Quasinormal modes of charged fermions in linear dilaton black hole spacetime: Exact frequencies
TTurk J Phys(2021) TBA: tba1 – tba2 © T ¨UB˙ITAKdoi:10.3906/fiz-10.3906/fiz-2012-6
Quasinormal modes of charged fermions in linear dilaton black hole spacetime:Exact frequencies ˙Izzet SAKALLI ∗ , G¨ulnihal TOKG ¨OZ HYUSEIN , Physics Department, Eastern Mediterranean University, Famagusta, North Cyprus99628 via Mersin 10, Turkey G¨uzeloba Mahallesi, ¨Ornekk¨oy Caddesi, ¨Ornekk¨oy Sitesi, 2993 Ada, 11KA Blok, No: 2, Muratpa¸sa07230 Antalya, TurkeyCorresponding Author: ˙Izzet SAKALLI Received: .201 • Accepted/Published Online: .201 • Final Version: ..201
Abstract:
We study charged massless fermionic perturbations in the background of 4-dimensional linear dilaton blackholes in Einstein-Maxwell-dilaton theory with double Liouville-type potentials. We present the analytical fermionicquasinormal modes, whose Dirac equations are solved in terms of hypergeometric functions. We also discuss the stabilityof these black holes under the charged fermionic perturbations.
Key words:
Quasinormal modes, fermionic perturbations, dilaton, Dirac equation, Newman-Penrose, Liouville-typepotential, hypergeometric functions.
1. Introduction
At high enough energies, it is quite possible that gravity is not described by the action of Einstein’s generalrelativity theory. Today, the literature has a compelling evidence that in string theory (ST), gravity becomesa scalar-tensor [1]. The low-energy limit of the ST corresponds to Einstein’s gravitational attraction, whichis non-minimally coupled to a scalar dilaton field [2]. When the Einstein-Maxwell theory is combined witha dilaton field, the resulting spacetime solutions have important consequences. There have been significantstudies in the literature to find exact solutions of the Einstein-Maxwell-dilaton (EMD) theory. For example, inthe absence of a dilaton potential, EMD gravity’s charged dilaton black hole (BH) solutions were found by manyresearchers [3–11]. Essentially, the presence of the dilaton alters the arbitrary structure of spacetime and causesto curvature singularities with finite radii. The obtained dilaton BHs in general have non-asymptotically flat(NAF) structure. In recent years, even in the various extensions of general relativity, there has been an activeinterest in the NAF spacetimes. Among them, probably the most important ones are the asymptotic anti-deSitter (AdS) BHs [12], which play a vital role in string and quantum gravity theories as well as for the AdS/CFT(conformal field theory) correspondence, which is sometimes called Maldacena duality or gauge/gravity duality[13]. The BH solutions of EMD theory were studied by many researchers: the uncharged EMD BH solutionswere found in [14–16], while the charged EMD BH solutions were considered in [17, 18]. With the inclusion ∗ Correspondence: [email protected] work is licensed under a Creative Commons Attribution 4.0 International License. a r X i v : . [ h e p - t h ] F e b AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys of the Liouville-type potentials, the static charged BH solutions were found by [19–21]. The generalizationto dyonic (having both electric and magnetic charges) BH solutions in 4-dimensional and higher-dimensionalEMD gravity with single and double Liouville-type potentials were also found in [22]. In fact, similar to theHiggs potential, the double Liouville-type potentials have the ability to admit local extremes and critical points.However, a single Liouville potential lacks from these features. Besides, the double Liouville-type potentials alsoappear when the higher-dimensional theories are compressed into a 4-dimensional spacetime. Many of the BHsolutions with the double Liouville-type potentials to date can be seen from [23] and references therein. Amongthem Mazharimousavi et al’s BH [24] in the EMD theory with the double Liouville-type potentials has someunique features. First of all, those BHs cover Reissner-Nordstr¨om (RN) type BHs and Bertotti-Robinson (BR)spacetimes [25] interpolated within the same metric. Remarkably, in between the two spacetimes, there existsa linear dilaton BH (LDBH) [26] solution for the specific values of the dilaton, double Liouville-type potentials,and EMD parameters. The particular motivation of this work is to compute the quasinormal modes (QNMs)[27, 28] for the charged fermionic field perturbations [29–32] in the spacetime of those LDBHs. Since the