Black holes in Gauss-Bonnet and Chern-Simons-scalar theory
aa r X i v : . [ g r- q c ] M a r Black holes in Gauss-Bonnet andChern-Simons-scalar theory
Yun Soo Myung a and De-Cheng Zou a,b a Institute of Basic Sciences and Department of Computer Simulation,Inje University, Gimhae 50834, Korea b Center for Gravitation and Cosmology and College of Physical Science andTechnology, Yangzhou University, Yangzhou 225009, China
Abstract
We carry out the stability analysis of the Schwarzschild black holein Gauss-Bonnet and Chern-Simons-scalar theory. Here, we introducetwo quadratic scalar couplings ( φ , φ ) to Gauss-Bonnet and Chern-Simons terms, where the former term is parity-even, while the latterone is parity-odd. The perturbation equation for the scalar φ is theKlein-Gordon equation with an effective mass, while the perturba-tion equation for φ is coupled to the parity-odd metric perturbation,providing a system of two coupled equations. It turns out that theSchwarzschild black hole is unstable against φ perturbation, leadingto scalarized black holes, while the black hole is stable against φ andmetric perturbations, implying no scalarized black holes. e-mail address: [email protected] e-mail address: [email protected] Introduction
Black holes with scalar hair (spontaneous scalarization) are obtained dy-namically in a different class of models where the scalar field couples eitherto the topological invariant, Gauss-Bonnet term [1, 2, 3] or to Maxwell ki-netic term [4, 5, 6] with a coupling function f ( φ ). In this case, the ap-pearance of static scalarized solutions is closely connected to the instabilityof Schwarzschild [7] or Reissner-Nordstr¨om solution [8] without scalar hair.Here, the coupling constant α plays the role of a spectral parameter (effectivemass) in the linearized equation. However, the other topological invariantof Chern-Simon term does not activate the static scalarized black hole, butit could develop the scalarized Kerr solution with linear coupling [9] andscalarized Schwarzschild-NUT solution with quadratic coupling [10]. It wasknown that the Chern-Simons term is a parity-odd (violating) one and thus,non-rotating black holes are not modified because these are parity-even so-lution [11]. The Kerr and Schwarzschild-NUT solutions are parity-odd andthus, they do acquire modifications from the Chern-Simons term. We notethat Schwarzschild black hole was stable in the Einstein-Chern-Simons grav-ity with linear coupling [12, 13, 14, 15], which may explain why scalarizedstatic black holes do not exist.Similarly, cosmology with the Gauss-Bonnet term could affect the back-ground evolution of parity-even and results in the tensor-to-scalar ratio r ,leading to violation of the consistency relation [16, 17]. On the other hand,the Chern-Simons term did not appear in the background and scalar per-turbations, but this appears in tensor perturbations as circularly polarizedmodes because these modes belong to the parity-odd case.An effective action including both topological invariants with different lin-ear couplings could be obtained from some superstring models [18] and theheterotic strings [19]. Inspired by this, we introduce a new action (1) whichincludes both topological invariants with different quadratic couplings. Wewill