Eqs. (4.9)–(4.11), (4.13) and (4.14) into the CDEs (4.12), we obtain (cid:40) √ ∆ ∂ r − iω ( r + (1 + L ) a ) (cid:112) ∆ (1 + L ) + ∆ r √ ∆ − ima √ ∆ (cid:41) Ψ ( r, θ ) +1 √ L (cid:26) ∂ θ − m sin θ − aω √ L sin θ + cot θ (cid:27) Ψ ( r, θ ) = iµ (cid:16) r − ia √ L cos θ (cid:17) Φ ( r, θ ) , (4.15) − (cid:40) √ ∆ ∂ r + iω ( r + (1 + L ) a ) (cid:112) ∆ (1 + L ) + ∆ r √ ∆ + ima √ ∆ (cid:41) Ψ ( r, θ ) +1 √ L (cid:26) ∂ θ − m sin θ − aω √ L sin θ + cot θ (cid:27) Ψ ( r, θ ) = iµ (cid:16) r − ia √ L cos θ (cid:17) Φ ( r, θ ) , (4.16)12 √ ∆ ∂ r − iω ( r + (1 + L ) a ) (cid:112) ∆ (1 + L ) + ∆ r √ ∆ − ima √ ∆ (cid:41) Φ ( r, θ ) − √ L (cid:26) ∂ θ + m sin θ + aω √ L sin θ + cot θ (cid:27) Φ ( r, θ ) = iµ (cid:16) r + ia √ L cos θ (cid:17) Ψ ( r, θ ) , (4.17) − (cid:40) √ ∆ ∂ r + iω ( r + (1 + L ) a ) (cid:112) ∆ (1 + L ) + ∆ r √ ∆ + ima √ ∆ (cid:41) Φ ( r, θ ) − √ L (cid:26) ∂ θ − m sin θ − aω √ L sin θ + cot θ (cid:27) Φ ( r, θ ) = iµ (cid:16) r + ia √ L cos θ (cid:17) Ψ ( r, θ ) . (4.18)Since the functions Ψ i ( r, θ ) and Φ i ( r, θ ) depend on the radial and angular variables, onecan separate them by the following ansatzesΨ ( r, θ ) = (cid:60) + ( r ) ℵ + ( θ ) , (4.19)Ψ ( r, θ ) = (cid:60) − ( r ) ℵ − ( θ ) , (4.20)where (cid:60) ± ( r ) = ∆ − / p ( r ) ± / . Using the tortoise coordinate ( r ∗ ) as ddr ∗ = ∆ √ Lr + a (1+ L ) ddr ,Eqs. (4.15)–(4.18) yield the following two one-dimensional Schr¨odinger-like radial equations: (cid:26) ddr ∗ − i(cid:36) (cid:27) p +1 / = λ √ L √ ∆ K p − / , (4.21) − (cid:26) ddr ∗ + i(cid:36) (cid:27) p − / = λ √ L √ ∆ K p +1 / , (4.22)in which K = r + a (1 + L ) √ L , (cid:36) = ω + maK . (4.23)Setting the eigenvalue λ = − (cid:0) l + (cid:1) for the angular equations as L † ℵ − ( θ ) ℵ + ( θ ) = − λ, Lℵ + ( θ ) ℵ − ( θ ) = λ, (4.24)where L and L † are the angular operators 13 = ∂ θ + m sin θ + cot θ aω √ L sin θ, (4.25) L † = ∂ θ − m sin θ + cot θ − aω √ L sin θ, (4.26)one can have the spin-weighted spherical harmonics [68] for the angular equations. Moreover,if we let Z + = p +1 / + p − / , Z − = p − / − p +1 / , (4.27)equations (4.21) and (4.22) can be transformed to one-dimensional Schr¨odinger-like waveequations: (cid:18) d dr ∗ + (cid:36) (cid:19) Z ± = V ± eff Z ± . (4.28)From now on, for the sake of simplicity, we consider the massless ( µ = 0) fermions. Inthis case, the effective potentials for the propagating Dirac fields become V ± eff = ∆ K (cid:40) λ K (1 + L ) ± ddr (cid:32) λ √ ∆ K (cid:33)(cid:41) . (4.29)The behaviors of the effective potentials (4.29) are depicted in Figs. 2 and 3, which standfor spin-up and spin-down particles, respectively. V. GREYBODY RADIATION IN KERR SPACETIME OF BGMA. GFs of Bosons
Studying GFs provides important clues on the quantum structure of black holes. Deriva-tion of the GF can be conducted as ([69]) σ (cid:96) ( ω ) ≥ sec h (cid:18)(cid:90) + ∞−∞ ℘dr ∗ (cid:19) , (5.1)where ℘ = (cid:113) ( h ´) + ( ω − V eff − h ) h . (5.2)14 + eff FIG. 2: Plots of V + eff versus r for the spin-( +1 / ) particles. The physical parameters arechosen as; M = a = 1 , and λ = − . h ( r ∗ ) (cid:31) h ( −∞ ) = h ( ∞ ) = ω . Without loss of generality, if one simply sets h = ω, thus GF formula (5.1) reduces to σ (cid:96) ( ω ) ≥ sec h (cid:18)(cid:90) + ∞−∞ V eff ω dr ∗ (cid:19) . (5.3)To have an immaculately integration, let us consider the massless form of the bosoniceffective potential (3.10). Since the tortoise coordinate (3.8) varies from −∞ (the eventhorizon r h : lower boundary of the integral) and to + ∞ (spatial infinity: upper boundary ofthe integral) in Eq. (5.3), we get σ (cid:96) ( ω ) ≥ sec h (cid:18) ω (cid:90) + ∞ r h √ L ( r + (1 + L ) a ) (cid:20) ∆´ r ( r + (1 + L ) a ) +∆( r + (1 + L ) a ) − r ∆( r + (1 + L ) a ) + 4 M ramω ∆ √ L − m a ∆ + (cid:0) ω a (1 + L ) + λ (cid:1)(cid:21) dr (cid:19) . (5.4)15 − eff FIG. 3: Plots of V − eff versus r for fermions spin-( − / ) particles. The physical parametersare chosen as; M = a = 1 , and λ = − . σ (cid:96) ( ω ) ≥ sec h (cid:18) √ L ω (cid:19) (cid:26)(cid:18) − a r h + Mr h (1 + L ) − a (1 + L )5 r h + 2 M a r h (cid:19) − m a (1 + L ) (cid:18) r h + M r h + 4 M − a (1 + L )5 r h − M a (1 + L ) − M r h (cid:19) + (cid:0) ω a (1 + L ) + λ (cid:1) (cid:34) r h − (1 + L ) a r h + (1 + L ) a r h (cid:35) +4 M amω √ L (cid:18) r h + 2 M r h − (1 + L ) a − M r h − M a (1 + L ) − M r h (cid:19)(cid:27) . (5.5)In Fig. 4, the behaviors of the obtained bosonic GF of the Kerr-like black hole aredemonstrated. Thus, the effect of LSB on the bosonic GF is visualized. As can be seenfrom the plots, σ (cid:96) clearly decreases with the increasing LSB parameter. Namely, LSB playsa kind of fortifier role for the GF of spin-0 particles.16 l ( ω ( ω FIG. 4: Plots of σ l ( ω ) versus ω for the spin-0 particles. The physical parameters arechosen as; M = r = 1 , a = 0 .
03 and λ = 2. B. GFs of Fermions
In this section, we shall concentrate on the GF of the fermions to elicit the effect of theLSB on their emission from the Kerr-like black hole in the BGM. For this purpose, we usethe effective potentials (4.29) in Eq. (5.3): σ l ( ω ) (cid:23) sec h (cid:32) ω (cid:90) + ∞ r h (cid:40) λ dr ( r + a (1 + L )) √ L ± λ ddr (cid:32) (cid:112) ∆ (1 + L ) r + a (1 + L ) (cid:33)(cid:41) dr (cid:33) . (5.6)We then employ the classical term-by-term integration technique used for obtainingasymptotic expansions of integral, which requires the integrand to have an uniform asymp-totic expansion in the integration variable [70]. Thus, the evaluation of the integral (5.6)yields σ l ( ± ) ( ω ) (cid:23) sec h (cid:18) λ ω (cid:26) λ √ Lr h (cid:18) − a (1 + L )3 r h (cid:19) ± (cid:18) M − a (1 + L )2 r h (cid:19) ∓ (cid:20) M (cid:18) M r h + 38 M − a (1 + L ) r h (cid:19) + 1 r h − Mr h (cid:21)(cid:27)(cid:19) . (5.7)17 l + ω ( ( ω FIG. 5: Plots of σ l + ( ω ) versus ω for fermions with spin-( +1 / ). The physical parametersare chosen as; M = r = 1 , a = 0 .