QNMscan signal information about the stability of BHs, we shall also analyze the stability of the LDBHs under thecharged fermionic perturbations.Meanwhile, we would like to remind the reader for information purposes that QNMs are nothing butthe energy dissipation of a perturbed BH. Similar to an ordinary object, when a BH is perturbed it begins toring with its natural frequencies, which are the modes of its energy dissipation. These characteristic modesare called QNMs. The amplitudes of their oscillations decay in time. The amplitude of the oscillation can beapproximated by Ψ ≈ e iωt = e − ω I t cos( ω R t ) , (1.1)where ω = ω R + iω I is the frequency of the QNM [28]. Here, ω R and ω I are the frequencies of oscillatoryand exponential modes, respectively. After LIGO’s great successes about the gravitational wave measurements[33], the subject of QNMs has gained considerable importance in the direct identification of BHs. Today, thereare numerous well-known studies which show that the surrounding geometry of a BH experiences QNMs underperturbations (see for example [34–41] and references therein). One of the most important features to knowabout QNMs is that they allow us to analyze the quantum entropy/area spectrum of the BHs. In this regard,the reader is referred to [42–48] and references therein.The organization of the present paper is as follows: In Sec. 2, we review the LDBH solution, which isobtained from the 4-dimensional action of the EMD theory having the double Liouville-type potentials. In Sec.3, we first derive the spin- field equations by using 4-dimensional charged massless Dirac equations within theframework of Newman-Penrose (NP) formalism. Solution procedure to the obtained Dirac equations is givenin Sec. 4. Section 5 is devoted to the computations of the QNM frequencies of the charged fermions. Wesummarize our results in Sec. 6. (Throughout the paper, we use the geometrized units of G = c = (cid:126) = 1.)
2. LDBH spacetime in EMD theory with double Liouville-type potential S = (cid:90) d x √− g (cid:18) R − ∂ µ ϕ∂ µ ϕ − V ( ϕ ) − W ( ϕ ) (cid:0) F λσ F λσ (cid:1)(cid:19) , (2.1)2 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys where V ( ϕ ) = V e β ϕ + V e β ϕ , W ( ϕ ) = λ e − γ ϕ + λ e − γ ϕ , (2.2)where ϕ is the dilaton, V , are the double Liouville-type potentials, γ , denote the dilaton parameter, and λ , , β , are constants. Moreover, R is the Ricci scalar and the Maxwell 2-form is given by F = d A . (2.3)For the choice of pure magnetic potential A = − Q cos θdϕ, (2.4)where Q denotes the charge, we have F = Q sin θdθ ∧ dϕ. (2.5)It is worth noting that with the current choice of W ( ϕ ), the magnetic-electric symmetry, which exists in thestandard dilatonic coupling, namely λ ( λ ) = 0 , is not valid anymore. Thus, as highlighted in [24], the chargedLDBH spacetime is pure magnetic. Variations of the action (2.1) with respect to the gravitational field g µν andthe dilaton ϕ yield the following EMD field equations R µν = ∂ µ ϕ∂ ν ϕ + V g µν + W (cid:18) F µλ F λν − F λσ F λσ g µν (cid:19) , (2.6) ∇ ϕ − V (cid:48) − W (cid:48) (cid:0) F λσ F λσ (cid:1) = 0 , (2.7)where R µν is the Ricci tensor and the prime symbol ( (cid:48) ) denotes the derivative with respect to ϕ . Furthermore,the Maxwell equation is obtained with the variation with respect to A as d ( W (cid:63) F ) = 0 (2.8)in which the Hodge star ( (cid:63) ) refers to duality. After substituting the following metric (ansatz): ds = B ( r ) dt − dr B ( r ) − R ( r ) d Ω , (2.9)into Eqs.(2.6) and (2.7) and in the sequel of compelling calculations as made in [24], the field equations resultin the metric functions of the LDBH as follows B ( r ) = b ( r − r )( r − r ) r and R ( r ) = A r , (2.10)where b = 1 A − V + V ) > , and A = 2 λ Q , ( A : Real constant ) . (2.11)The inner and outer horizons of the LDBH are given by [24] r = 12 b (cid:16) c − (cid:112) c − ab (cid:17) and r = 12 b (cid:16) c + (cid:112) c − ab (cid:17) , (2.12)3 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys in which the physical parameters read c = 4 M and a = λ λ A , (2.13)where M denotes the quasilocal mass [49] and ϕ ( r ) = − √ r ) , β = β = √ , γ = − √ , γ = 1 √ , (2.14) V = V + V r , and W = λ r + λ r. (2.15)The above spacetime corresponds to a phase transition geometry, which changes the structure of spacetime fromRN to BR [24]. The case of c = 4 ab gives us the extremal ( r = r ) LDBHs whose congenerics can be seen in[26, 50].