use this new action to investigate the stability analysis of the Schwarzschildblack hole which is closely related to the spontaneous scalarization of blackhole. This will reveal the hidden roles of two topological invariants on thescalarization process.To make all things clear, we mention our notations. We will use naturalunits of G = c = ~ = 1 with signature ( − , + , + , +). The Riemann, Riccitensor, and Levi-Civita tensor are defined by R ρσµν = ∂ µ Γ ρνσ − ∂ ν Γ ρµσ + Γ ρµλ Γ λνσ − Γ ρνλ Γ λµσ , R µν = R ρµρν , ǫ trϕ ϕ = 1 √− g . GBCSS theory
We introduce newly the Gauss-Bonnet and Chern-Simons-scalar (GBCSS)theory in four dimensions whose action is given by S GBCSS = 116 π Z d x √− g " R −
12 ( ∂φ ) −
12 ( ∂φ ) + αφ R + βφ ∗ RR (1)where R is the Gauss-Bonnet term R = R − R µν R µν + R µνρσ R µνρσ (2)and ∗ RR is the Chern-Simons term ∗ RR = ∗ R η µνξ R ξηµν . (3)Here the dual Ricci tensor is defined by ∗ R η µνξ = 12 ǫ µνρσ R ηξρσ . (4)From now on, we call the EGBS (ECSS) theory for the case of β = 0( α =0). It is worth noting that the mass dimensions of coupling constants aregiven by [ α ] = [ β ] = −
2. Although both R and ∗ RR are topological(total derivatives) in four dimensions, but they become dynamical due tothe coupling to the scalars. However, there exists a difference between them.The Gauss-Bonnet term affects the property of the static solutions of parity-even source, while the Chern-Simons term gives different results only in therotating solution of parity-odd source. This explains that the scalarizedSchwarzschild black holes could be found from the Gauss-Bonnet coupling,but they might not be found from the Chern-Simons coupling. In otherwords, the Chern-Simons term does not activate a scalar monopole field inthe static black hole spacetime. Also, the appearance of scalarized blackholes is closely related to the instability of scalar-free black hole. Hence, itis important to investigate the stability of black holes without scalar hair inthe GBCSS theory.Varying for g µν , φ , and φ lead to the three equations as G µν = 12 ∂ µ φ ∂ ν φ + 12 ∂ µ φ ∂ ν φ − g µν [( ∂φ ) + ( ∂φ ) ] − α ∇ ρ ∇ σ ( φ ) P µρνσ − βC µν , (5) ∇ φ = − αR φ , (6) ∇ φ = − β ∗ RRφ , (7)3here P µρνσ -tensor takes the form P µρνσ = R µρνσ + g µσ R νρ − g µν R ρσ + g νρ R µσ − g ρσ R µν + R g µν g ρσ − g µσ g νρ ) , (8)which corresponds to the divergence-free part of the Riemann tensor ( ∇ µ P µρνσ =0). Here the Cotton tensor C µν is given by C µν = ∇ ρ ( φ ) ǫ ρσγ ( µ ∇ γ R ν ) σ + 12 ∇ ρ ∇ σ ( φ ) ǫ ργδ ( ν R σ µ ) γδ . (9)Using the trace of (5) R = 12 ( ∂φ ) + 12 ( ∂φ ) − α ∇ ρ ∇ σ ( φ ) G ρσ , (10)we rewrite (5) as the Ricci-tensor equation R µν = 12 ∂ µ φ ∂ ν φ + 12 ∂ µ φ ∂ ν φ − α ∇ ρ ∇ σ ( φ ) h R µρνσ + g µσ R νρ + g νρ R µσ i − βC µν (11)which may be suitable for analyzing the stability of the black holes.Choosing the background quantities (by denoting the “overbar”)¯ φ = 0 , ¯ φ = const , (12)Eqs.