03 and λ = − . +1 / ) and spin-( − / ) under the influence of LSB effect aredepicted in Figs. 5 and 6, respectively. VI. EFFECT OF LSB PARAMETER ON QNMs
Non-trivial information about thermalisation in quantum field theory are obtained bystudying small perturbations of a black hole away form the equilibrium. Such perturbationsare described by QNMs [71, 72]. These special oscillations are similar to normal modes ofa closed system. However, since the perturbation can fall into the black hole or radiateto infinity, the corresponding frequencies are complex [73]. The oscillation frequency isdefined by the real part and the rate of specific damping mode as a result of a radiation isdetermined by imaginary part. Thus, by getting the QNMs in the BGM, the comparison oftheoretical predictions with the experimental data supplied by ”future” LIGO and VIRGOtype experiments would help us to numerically constrain the LSB parameter. Thus, ingeneral, it is important to accumulate data on QNMs of black holes in various theories ofgravity [74]. 18 l − ( ω ω ( FIG. 6: Plots of σ l − ( ω ) versus ω for the spin-( − / ) particles. The physical parameters arechosen as; M = r = 1 , a = 0 .
03 and λ = − . d Zdr ∗ + V Geff Z = 0 , (6.1)where Z function is assumed to have a time-dependence e iωt , V Geff is the generic effectivepotential, and r ∗ is the tortoise coordinate as being stated above. A. Scalar QNMs
Comparing with the numerical results, the WKB approach [75] is known to lead to goodpredictions for obtaining the QNMs. In this method, V Geff is written by using the tortoisecoordinate so as to be constant at r ∗ → r ∗ → + ∞ (spatial infinity).The maximum value of V Geff , which we symbolize it as V from now on, is achieved at r ∗ .Three regions are defined as follows: region- I from −∞ to r , which is the first turning19oint where the potential becomes zero, region- II from r to r , namely the second turningpoint, and region- III from r to + ∞ . In region- II , the Taylor expansion is made over r ∗ .In regions- I and - III , the solution can be approximated by an exponential function: Z ∼ exp (cid:20) ζ ∞ (cid:88) n =0 ζ n Ξ n ( x ) (cid:21) , ζ → . (6.2)This expression can be substituted in Eq. (6.1) to get Ξ n as a function of the potentialand its derivative. We then impose the boundary conditions of the QNMs: Z ∼ e − iωr ∗ r ∗ → −∞ , (6.3) Z ∼ e iωr ∗ r ∗ → + ∞ , (6.4)and match the solutions of regions- I and - III with the solution of region- II at the turningpoints, r and r , respectively. The WKB approximation can be extended from the thirdto sixth order. This allows us to obtain the complex frequencies of the QNMs from thefollowing expression [75]: ω = (cid:20) V + (cid:113) V ´´0 Λ ( n ) − i (cid:18) n + 12 (cid:19) (cid:113) − V ´´0 (1 + Ω ( n )) (cid:21) , (6.5)where Λ ( n ) = 1 (cid:112) − V ´´0 (cid:34) (cid:32) V (4)0 V ´´0 (cid:33) (cid:18)
14 + α (cid:19) − (cid:18) V ´´´0 V ´´0 (cid:19) (cid:0) α (cid:1)(cid:35) , (6.6)andΩ ( n ) = 1 − V ´´0 (cid:34) (cid:18) V ´´´0 V ´´0 (cid:19) (cid:0)
77 + 188 α (cid:1) − (cid:32) V ´´´20 V (4)0 V ´´30 (cid:33) (cid:0)
51 + 100 α (cid:1) +12304 (cid:32) V (4)0 V ´´0 (cid:33) (cid:0)
67 + 68 α (cid:1) + 1288 (cid:32) V ´´´0 V (5)0 V ´´20 (cid:33) (cid:0)
19 + 28 α (cid:1) − (cid:32) V (6)0 V ´´0 (cid:33) (cid:0) α (cid:1) , (6.7)where the prime symbol denotes the differentiation with respect to r ∗ . The value of r ∗ isdetermined, and α = n + . With the help of the effective potential (3.10), one can easily get V Geff . After making straightforward calculations and numerical analysis, we have obtainedthe bosonic QNMs, which are tabulated in Table I for the angular momentum l = 2. In20able I, the case of m = 0 for the first (fundamental) overtone n = 0 is considered (highertones also show similar results that are parallel to the behaviors of the n = 0 mode). Therevealed knowledge from Table I is that the oscillations decrease when the LSB parameterincreases. But for the damping rate, there is no ostensive information about the LSBeffect. At the beginning ( L ≈ − . B. Dirac QNMs