3. Charged Dirac equation in LDBH geometry
In the NP formalism [51], massless Dirac equations with charge coupling are given as follows [29, 52] (cid:2) D + iql j A j + ε − ρ (cid:3) F + (cid:2) δ + iqm j A j + π − α (cid:3) F = 0 , (cid:2) δ + iqm j A j + β − τ (cid:3) F + (cid:2) ∆ + iqn j A j + µ − γ (cid:3) F = 0 , (cid:2) D + iql j A j + ε − ρ (cid:3) (cid:101) G − (cid:2) δ + iqm j A j + π − α (cid:3) (cid:101) G = 0 , (cid:2) ∆ + iqn j A j + µ − γ (cid:3) (cid:101) G − (cid:2) δ + iqm j A j + β − τ (cid:3) (cid:101) G = 0 , (3.1)where q is the charge of the fermion and A j represents the j th component of the vector potential of thebackground electromagnetic field (2.4). The wave functions F , F , (cid:101) G , (cid:101) G represent the Dirac spinors while α, β, γ, ε, µ, π, ρ, τ are the spin (Ricci rotation) coefficients. The directional derivatives for NP tetrads aredefined as D = l j ∇ j , ∆ = n j ∇ j , δ = m j ∇ j , ¯ δ = ¯ m j ∇ j , (3.2)In the meantime, a bar over a quantity stands for the complex conjugation. We choose a complex null tetrad { l, n, m, m } (the covariant one-forms) for the LDBH geometry as l j = 1 √ (cid:34)(cid:112) B ( r ) , − (cid:112) B ( r ) , , (cid:35) ,n j = 1 √ (cid:34)(cid:112) B ( r ) , (cid:112) B ( r ) , , (cid:35) ,m j = A (cid:114) r , , , i sin θ ] ,m j = A (cid:114) r , , , − i sin θ ] , (3.3)4 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys which all together satisfy the orthogonality conditions: l.n = − m.m = 1. The non-zero spin coefficients areobtained as ρ = µ = − − √ (cid:112) B ( r ) r ,(cid:15) = γ = b (cid:112) B ( r ) (cid:16) − r r r (cid:17) ,α = − β = cot θ A √ r . (3.4)The form of the Dirac equations (3.1) suggests that the spinors can be chosen as follows F = f ( r ) A ( θ ) e i ( kt + mϕ ) , (cid:101) G = g ( r ) A ( θ ) e i ( kt + mϕ ) ,F = f ( r ) A ( θ ) e i ( kt + mϕ ) , (cid:101) G = g ( r ) A ( θ ) e i ( kt + mϕ ) , (3.5)where k is the frequency of the wave corresponding to the Dirac particle and m is the azimuthal quantumnumber.