(5)-(7) admit the spherically symmetric Schwarzschild spacetime ds = ¯ g µν dx µ dx ν = − f ( r ) dt + dr f ( r ) + r ( dϕ + sin ϕ dϕ ) (13)with the metric function f ( r ) = 1 − Mr = 1 − r + r . (14)In this case, one has¯ R = 48 M r , ∗ ¯ R ¯ R = 0 , ¯ C µν = 0 , ¯ R = ¯ R µν = 0 , ¯ R µρνσ = 0 . (15)Now let us introduce the perturbations around the background as g µν = ¯ g µν + h µν , φ = 0 + δφ , φ = ¯ φ + δφ . (16)4he linearized equation to (11) can be written by δR µν ( h ) = − βδC µν ( δφ ) (17)where δR µν ( h ) and δC µν ( δφ ) take the forms δR µν ( h ) = 12 (cid:0) ¯ ∇ γ ¯ ∇ µ h νγ + ¯ ∇ γ ¯ ∇ ν h µγ − ¯ ∇ h µν − ¯ ∇ µ ¯ ∇ ν h (cid:1) , (18) δC µν ( δφ ) = ¯ φ ¯ ∇ ρ ¯ ∇ σ δφ ǫ ργδ ( ν ¯ R σ µ ) γδ . (19)We observe from (7) and (19) that ¯ φ = ¯ φ = 0 gives the same solution(13), but it gives no coupling to the perturbed Einstein equation. Also, wenote from (6) that ‘ ¯ R = 0’ does not admit ¯ φ = const solution for the φ coupling. From Eqs.(6) and (7), we obtain the linearized scalar equation for δφ (cid:16) ¯ ∇ + 96 αM r (cid:17) δφ = 0 , (20)while δφ couples to the metric perturbation h µν as¯ ∇ δφ + 4 β ¯ φ ǫ µνρσ ¯ R ηξµν ¯ ∇ ρ ¯ ∇ η h ξσ = 0 . (21)Before we proceed, we would like to focus on two linearized theories. Inthe single scalar coupling of φ = φ = φ , choosing ¯ φ = 0 provides theSchwarzschild solution. However, the linearized equations around the blackhole lead to δR µν = 0 , (22) (cid:16) ¯ ∇ + 48 αM r (cid:17) δφ = 0 , (23)which lead to the linearized version for the EGBS theory because the lin-earized Cotton term ( δC µν ) decouples from (22).For the other case of linear couplings ( αφ and βφ ) with possessing ashift symmetry of φ → φ + c and φ → φ + c , the Schwarzschild blackhole (13) is not a solution because of non-zero Gauss-Bonnet term. In theECSS (linear coupling) theory, it turned out that the Schwarzschild blackhole is stable against the metric and scalar perturbations [12, 13, 14, 15]. δφ We wish to start with stability analysis by noting that δφ is completely de-coupled from other fields. Using the tortoise coordinate ( r ∗ = R dr/f ( r )) and5 φ = u ( r ) r Y e − iωt with Y ≡ Y lm ( ϕ , ϕ ) spherical harmonics, the linearizedequation (20) becomes d udr ∗ + h ω − V u ( r ) i u ( r ) = 0 , (24)where the potential V u ( r ) is given by V u ( r ) = f ( r ) (cid:16) λr + 2 Mr − αM r (cid:17) (25)with λ = l ( l + 1). We note that the last term in (25) plays the role of aneffective mass with [ α − ] = 2. This term contributes to potential large nega-tively near the horizon, while its contribution becomes small neglectfully as r increases. From now on, we focus on the l = 0-mode of u ( r ) because it is re-sponsible for analyzing the stability analysis and obtaining scalarized blackholes. The s ( l = 0)-mode potential V l =0u ( r ) develops negative region out-side the horizon, depending the value of coupling constant α . The sufficientcondition for the instability is given by Z ∞ r + =2 M h V l =0u ( r ) f ( r ) i dr < → M α < → < r + √ α < . . (26)However, (26) is not a necessary and sufficient condition for instability. Todetermine the threshold of instability precisely, one has to solve the linearizedequation numerically d udr ∗ − h Ω + V l =0u ( r ) i u ( r ) = 0 , (27)which may allow an exponentially growing mode of e Ω t as an unstable mode.We obtain the unstable bound for the scalar perturbation0 < r + √ α < . , (28)which implies that the threshold of instability is determined by 1 / √ α =1 / √ α th = 3 . α is given by α > r . , (29)which is smaller than the sufficient condition for instability ( α > r / . α when obtaining