Dirac Equation on Kerr--Newman spacetime and Heun functions
DDirac Equation on Kerr–Newman spacetime and Heunfunctions
Ciprian Dariescu, Marina-Aura Dariescu, Cristian Stelea, , Faculty of Physics, “Alexandru Ioan Cuza” University of Iasi11 Bd. Carol I, Iasi, 700506, Romania Science Research Department, Institute of Interdisciplinary Research, “Alexandru IoanCuza” University of Iasi11 Bd. Carol I, Iasi, 700506, Romania
Abstract
By employing a pseudo-orthonormal coordinate-free approach, the Dirac equationfor particles in the Kerr–Newman spacetime is separated into its radial and angularparts. In the massless case to which a special attention is given, the general Heun-type equations turn into their confluent form. We show how one recovers some resultspreviously obtained in literature, by other means.
Keywords : Dirac Equation; Heun functions; Kerr–Newman spacetime.
PACS: E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] a r X i v : . [ h e p - t h ] F e b Introduction
After Carter found that the scalar wave function is separable in the Kerr–Newman–de Sittergeometries [1], the solutions to the Teukolsky equations [2], for massless fields in the Kerrmetrics, have been analytically expressed in the form of series of various functions [3]; [4].Starting with the work of Chandrasekhar [5], general properties of a massive Dirac fieldequation in the Kerr background have been extensively studied.The recent interest in the so-called quasinormal modes of a Dirac field in the Kerr back-ground is motivated by the detection of gravitational waves [6], [7], [8], whose phase can bedescribed in terms of the proper oscillation frequencies of the black hole.In terms of techniques, after the Dirac equation in the Kerr-Newman background wasseparated [9], [10]], using the Kinnersley tetrad [11], the Newman-Penrose formalism[12] hasbeen considered as a valuable tool for dealing with this subject [13]. This formalism aswell as the Geroch–Held–Penrose variant have been used for the Teukolksy Master Equationdescribing any massless field of different spins, in the Kerr black hole and for an arbitraryvacuum spacetime [14], [15].In [16], [17] it was shown that, for Kerr-de Sitter and Kerr-Newman-de Sitter geometriesboth angular and radial equations for the Teukolsky equation, for massless fields, are trans-formed into Heun’s equation [18], [19] and analytic solutions can be derived in the form ofseries of hypergeometric functions.The massive case was tackled within the WKB approach [20] or numerically, using theconvergent Frobenius method [21]. Very recently, in [21], after a tedious calculation, usinga generalised Kinnersley null tetrad in the Newman-Penrose formalism, the Dirac equationfor a massive fermion has been separated in its radial and angular parts, the solutions beingexpressed in terms of generalised Heun functions.Our work is proposing an alternative, free of coordinates, method based on Cartan’sformalism. Thus, we are computing all the geometrical essentias for dealing with the Diracequation in its SO (3 , × U (1) gauge covariant formulation. Our approach is generalizing thetheory developed in [22], where, for massless fermions on the Kerr space-time, the authorsare switching between canonical and pseudo-orthonormal basis and the solutions are derivedusing numerical techniques.By imposing the necessary condition for a polynomial form of the Heun confluent func-tions [18], [19], we obtain the resonant frequencies, which are of a crucial importance forgetting information on the black holes