4. Solution of charged Dirac equation in LDBH geometry
After substituting the spin coefficients (3.4) and the spinors (3.5) into the Dirac equations (3.1), one gets1 f (cid:101) Zf − ( LA ) A = 0 , f (cid:101) Zf + (cid:0) L † A (cid:1) A = 0 , g (cid:101) Zg − (cid:0) L † A (cid:1) A = 0 , g (cid:101) Zg + ( LA ) A = 0 . (4.1)The radial operators appear in the above simplified Dirac equations are (cid:101) Z = A √ Λ ∂ r + H + iAkr √ Λ , (cid:101) Z = A √ Λ ∂ r + H − iAkr √ Λ , (4.2)in which Λ = b ( r − r )( r − r ) , (4.3)5 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys H = A r (cid:20) b ( r − r r )2 √ Λ + √ Λ (cid:21) . (4.4)The angular operators are L = ∂ θ + m sin θ + (cid:18) − p (cid:19) cot θ,L † = ∂ θ − m sin θ + (cid:18)
12 + p (cid:19) cot θ, (4.5)where p = qQ . Further, choosing f = g , f = g , A = A , A = A , (4.6)and introducing a real eigenvalue λ which is the separation constant of the complete Dirac equations, one canseparate the Dirac equations into two sets of master equations. The radial master equations read (cid:101) Zg = − λg , (cid:101) Zg = − λg , (4.7)and the angular master equations become L † A = λA ,LA = − λA . (4.8) The laddering operators L and L † [53] govern the spin-weighted spheroidal harmonics as (cid:16) ∂ θ − m sin θ − s cot θ (cid:17) ( s Y ml ( θ )) = − (cid:112) ( l − s ) ( l + s + 1) s +1 Y ml ( θ ) , (cid:16) ∂ θ + m sin θ + s cot θ (cid:17) ( s Y ml ( θ )) = (cid:112) ( l + s ) ( l − s + 1) s − Y ml ( θ ) . (4.9)Eigenfunctions s Y ml ( θ ), called the spin-weighted spheroidal harmonics, are complete and orthogonal, and havean explicit form [54]: s Y ml ( θ ) = (cid:115) l + 14 π ( l + m )! ( l − m )!( l + s )! ( l − s )! (cid:18) sin θ (cid:19) l × l (cid:88) ( − r = − l l + m − r (cid:18) l − sr − s (cid:19)(cid:18) l + sr − m (cid:19) (cid:18) cot θ (cid:19) r − m − s , (4.10)where l and s are the angular quantum number and the spin-weight, respectively. They satisfy the followingexpressions: l = | s | , | s | + 1 , | s | + 2 , .... and − l < m < + l. (4.11)6 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys
Comparison between the angular master equations (4.8) and (4.9) leads us to identify A = − ( + p ) Y ml ,A = ( − p ) Y ml . (4.12)Since l and s both must be integers or half-integers, we impose the ”Dirac quantization condition” [55]:2 qQ = 2 p = n, n = 0 , ± , ± ...., (4.13)and obtain the eigenvalue of the spin-weighted spheroidal harmonic equation as λ = − (cid:115)(cid:18) l + 12 (cid:19) − p , (cid:18) Real: (cid:18) l + 12 (cid:19) ≥ p (cid:19) , ∴ λ = (cid:18) l + 12 (cid:19) − p . (4.14) Letting g = G A ( r Λ) ,g = G A ( r Λ) , (4.15)and substituting them into the radial master equations (4.7), we obtain G (cid:48) − ikB = − λG A √ Λ ,G (cid:48) + ikB = − λG A √ Λ . (4.16)Introducing the tortoise coordinate [26] dr ∗ = drB −→ r ∗ = 1 b ( r − r ) ln (cid:20) ( r − r ) r ( r − r ) r (cid:21) , (4.17)one can rewrite Eq. (4.16) as G ,r ∗ − ik = − λ √ Λ G Ar ,G ,r ∗ + ik = − λ √ Λ G Ar . (4.18)Setting the solutions of the above equations into the following forms G = P + P ,G = P − P , (4.19)7 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys we get two decoupled