staticsolutions: 1 / √ α s = r + / √ α ∈ [3 . , . , . , . , · · · ] where weidentify the first value with the threshold of instability. These solutions arelabelled by the order number n = 0 (fundamental branch), 1 , , , · · · (excited branches) which is identified with the number of nodes for δφ ( z ) = u ( z ) /z with z = r/r + . Actually, it may represent the n = 0 , , , , , · · · scalarized black holes found when solving full equations (5)-(7). This impliesthat the appearance of n = 0 scalarized black hole is closely related to thethreshold of instability for Schwarzschild black hole. δφ and h µν The metric perturbation h µν is classified according to the transformationproperties under parity, namely odd sector ( h , h ) and even sector ( H , H , H , K )as h µν ( t, r, ϕ , ϕ ) = e − iωt H ( r ) Y H ( r ) Y − h ( r ) ∂ ϕ Y sin ϕ h ( r ) sin ϕ ∂ ϕ Y ∗ H ( r ) Y − h ( r ) ∂ ϕ Y sin ϕ h ( r ) sin ϕ ∂ ϕ Y ∗ ∗ r Y K ( r ) 0 ∗ ∗ ∗ r sin ϕ Y K ( r ) (30)with ∗ symmetrizations. The form of δφ is given by δφ = ψ ( r ) r Y e − iωt . (31)Plugging Eqs. (30) and (31) into Eq.(17), we find ten perturbation equationsas appeared in Appendix. It is important to note that ten perturbationequations imply twenty constraints like E i = 0 , O i = 0 , for i = 1 , · · · , , (32)where E i with i = 1 , · · · ,
10 are functions of ( H , H , H , K ) and O i with i = 1 , · · · ,
10 are functions of ( h , h , ψ ). This implies that ten perturbationequations can be classified into two parties: odd-parity equations ( { O i = 0 } )and even-parity equations ( { E i = 0 } ). We emphasize that there are couplingsbetween h µν and δφ in the odd-parity equations.First of all, we consider the even-parity ( { E i = 0 } ) because it correspondsto even-parity sector of Einstein gravity ( δR µν = 0). It is well known thatthis case reduces to a single second-order equation for a field defined byˆ M = 1 p ( r ) q ( r ) − h ( r ) (cid:26) p ( r ) K ( r ) − H ω (cid:27) , (33)7here q ( r ) = ˜ λ (˜ λ + 1) r + 3˜ λM r + 6 M r (˜ λr + 3 M ) , h ( r ) = i ( − ˜ λr + 3˜ λM r + 3 M )( r − M )(˜ λr + 3 M ) ,p ( r ) = − ir r − M , ˜ λ = λ − . (34)Here, we obtain the Zerilli equation [20] d ˆ Mdr ∗ + h ω − V Z ( r ) i ˆ M = 0 , (35)where the Zerilli potential is given by V Z ( r ) = f ( r ) " λ (˜ λ + 1) r + 6˜ λ M r + 18˜ λM r + 18 M r (˜ λr + 3 M ) . (36)All potentials V Z ( r ) for l ≥ λ ≥
2) are always positive for whole range of r + ≤ r ≤ ∞ , which implies that the even-parity perturbation is stable.On the other hand, for odd-parity sector ( δR µν = − βδC µν ), the firstfive equations ( { O i = 0 } , i = 1 , · · · ,
5) provide three relevant equations with˜ β = 8 ¯ φ β : O = 0( O = 0) r ( − M + λr ) h − rf (cid:16) iωr h − βM ψ + iωr h ′ +3 ˜ βM rψ ′ + r h ′′ (cid:17) = 0 , (37) O = 0( O = 0) − iωr (cid:16) h − iωrh − rh ′ (cid:17) + r f ( λ − h + 3 i ˜ βωM ψ = 0 , (38) O = 0 iωr h − (2 M − r ) n M h − (2 M − r ) rh ′ o = 0 . (39)We note that all remaining equations O i = 0 with i = 6 , · · · ,