In this sub-section, we shall apply the methodology applied for QNM bosons in theprevious section to the fermions. To this end, we consider the potentials obtained in Eq.(4.29) and use them in Eqs. (6.5)–(6.7). Table II constitutes the main results of this part:the numerically computed QNM frequencies for varying values of the LSB parameter forthe fix rotating parameter a = 0 . l and n values:hence, the characteristic fermionic QNMs are different from the bosonic ones.21 ω bosons l n L ω fermions VII. CONCLUSION
Very recently, it has been shown [27] that Kerr-like black hole solutions are existed in theBGM. For the first time with this study, we have studied the GFs and QNMs of the scalaran fermionic fields for the asymptotically flat black holes in Kerr-like black hole. To thisend, we have studied the perturbations of the scalar and Dirac fields, respectively. We havesummarized the results of our study in Tables I and II.We have computed GFs for both spin-0 and spin-( ± / ) particles. As a result of ouranalysis, we have seen that while the increase in the LSB parameter for scalar waves decreasesthe GFs, however for the fermionic waves, by making the opposite effect, the increases in theLSB cause also the increment in the GFs. Those remarkable behaviors are clearly depictedin Figs. 4–6. Moreover, for the bosonic QNMs, the oscillations decrease when the LSB22arameter increases. But for the damping rate, there is no palpable behavior about theLSB effect. However, these results make sense when looking at the plots of the potentialFig. 1, takes negative and positive values. For the fermionic case, we have deduced fromTable II that the fermionic QNMs (both the oscillatory and damping parts of the complexfrequency) tend to decrease once the LSB paramete increases. On the contrary, QNMfrequencies increase with the increment of l and n values: hence, the spin-( ± / ) QNMsexhibit different character compared with the spin-0 ones. Beyon all those, one may requireto perform a full-time domain analysis in order to understand the complete stability featureof the spacetime under the perturbations. However, the present study therefore can onlygive the qualitative nature of variations of the QNM frequencies with the LSB parameter asfar as the bosonic/fermionic perturbations are concerned.One of the most powerful uses of BGMs is to potentially explain dark energy, which is thephenomenon responsible for the observed accelerated expansion of the universe. Therefore,the GF/QNM analyses of the black hole in the AdS background within the framework ofthe BGM will be an important future extension of the present work. This may also beimportant to understand the AdS/CFT conjecture with LIV, since QNMs are responsiblefrom the stability in the CFT side. This is the next stage of study that interests us. [1] D. Mattingly, Living Rev. Relat. , 5 (2005).[2] S. Liberati and L. Maccione, Ann. Rev. Nucl. Part. Sci. , 245-267 (2009).[3] O. Bertolami, J. P´aramos, Phys. Rev. D , 044001 (2005).[4] V. A. Kostelecky, S. Samuel, Phys. Rev. D , 683 (1989); Phys. Rev. Lett. , 1811 (1991).[5] V. A. Kostelecky, R. Potting, Phys. Rev. D , 3923 (1995).[6] H. B. Nielsen and M. Ninomiya, Nucl. Phys. B , 153 (1978).[7] J. Ellis, M. K. Gaillard, D. V. Nanopoulos, and S. Rudaz, Nucl. Phys. B , 61 (1980).[8] H. B. Nielsen and I. Picek, Nucl. Phys. B , 269 (1983).[9] J. Nishimura, G. Vernizzi, JHEP (2000).[10] R. Gambini, J. Pullin, Phys.Rev. D , 116008 (1999).[12] D. Colladay, V. A. Kosteleck´y, Phys. Rev. D , 116002 (1998).
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