radial equations, which correspond to 1-dimensional Schr¨odinger equation or the so-calledZerilli equation [26] : P j,r ∗ r ∗ + k P j = V j P j , j = 1 , V j = λ BR + ( − j λb r (cid:114) BR [ r ( r + r ) − r r ] . (4.21)The near-horizon and asymptotic limits of the potentials are as followslim r → r V j ≡ lim r ∗ →−∞ V j = 0 , lim r →∞ V j ≡ lim r ∗ →∞ V j = λ bA . (4.22)For the outer region of LDBH, the behaviors of potentials V j =1 , according to the various angular quantumnumbers are depicted in Figures 1 and 2, respectively. l = ll == V r Event Horizon
Figure 1 . V versus r graph. The physical parameters are chosen as A = b = r = 2 r = 2 p = 1. Plot is governed byEq. (4.21). Near-horizon ( r → r ) solutions of the radial equations (4.20) read P j = C j e ikr ∗ + C j e − ikr ∗ , (4.23)and thus we have G = P + P −→ G = D e − ikr ∗ + D e ikr ∗ , (4.24) G = P − P −→ G = D e − ikr ∗ + D e ikr ∗ . AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys lll === r V Event Horizon
Figure 2 . V versus r graph. The physical parameters are chosen as A = b = r = 2 r = 2 p = 1. Plot is governed byEq. (4.21). We impose one of the QNM conditions that the wave should be purely ingoing at the horizon by letting D = D = 0; thence G j = D j e − ikr ∗ . (4.25)Asymptotic ( r → ∞ ) solutions of the radial equations (4.20) are found as P j = (cid:101) C j e iηr ∗ + (cid:101) C j e − iηr ∗ , (4.26)where η = 1 A (cid:112) k A − λ b. (4.27)For k > λ bA , the Dirac waves can propagate to spatial infinity without fading. The asymptotic solutions tothe radial equations are then found to be G = P + P −→ G = (cid:101) D e iηr ∗ + (cid:101) D e − iηr ∗ ,G = P − P −→ G = (cid:101) D e iηr ∗ + (cid:101) D e − iηr ∗ . (4.28)We impose the other QNM condition which requires the propagating waves to be purely outgoing at spatialinfinity. To this end, we simply set (cid:101) D = (cid:101) D = 0 in Eq. (4.28) and obtain G j = (cid:101) D j e iηr ∗ . (4.29)At the asymptotic region, the tortoise coordinate becomes r ∗ = 1 b ( r − r ) ln (cid:20) ( r − r ) r ( r − r ) r (cid:21) ≈ b ( r − r ) ln r r − r . (4.30)9 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys
Therefore, one can see that ηr ∗ ≈ (cid:101) αr − r ln r r − r , (4.31)where (cid:101) α = ηb and e iηr ∗ = r i (cid:101) α . By this way, the asymptotic solutions (4.29) can be expressed in terms of theradial coordinate as follows G j = (cid:101) D j r i (cid:101) α . (4.32)An alternative way of getting the latter result is the decoupling of G j from Eq. (4.16):Λ G (cid:48)(cid:48) j − b ( r + r − r )2 G (cid:48) j + (cid:26) ( − j ik (cid:20) b ( r r − r )Λ − b ( r + r − r )2Λ (cid:21) + k r Λ − λ A (cid:27) G j = 0 . (4.33)As r → ∞ , Eq. (4.33) takes the following form r G (cid:48)(cid:48) j + rG (cid:48) j + ( k A − λ bA b = (cid:101) α ) G j = 0 . (4.34)whose solutions are found to be G j = (cid:101) D j r i (cid:101) α + (cid:98) C j r − i (cid:101) α . (4.35)Since e iηr ∗ = r i (cid:101) α and the wave should be purely outgoing at the spatial infinity, we set (cid:98) C j = 0 and thus find G j = (cid:101) D j r i (cid:101) α which is nothing but the same result that was obtained in Eq. (4.32).