10 are redun-dant. Introducing a new field Q = 2 f h / ( ˜ βr ), (37)-(39) become one coupledsecond-order equation d Qdr ∗ + h ω − V Q ( r ) i Q = 6 iωM f ( r ) r ψ, (40)where V Q ( r ) represents the Regge-Wheeler potential [21] for odd-parity per-turbation as V Q ( r ) = f ( r ) (cid:16) λr − Mr (cid:17) . (41)8lso, the linearized equation (21) for the scalar δφ becomes a coupledsecond-order equation d ψdr ∗ + h ω − V ψ ( r ) i ψ = − iM ˜ β ( l + 2)( l + 1) l ( l − f ( r ) ωr Q, (42)where V ψ ( r ) denotes a potential for ψV ψ ( r ) = f ( r ) n λr (cid:16) M ˜ β r (cid:17) + 2 Mr o . (43)Eqs.(40) and (42) represent an important property of CS coupling to thescalar φ .For s ( l = 0)-mode, one has a decoupled equation for ψ from (42) as d ψdr ∗ + h ω − V l =0 ψ ( r ) i ψ = 0 , (44)where the potential takes the form V l =0 ψ ( r ) = f ( r ) h Mr i . (45)The s ( l = 0)-mode of ψ is stable because V l =0 ψ ( r ) is positive definite outsidethe horizon. The l = 1-mode equation leads to d ψdr ∗ + h ω − V l =1 ψ ( r ) i ψ = 0 , (46)where the corresponding potential is given by V l =1 ψ ( r ) = f ( r ) h r (cid:16) M ˜ β r (cid:17) + 2 Mr i . (47)Also, the l = 1-mode of ψ is stable because V l =1 ψ is positive definite outsidethe horizon. Actually, the l = 1-mode is important because if one includes V ( φ ) = µ φ /
2, it corresponds to the most prominent superradiant insta-bility in the rotating black hole [15]. Furthermore, it proved that all highermodes with l ≥ stability of Schwarzschild black hole in the Chern-Simons coupling isclosely related to the disappearance of black holes with scalar hair φ ( r ).We will explore this connection in the next section.9 Scalarized Black holes
Let us develop scalarized black hole solutions by making use of (5)-(7). Con-sidering a spherically symmetric line element ds = − A ( r ) dt + dr B ( r ) + r ( dϕ + sin ϕ dϕ ) . (48)From (5), we display its ( t, t ) and ( r, r ) components2 h r + 4 α (1 − B ) φ φ ′ i + 12 h B (cid:0) r ( φ ′ + φ ′ ) (cid:1) − i − αB ( B − (cid:0) φ ′ + φ φ ′′ (cid:1) = 0 , (49)2 A ′ h r + 4 α (1 − B ) φ φ ′ i − A B h B ( − r ( φ ′ + φ ′ )) i = 0 . (50)Two scalar equations (6) and (7) are given by φ ′′ + (cid:18) r + A ′ A + B ′ B (cid:19) φ ′ − αφ A ′ r A B (cid:0) B A ′ + AB ′ − A ′ B − ABB ′ (cid:1) + 8 αφ r A ( B − A ′′ = 0 , (51) φ ′′ + (cid:18) r + A ′ A + B ′ B (cid:19) φ ′ = 0 . (52)In the above, we observe the absence of Chern-Simons terms ( ∗ RR = C µν =0), implying that there is no source to develop scalar hair φ ( r ). We notethat (49)-(52) with φ = 0 reduce to the corresponding equations in Ref. [22],where black holes with scalar hair φ ( r ) was obtained in the EGBS theory.In this section, we are interested in obtaining scalarized black holes withtwo scalar hairs φ and φ . Near the horizon, one may consider power-seriesexpansion of solution in terms of ( r − r h ) as A ( r ) = ∞ X n =1 a n ( r − r h ) n , B ( r ) = ∞ X n =1 b n ( r − r h ) n ,φ ( r ) = ∞ X n =0 φ ,n ( r − r h ) n , φ ( r ) = ∞ X n =0 φ ,n ( r − r h ) n . (53)where { a n , b n , φ ,n , φ ,n } are constant coefficients. Substituting (53) into (49)-(52), b , φ , and a can be solved for r h , φ , and α as b = r h (cid:16) r h − q r h − α φ , (cid:17) φ , α , φ , = − r h − q r h − α φ , αφ , ,φ , = φ , = · · · = 0 , φ , = const (54)10nd so on. a does not display here because of its complicated form. Hence,we check that the scalar field φ becomes trivial because it is const. Thismeans that it is hard to construct scalarized black holes with scalar hair φ ( r ). Actually, the n = 0 scalarized black hole is found in Fig. 1. Here,we choose a rescaling of A ( r ) such that a approaches 1 / V ( φ ) = µ φ / φ ( r ) are not allowed becausethere is no way of escaping from the no-hair theorem [23]. A H r L (cid:144) B H r L f H r L Φ H r L