interacting with different quantum fields [23].In the last years, the Heun functions in either their general or confluent forms have beenobtained by many authors, as for example [25] - [36] and the references therein.2he structure of this paper is as follows: In section 2, we present all the necessaryingredients for writting down the massive Dirac equation in the Kerr-Newman background.We show that, by using an orthonormal tetrad adapted to the Kerr-Newman metric, one canseparate the massive Dirac equation. For slowly rotating objects, the solutions of the radialequations can be expressed in terms of the confluent Heun functions. As an application, wecompute the modal radial current vector. In section 3, we turn our attention to the masslessDirac fermions and show that the Dirac equations can be solved exactly in the Kerr case andalso in the extremal case, a result previously known in literature, obtained by other means.The final section is dedicated to conclusions. SO (3 , × U (1) − gauge covariant Dirac Equation Let us start with the four-dimensional Kerr–Newman metric in the usual Boyer–Lindquistcoordinates, ds = ρ ∆ ( dr ) + ρ ( dθ ) + sin θρ (cid:2) a dt − ( r + a ) dϕ (cid:3) − ∆ ρ (cid:2) dt − a sin θ dϕ (cid:3) , (1)where ∆ = r − M r + a + Q , ρ = r + a cos θ and M , Q and a are the black hole’smass, charge and angular momentum per unit mass. The electromagnetic background of theblack hole is given by the four-vector potential, in coordinate basis, A i dx i = Qrρ (cid:0) dt − a sin θdϕ (cid:1) . (2)Within a SO (3 , − gauge covariant formulation, we introduce the pseudo-orthonormalframe { E a } a =1 , , whose corresponding dual base isΩ = ρ dθ , Ω = sin θρ ( r + a ) dϕ − a sin θρ dt , Ω = ρ √ ∆ dr , Ω = − a √ ∆ ρ sin θ dϕ + √ ∆ ρ dt , (3)leading to the expressions dθ = 1 ρ Ω , dϕ = 1 ρ sin θ Ω + aρ √ ∆ Ω ,dr = √ ∆ ρ Ω , dt = aρ sin θ Ω + r + a ρ √ ∆ Ω . Thus, using the relations g ik dx i dx k = g ik E ia E kb Ω a Ω b = δ ab Ω a Ω b , i.e. dx i = E ia Ω a , one may3rite down the pseudo-orthonormal frame E = 1 ρ ∂ θ , E = 1 ρ sin θ ∂ ϕ + aρ sin θ ∂ t ,E = √ ∆ ρ ∂ r , E = aρ √ ∆ ∂ ϕ + r + a ρ √ ∆ ∂ t . (4)Using (3), the first Cartan’s equation, d Ω a = Γ a. [ bc ] Ω b ∧ Ω c , (5)with 1 ≤ b < c ≤ a. [ bc ] = Γ a.bc − Γ a.cb , can be explicitely worked out as d Ω = − √ ∆ ρ ρ, Ω ∧ Ω ,d Ω = 1sin θ (cid:18) sin θρ (cid:19) , Ω ∧ Ω − r √ ∆ ρ Ω ∧ Ω + 2 ar sin θρ Ω ∧ Ω ,d Ω = ρ, ρ Ω ∧ Ω ,d Ω = − a √ ∆ ρ cos θ Ω ∧ Ω − a ρ sin θ cos θ Ω ∧ Ω + (cid:32) √ ∆ ρ (cid:33) , Ω ∧ Ω , where ( · ) , and ( · ) , are the derivatives with respect to θ and r , leading to the followingcomplete list of non-zero connection coefficients in the Cartan frames { Ω a , E a } a =1 , :Γ = − Γ = − θ (cid:18) sin θρ (cid:19) , , Γ = − Γ = − Γ = − a √ ∆ ρ cos θ , Γ = − Γ = √ ∆ ρ ρ, , Γ = − Γ = − ρ, ρ , Γ = − Γ = r √ ∆ ρ , Γ = − Γ = arρ sin θ , Γ = − Γ = − a √ ∆ ρ cos θ , Γ = − Γ = − a ρ sin θ cos θ , Γ = − Γ = Γ = a √ ∆ ρ cos θ , Γ = − Γ = − Γ = − arρ sin θ , Γ = − Γ = − Γ = − arρ sin θ , Γ = − Γ = (cid:32) √ ∆ ρ (cid:33) , . (6)Now, one has all the essentials to write down the SO (3 , × U (1) gauge-covariant Diracequation for the fermion of mass µ , γ a Ψ ; a + µ Ψ = 0 , (7)4here “;” stands for the covariant derivativeΨ ; a = Ψ | a + 14 Γ bca γ b γ c Ψ − iqA a Ψ , (8)with Ψ | a = E a Ψ.In view of the relations (6), the term expressing the Ricci spin-connection14 Γ bca γ a γ b γ c = 12 [Γ + Γ − Γ ] γ + 12 [Γ + Γ − Γ ] γ + i γ γ + i γ γ has the concrete expression14 Γ bca γ a γ b γ c = 12 (cid:20) cot θρ − a ρ sin θ cos θ (cid:21) γ + 12 (cid:34) ( √ ∆) , ρ + r √ ∆ ρ (cid:35) γ + iar ρ sin θ γ γ + ia √ ∆2 ρ cos θ γ γ , where γ = − iγ γ γ γ , while the kinetic term reads γ a Ψ | a = 1 ρ γ Ψ , + γ (cid:20) ρ sin θ Ψ , + aρ sin θ Ψ , (cid:21) + √ ∆ ρ γ Ψ , + γ (cid:20) aρ √ ∆ Ψ , + r + a ρ √ ∆ Ψ , (cid:21) . Putting everything together, the Dirac equation (7) has the explicit form γ (cid:26) ρ Ψ , + (cid:18) cot θ ρ − a ρ sin θ cos θ + iar ρ sin θγ (cid:19) Ψ (cid:21) + γ (cid:26) ρ sin θ Ψ , + aρ sin θ Ψ , (cid:27) + γ (cid:40) √ ∆ ρ Ψ , + (cid:32) ( √ ∆) , ρ + r √ ∆2 ρ + ia √ ∆2 ρ cos θγ (cid:33) Ψ (cid:41) + γ (cid:26) aρ √ ∆ Ψ , + r + a ρ √ ∆ Ψ , (cid:27) − iqγ A Ψ + µ Ψ = 0 , (9)where the proper component of the four-potential, coming from A ( c ) i dx i = A Ω , with A ( c ) i given in (2), reads A = Qrρ √ ∆ . (10)For ease of calculations, the choice for γ a matrices is important and we are going toemploy the Weyl’s representation γ µ = − iβ α µ , γ = − iβ , (11)5ith α µ = (cid:18) σ µ − σ µ (cid:19) , β = (cid:18) − I − I (cid:19) , so that γ = − iγ γ γ γ = (cid:18) I − I (cid:19) . Thus, for the bi-spinor written in terms of two components spinors asΨ = (cid:20) ζη (cid:21) , (12)the general equation (9) leads to the following system of coupled equations for the spinors ζ and η : σ (cid:20) ρ ζ, + (cid:18) cot θ ρ + ia sin θ ρ ρ + (cid:19) ζ (cid:21) + σ (cid:20) ρ sin θ ζ, + aρ sin θζ, (cid:21) + σ (cid:34) √ ∆ ρ ζ, + (cid:32) ( √ ∆) , ρ + √ ∆2 ρ ρ + (cid:33) ζ (cid:35) + aρ √ ∆ ζ, + r + a ρ √ ∆ ζ, − iqA ζ − iµη = 0 (13)and σ (cid:20) ρ η, + (cid:18) cot θ ρ − ia sin θ ρ ρ − (cid:19) η (cid:21) + σ (cid:20) ρ sin θ η, + aρ sin θη, (cid:21) + σ (cid:34) √ ∆ ρ η, + (cid:32) ( √ ∆) , ρ + √ ∆2 ρ ρ − (cid:33) η (cid:35) − aρ √ ∆ η, − r + a ρ √ ∆ η, + iqA η + iµζ = 0 , (14)where ρ ± = r ± ia cos θ and ρ = ρ + ρ − .Due to the time independence and symmetry of the spacetime, we can assume that thewave function can be written as ζ = ∆ − / ρ − / − e i ( mϕ − ωt ) X ( ρ, θ ) , η = ∆ − / ρ − / e i ( mϕ − ωt ) Y ( ρ, θ ) , (15)where the factors ∆ − / ρ − / ± have been introduced in order to pull some terms out of equa-tions (13) and (14).With the new functions X ( ρ, θ ) and Y ( ρ, θ ), the equations (13) and (14) can be put intothe transparent form σ D θ X + iσ HX + σ √ ∆ X, + iKX − iµρ − Y = 0 ,σ D θ Y + iσ HY + σ √ ∆ Y, − iKY + iµρ + X = 0 , (16)6here we have introduced the operators D θ = ∂∂θ + cot θ , H = m sin θ − ωa sin θ ,K = 1 √ ∆ (cid:2) ma − ω ( r + a ) − qQr (cid:3) . (17)Finally, by applying the separation ansatz X = R ( r ) T ( θ ) , X = R ( r ) T ( θ ) , Y = R ( r ) T ( θ ) , Y = R ( r ) T ( θ ) , (18)one gets the system R [ D θ − H ] T − T (cid:104) √ ∆ ∂ r − iK (cid:105) R − iµρ − R T = 0 R [ D θ + H ] T + T (cid:104) √ ∆ ∂ r + iK (cid:105) R − iµρ − R T = 0 R [ D θ − H ] T − T (cid:104) √ ∆ ∂ r + iK (cid:105) R + iµρ + R T = 0 R [ D θ + H ] T + T (cid:104) √ ∆ ∂ r − iK (cid:105) R + iµρ + R T = 0 , (19)which leads to the radial and angular equations (cid:104) √ ∆ ∂ r + iK (cid:105) R = ( λ + iµr ) R , (cid:104) √ ∆ ∂ r − iK (cid:105) R = ( λ − iµr ) R , [ D θ − H ] T = ( λ + µa cos θ ) T , [ D θ + H ] T = ( − λ + µa cos θ ) T , (20)where λ is a separation constant.The first-order angular equations may be combined to obtain the so-called Chandrasekhar-Page angular equation and have been discussed in detail in [37].From the radial equations in (20), one gets the following second order differential equationfor the R component:∆ R (cid:48)(cid:48) + (cid:20) r − M − iµ ∆ λ + iµr (cid:21) R (cid:48) + (cid:34) i √ ∆ K (cid:48) + µK √ ∆ λ + iµr + K − λ − µ r (cid:35) R = 0 , (21)and i → − i , for R .Similar relations have been obtained in [21], by a different approach, namely using theNewman–Penrose formalism. In the generalised Kinnersley frame, the null tetrad have beenconstructed directly from the tangent vectors of the principal null geodesics. Even thoughthe radial and angular equations coming from (20) have been reduced to generalised Heundifferential equations [18], [19], the solutions are not physically transparent since they look7uite complicated and there are many open questions especially related to their normalizationor to the behavior around the singular points.However, for large values of the coordinate r , the equation (21), with K given in (17),reads r (cid:18) − Mr (cid:19) R (cid:48)(cid:48) + (cid:18) M − a + Q r (cid:19) R (cid:48) + (cid:20) − iωr − iqQ + 2 i Ω r ( r − M ) r − M + Ω r r − M − λ − µ r (cid:21) R = 0 , (22)with the notation Ω = ω + qQr + a r (cid:16) ω − ma (cid:17) , (23)where one may identify the fermion’s quanta energy, ω , the standard Coulomb energy, qQ/r ,and the internal centrifugal energy with the quantum resonant correction, i.e. ω − m/a .To first order in a , meaning a slowly rotating object, for whichΩ ≈ ω + qQr − mar , (24)and Ω ≈ (cid:18) ω + qQr (cid:19) − ωmar , the solutions of (22) are given in terms of the Heun Confluent functions [18], [19] as: R ∼ e ipr ( r − M ) + γ r / (cid:110) C r β/ HeunC (cid:104) α, β, γ, δ, η, r M (cid:105) + C r − β/ HeunC (cid:104) α, − β, γ, δ, η, r M (cid:105)(cid:111) (25)with the parameters written in the physical transparent form as: α = 4 ipM , β = (cid:114) − imaM ≈ − ima M ,γ = 4 iM (cid:34)(cid:18) Ω ∗ + i M (cid:19) + 364 M (cid:35) / ≈ iM Ω ∗ ,δ = 8 M (cid:20) ω (cid:18) ω + qQ M (cid:19) − µ (cid:21) , η = 58 − λ , (26)where p = ω − µ and Ω ∗ is the energy computed on the Schwarzschild horizon, i.e.Ω ∗ = ω + qQ M − ma M . The second component, R , is given by the complex conjugated expression of (25).8ne may notice that, for ∆ ≈ r ( r − M ) and ρ + ≈ ρ − ≈ r , the first component in ζ defined in (15) reads ζ = e αx/ e imϕ e − iωt ( x − γ/ x ± β/ HeunC [ α, ± β, γ, δ, η, x ] , with x = r/ (2 M ).Moreover, since | R | = | R | , the modal radial current (of quantum origin), computedas j r = i ¯Ψ γ Ψ = Ψ † α Ψ = ζ † σ ζ − η † σ η , vanishes. The only non-vanishing component is the azimuthal one, which is given by theexpression j ϕ = i ¯Ψ γ Ψ = Im (cid:2) e αx ( x − γ x β HeunC [ α, β, γ, δ, η, x ] (cid:3) T T = x / Im (cid:26) exp (cid:20) ipM x + 4 iM Ω ∗ ln( x − − ima M ln( x ) (cid:21) [ HeunC ] (cid:27) T T (27)The current has the generic representation given in the figure 1, for x >
1, i.e. r > M .One may notice the oscillating behavior, with both positive and negative regions, vanishingat infinity. Also, there is a dominant positive maximum, just after the (Schwarzschild)horizon r = 2 M of the slowly rotating black hole, where the Heun functions have a regularsingularity.For the asymptotic behavior in the neighborhood of the singular point at infinity, wherethe two solutions of the confluent Heun equation exist, one may use the formula [23] HeunC [ α, β, γ, δ, η, x ] ≈ D x − [ β + γ +22 + δα ] + D e − αx x − [ β + γ +22 − δα ]= e − αx x − β + γ +22 (cid:110) D e αx x − δα + D e − αx x δα (cid:111) = De − αx x − β + γ +22 sin (cid:20) − iαx iδα ln x + σ (cid:21) , (28)so that the two independent solutions in (25) are given by the simple expression R = D sin (cid:26) pr + 