5. Exact QNM frequencies
To find the analytical solutions of G j , we first introduce G j = H j e ( − j +1 ikr ∗ , (5.1)and change the variable to z = − x = − r − r r − r . (5.2)After inserting Eq. (5.2) in Eq. (4.33), we obtain z (1 − z ) H (cid:48)(cid:48) j + [ c − (1 + a + b ) z ] H (cid:48) j − ab H j = 0 , (5.3)which is the Euler’s hypergeometric differential equation [56] with a = i (cid:20) ( − j +1 kb + (cid:101) α (cid:21) , b = i (cid:20) ( − j +1 kb − (cid:101) α (cid:21) , c = 12 + 2 ik ( − j +1 r b ( r − r ) . (5.4)The general solution of Eq. (5.3) is given by [56] H j = C F ( a , b ; c ; z ) + C z − c F ( (cid:101) a , (cid:101) b ; (cid:101) c ; z ) , (5.5)10 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys where C and C are the integration constants and F ( a , b ; c ; z ) and F ( (cid:101) a , (cid:101) b ; (cid:101) c ; z ) are the hypergeometric (orGaussian) functions with (cid:101) a = a − c + 1 = 12 − i (cid:20) k ( − j +1 r + r b ( r − r ) + (cid:101) α (cid:21) , (cid:101) b = b − c + 1 = 12 − i (cid:20) k ( − j +1 r + r b ( r − r ) − (cid:101) α (cid:21) , (cid:101) c = 2 − c = 32 − ik ( − j +1 r b ( r − r ) . (5.6)In the near-horizon region: r → r , r ∗ → −∞ ⇒ z ≈ e r ∗ →
0. By recalling that F ( a , b ; c ; 0) = 1 [56],the physical solutions (5.1), which fulfill the QNM condition that only ingoing waves can propagate near thehorizon, are found to be as follows G = H e − ikr ∗ = C e − ikr ∗ F ( a , b ; c ; z ) ,G = H e ikr ∗ = C z − c e + ikr ∗ F ( (cid:101) a , (cid:101) b ; (cid:101) c ; z ) . (5.7)By using one of the special features of the hypergeometric functions [56]: F ( a , b ; c ; y ) = Γ( c )Γ( b − a )Γ( b )Γ( c − a ) ( − y ) − a F ( a , a + 1 − c ; a + 1 − b ; 1 /y )+ Γ( c )Γ( a − b )Γ( a )Γ( c − b ) ( − y ) − b F ( b , b + 1 − c ; b + 1 − a ; 1 /y ) , (5.8)one can extend the solutions (5.7) to spatial infinity ( r → ∞ , r ∗ → ∞ ⇒ z →
0) as follows G = C z − c e ikr ∗ (cid:20) Γ( (cid:101) c )Γ( (cid:101) b − (cid:101) a )Γ( (cid:101) b )Γ( c − (cid:101) a ) ( x ) − (cid:101) a F ( (cid:101) a , (cid:101) a + 1 − (cid:101) c ; (cid:101) a + 1 − (cid:101) b ; 1 /z )+ Γ( (cid:101) c )Γ( (cid:101) a − (cid:101) b )Γ( (cid:101) a )Γ( (cid:101) c − (cid:101) b ) ( x ) − (cid:101) b F ( (cid:101) b , (cid:101) b + 1 − (cid:101) c ; (cid:101) b + 1 − (cid:101) a ; 1 /z ) (cid:21) ,G = C e − ikr ∗ (cid:20) Γ( c )Γ( b − a )Γ( b )Γ( c − a ) ( x ) − a F ( a , a + 1 − c ; a + 1 − b ; 1 /z )+ Γ( c )Γ( a − b )Γ a )Γ( c − b ) ( x ) − b F ( b , b + 1 − c ; b + 1 − a ; 1 /z ) (cid:21) . (5.9)Since x = r − r r − r , and thus x (cid:39) r (cid:39) e br ∗ at the infinity, we have G ≈ C Γ( (cid:101) c )Γ( (cid:101) b − (cid:101) a )Γ( (cid:101) b )Γ( c − (cid:101) a ) r i (cid:101) α + C Γ( (cid:101) c )Γ( (cid:101) a − (cid:101) b )Γ( (cid:101) a )Γ( (cid:101) c − (cid:101) b ) r − i (cid:101) α . (5.10) G ≈ C Γ( c )Γ( b − a )Γ( b )Γ( c − a ) r i (cid:101) α + C Γ( c )Γ( a − b )Γ( a )Γ( c − b ) r − i (cid:101) α . (5.11)11 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys
The correspondence between the asymptotic solutions (4.28) and the above solutions (5.11) yields (cid:101) D = C Γ( (cid:101) c )Γ( (cid:101) b − (cid:101) a )Γ( (cid:101) b )Γ( c − (cid:101) a ) , (cid:101) D = C Γ( c )Γ( b − a )Γ( b )Γ( c − a ) , (cid:101) D = C Γ( (cid:101) c )Γ( (cid:101) a − (cid:101) b )Γ( (cid:101) a )Γ( (cid:101) c − (cid:101) b ) , (cid:101) D = C Γ( c )Γ( a − b )Γ( a )Γ( c − b ) . (5.12)Therefore, to have only pure outgoing waves at the spatial infinity (i.e., (cid:101) D , (cid:101) D = 0), we appeal to the polestructure of the Gamma functions. The Gamma functions Γ( (cid:101) x ) have poles at (cid:101) x = − n for n = 0 , , a = i (cid:20) − kb + (cid:101) α (cid:21) = − n, c − b = 12 + i (cid:20) − k r + r b ( r − r ) − (cid:101) α (cid:21) = − n, (cid:101) a = 12 − i (cid:20) k r + r b ( r − r ) + (cid:101) α (cid:21) = − n, (cid:101) c − (cid:101) b = 1 + i (cid:20) k r + r b ( r − r ) − (cid:101) α (cid:21) = − n. (5.13)Recall that the time dependence of the QNMs is governed by e ikt (see Eq.(3.5)). Therefore, having QNMs isconditional on Im ( k ) >