10 20 30 40 50 r0.0000.0050.0100.0150.0200.025 Φ H r L Figure 1: Metric functions [ A ( r ) , f ( r ) , /B ( r )] and scalar fields [ φ ( r ) , φ ( r )]as functions of r ∈ [1 ,
50] for the n = 0 scalarized black hole. The horizon islocated at r = r h = 1 and α is chosen 0 . > α th = 0 . n = 0scalarized black hole.Finally, let us discuss the stability issue of the n = 0 scalarized blackhole in the GBCSS theory. As was mention in [24], the difference betweenexponential and quadratic couplings in the EGBS theory is that the n = 0scalarized black hole is stable for the exponential coupling, while the n = 0scalarized black hole is unstable for the quadratic coupling. Recently, it isshown that the quadratic term controls the onset of the instability giving the n = 0 scalarized black hole, while the higher-order coupling terms includingthe exponential coupling control the nonlinearities quenching the instabilityand thus, control the stability of the n = 0 black hole [25]. Therefore, weexpect that the n = 0 scalarized black hole is unstable against perturbationsin the GBCSS theory since this theory has a quadratic coupling term. We state that the appearance of scalarized black holes is directly related tothe instability of black hole without scalar hair. We have found that the11chwarzschild (static) black hole is unstable against the s ( l = 0)-mode per-turbation of δφ coupled to the Gauss-Bonnet term, implying the appearanceof scalarized black holes in the EGBS theory. On the other hand, the staticblack hole is stable against all modes of δφ coupled to the Chern-Simonsterm, implying the disappearance of scalarized black holes in the ECSS the-ory. This is so because the Chern-Simons term could not activate a scalarmonopole field φ ( r ) in the static black hole spacetime. This explains whythe scalarized static black holes could not be found in the ECSS theory.In the single scalar coupling of φ = φ = φ , choosing ¯ φ = 0 providesthe Schwarzschild solution. In this case, the linearized equations aroundthe black hole lead to those of the EGBS theory, suggesting the appearanceof scalarized black holes. In this case, the role of the Chern-Simons termdisappears.Furthermore, we note that the Kerr black hole with scalar hair could beobtained from the superradiant instability of Kerr black hole in the Einstein-Klein-Gordon theory [26]. Here the threshold instability is given by the n = 0 , l = m modes of a perturbed scalar and these l = m clouds can bepromoted to Kerr with scalar hair in the full Einstein-Klein-Gordon theory.Also, the non-perturbative spinning black holes could be obtained from theECSS (linear coupling) theory [9]. Its linearized scalar equation takes theform of ¯ ∇ δφ + βδ ( ∗ RR ) = 0 in the Kerr black hole background. In thiscase, however, we do not know the stability issue of Kerr black hole and thus,connection between non-perturbative spinning black holes and instability ofKerr black hole is missed.In our model (1), we may obtain the Kerr black hole solution to (5)-(7)when setting ¯ φ = ¯ φ = 0 with a = J/M because ∗ ¯ R ¯ R = 96 aM r cos ϕ (3 r − a cos ϕ )( r − a cos ϕ )( r + a cos ϕ ) = 0 (55)in Boyer-Lindquist coordinates. In this case, the linearized Einstein equation( δR µν = 0) is completely decoupled from the perturbed scalar δφ , indicatingno instability from metric perturbations. A relevant equation is the linearizedscalar equation of ( ¯ ∇ + 2 β ∗ ¯ R ¯ R ) δφ = 0, which is a Teukolsky-like equationwith an effective mass of − β ∗ ¯ R ¯ R . For β ≤
1, recently, the Kerr black holemay be shown to be unstable against the l = m = 1 and l = m = 2 modes of δφ for the spin parameter a = 0 . s -mode of l = m = 0 was not chosen for the modal stability analysis in thisapproach. Acknowledgments
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MOE) (No. NRF-2017R1A2B4002057).13 ppendix: Ten perturbation equations ( t, t ) : e − iωt E Y = 0( t, r ) : e − iωt E Y = 0( t, ϕ ) : e − iωt (cid:16) E ∂ ϕ Y + O ∂ ϕ Y (cid:17) = 0( t, ϕ ) : e − iωt (cid:16) E ∂ ϕ Y + O ∂ ϕ Y (cid:17) = 0( r, r ); e − iωt E Y = 0( r, ϕ ) : e − iωt (cid:16) E ∂ ϕ Y + O ∂ ϕ Y (cid:17) = 0 (56)( r, ϕ ) : e − iωt (cid:16) E ∂ ϕ Y + O ∂ ϕ Y (cid:17) = 0( ϕ , ϕ ) : e − iωt (cid:16) E Y + E ∂ ϕ Y + O ∂ ϕ Y + O ∂ ϕ ∂ ϕ Y (cid:17) = 0( ϕ , ϕ ) : e − iωt (cid:16) E ∂ ϕ Y + E ∂ ϕ ∂ ϕ Y + O Y + O ∂ ϕ Y (cid:17) = 0( ϕ , ϕ ) : e − iωt (cid:16) E Y + E ∂ ϕ Y + E ∂ ϕ Y + O ∂ ϕ Y + O ∂ ϕ ∂ ϕ Y (cid:17) = 0 . The perturbation equations (56) imply twenty constraints like E i = 0 , O i = 0 , for i = 1 , · · · , . (57)14he explicit forms of E i and O i are as follows: E = 12 r h r ω K ( r ) + r ( − M f − + r λ ) H ( r ) + 2 i (3 M − r ) r ωH ( r ) − rf (2 M − r ω ) H ( r ) + 2 M r f K ′ ( r ) + r (5 M − r ) H ′ ( r ) − ir f ωH ′ ( r ) − M r f H ′ ( r ) − r f H ′′ ( r ) i ,E = i r h r − M ) ωf − K ( r ) − ωrf H ( r ) + 4 ωr K ′ ( r ) − iλH ( r ) i ,E = i r h ωr K ( r ) − iM H ( r ) + ωr f H ( r ) − ir f H ′ i ,E = 12 r f h M (2 r − M ) f − H ( r ) + 2 iM ωr H ( r ) − f (cid:16) M − M r + ω r − r f λ (cid:17) H ( r ) − r (6 M − M r + 2 r ) K ′ ( r ) − M r H ′ +2 iωr f H ′ + rf (6 M − M r + 2 r ) H ′ − r f K ′′ ( r ) + r f H ′′ i ,E = − r f h ( r − M ) rf − H ( r ) − iωr H ( r ) − (2 M − M r + r ) H ( r )+ r f K ′ ( r ) − r H ′ i ,E = − r h M rf − H ( r ) − iωr H ( r ) + 2(2 M + M r − r ) H ( r )+ r ( ω r f − + λ + 2) K ( r ) − r (3 M − r ) K ′ ( r ) − r H ′ ( r ) − r f H ′ ( r ) + r f K ′′ ( r ) i ,E = 12 r h rf − H ( r ) − rf H ( r ) i , E = − E cot ϕ , E = E sin ϕ ,E = E cos ϕ sin ϕ , = csc ϕ r h ir f n ωh ( r ) + ωrh ′ ( r ) − irh ′′ ( r ) o + (4 M − λr ) h ( r ) − βr M f n ψ ( r ) − rψ ′ ( r ) oi ,O = − r h ir f sin ϕ n ωh ( r ) + ωrh ′ ( r ) − irh ′′ ( r ) o − ( rf + 2 M cos 2 ϕ ) csc ϕ h ( r )+ r (csc ϕ − λ sin ϕ ) h ( r ) − βr M f sin ϕ n ψ ( r ) − rψ ′ ( r ) oi ,O = csc ϕ r f h iωr h ( r ) − iωr h ′ ( r ) + (2 rf + ω r − λrf ) h ( r ) − i ˜ βr kM ψ ( r ) i ,O = − r f h iωr sin ϕ { h ( r ) − rh ′ } + n − rf cos ϕ cot ϕ + ( rf + ω r ) sin ϕ + rf (csc ϕ − λ sin ϕ ) o h ( r ) − i ˜ βr kM sin ϕ ψ ( r ) i ,O = csc ϕ cot ϕ r f h iωr h ( r ) + rf n M h ( r ) + r f h ′ ( r ) oi ,O = − O tan ϕ ,O = O λ h sin ϕ tan ϕ i ,O = O sin ϕ tan ϕ ,O = − O sin ϕ ,O = O with ˜ β = 8 ¯ φ β . 16 eferences [1] G. 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