0, which guaranties the stability [40]. The conditions under which frequencies obtainedfrom G and G spinor solutions produce stable modes (i.e., QNMs) and which ones generate the unstablemodes are summarized in Tables 1 and 2:Frequencies Stable Modes (QNMs) Unstable Modes k = − I A ( β − [ Aβb (2 n + 1)+2 √ b (cid:113) A b ( n + ) + λ ( β − (cid:105) if A <
A < λ √ b ( n + ) if A >
A <
A > λ √ b ( n + ) k = I A ( β − [ Aβb ( n + 1)+ √ b (cid:112) A b ( n + 1) + λ ( β − (cid:105) if A >
A <
A < λ √ b ( n +1) if A <
A > λ √ b ( n +1) Table 1: QNM frequencies for G . The results are obtained from Eqs. (5.10) and (5.13).12 AKALLI and TOKG ¨OZ HYUSEIN/Turk J Phys
Frequencies Stable Modes (QNMs) Unstable Modes k = I A n ( λ − n A b ) if A >
A < − λ √ bn if A <
A > λ √ bn if A >
A > − λ √ bn if A <
A < λ √ b ( n k = − I A ( β − [ Aβb (2 n + 1)+2 √ b (cid:113) A b ( n + ) + λ ( β − (cid:105) if A <
A < λ √ b ( n + ) if A >
A <
A > λ √ b ( n + ) Table 2: QNM frequencies for G . The results are obtained from Eqs. (5.11) and (5.13).In Tables 1 and 2, the parameter of β is given by1 < β = r + r r − r < ∞ . (5.14)As can be seen from above, depending on the values of A, λ, b, n some QNM frequencies can have posi-tive/negative imaginary part, and therefore the LDBHs can be stable/unstable under charged fermionic pertur-bations.
6. Conclusion
In this paper, we have analytically computed the QNMs of charged fermionic perturbations for the 4-dimensionalLDBHs which are the solutions to the EMD theory with the double Lioville-type potentials. The fermionic fieldsare the solutions of the Dirac equation. For the LDBH geometry, we have found that the massless but chargedfermionic fields are governed by the Gaussian hypergeometric functions. We matched the exact solutionsobtained with the solutions found for near-horizon and asymptotic regions. Thus, we have shown that thereare two sets of frequencies for each component ( G and G ) of the fermionic fields. As being summarizedin Tables 1 and 2, in each set the frequencies obtained can belong to either the stable modes (QNMs) or theunstable modes depending on the relations between the parameters of A, λ, b, and n . Meanwhile, the readermay question what the advantage/disadvantage of perturbing the LDBH with fermionic fields instead of thebosonic fields is. If precise measurements of QNM frequencies become possible in the future and if the fermionicQNM data obtained for spin-up and spin-down fields (i.e., G and G ) coincide with (a kind of double check)the analytical expressions that we have derived, those results will likely reveal the presence of LDBHs moreaccurately than the bosonic QNM ones.As is known, rotating BH solutions are more considerable for testing relevant theories with the astrophys-ical observations. For this reason, in the near future, we plan to extend our study to the rotating LDBHs. Tothis end, we shall apply the Newman–Janis algorithm [57] to the metric (2.9) and perturb the obtained rotatingLDBH to reveal the effect of rotation on their QNMs. Acknowledgements
The authors are grateful to the Editor and anonymous Referees for their valuable comments and suggestions toimprove the paper.
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