2 Mp (cid:20) ω (cid:18) ω + qQ M (cid:19) − µ (cid:21) log (cid:16) r M (cid:17) + σ (cid:27) , (29)where σ ( ω ) is the phase shift, D = const and p = (cid:112) ω − µ .Thus, the first component of Ψ defined in (12), (15) and (18) has the following (physical)behavior for large r values ζ ≈ R r e i ( mϕ − ωt ) T ( θ ) , and similarly for the other three spinor’s component built with (18).Such analytical solutions of the radial part of the Dirac equation, computed far from theblack hole, are useful to investigate the scattering of charged massive fermions.9igure 1: The radial part of the current (27), for x > In the particular case of massless fermions, the Dirac equation can be solved exactly, itssolutions being given by the Heun Confluent functions. In view of the analyzis presented in the previous section, for µ = 0, the system (20) getsthe simplified form (cid:104) √ ∆ ∂ r + iK (cid:105) R = λR , (cid:104) √ ∆ ∂ r − iK (cid:105) R = λR , [ D θ − H ] T = λT , [ D θ + H ] T = − λT , (30)which firstly leads to the radial Teukolsky equations∆ R (cid:48)(cid:48) A + ( r − M ) R (cid:48) A + (cid:104) ± i √ ∆ K (cid:48) + K − λ (cid:105) R A = 0 , (31)where A = 1 ,
2, the prime denotes the derivative with respect to r and K can be writtenfrom (17) putting q = 0. Fermionic one-particle states in Kerr backgrounds have been considered in [38]. R = ∆ / e αz/ ( z − γ/ × (cid:8) C z β/ HeunC [ α, β, γ, δ, η, z ]+ C z − β/ HeunC [ α, − β, γ, δ, η, z ] (cid:9) , (32)of variable z = r − r − r + − r − , where r ± = M ± √ M − a are the outer and inner horizons and parameters α = 2 iω ( r + − r − ) , β = 12 + 2 i ( r + − r − ) (2 ωM r − − ma ) ,γ = −
12 + 2 i ( r + − r − ) (2 ωM r + − ma ) , δ = ω (4 M ω − i )( r + − r − ) ,η = ω (4 M ω − i ) r − − ω a − a M − a (cid:16) ωa − m (cid:17) − λ + 38 . (33)For the case under consideration with a < M , the two horizons are real, while for anoverspinning Kerr spacetime with a > M , the quantities r + and r − are complex. Thesolutions to Heun’s Confluent equations are computed as power series expansions aroundthe regular singular point z = 0, i.e. r = r − . The series converges for z <
1, where thesecond regular singularity is located. An analytic continuation of the HeunC function isobtained by expanding the solution around the regular singularity z = 1 (i.e. r = r + ), andoverlapping the series.For the polynomial form of the Heun functions, one has to impose the necessary condition[18], [19] δα = − (cid:20) n + 1 + β + γ (cid:21) , which gives us the resonant frequencies associated with the massless fermion.In view of the parameters in (33), it turns out that only the component multiplied by C gets a polynomial expression, the energy ω having the real and imaginary parts given by ω R = mar − ( r + + r − ) , ω I = (cid:18) n + 12 (cid:19) r + − r − r − ( r + + r − ) , (34)where m and n are the azimuthal and the principal quantum numbers.To first order in a /M , the above expressions become ω R ≈ ma , ω I ≈ (cid:18) n + 12 (cid:19) Ma , (35)and they depend only on the BH parameters.11ext, for a polynomial which truncates at the order n , once we set the n + 1 coefficientin the series expansion to vanish, we get the separation constant λ expressed in terms of theblack hole’s parameters.For the asymptotic behavior at infinity, one may use the formula (28) and the expression(32) turns into the simplified form R ≈ ∆ / r sin (cid:20) ωr + (cid:18) ωM − i (cid:19) log (cid:16) r M (cid:17) + σ (cid:21) ≈ ∆ / √ r exp (cid:110) i (cid:104) ωr + 2 ωM log (cid:16) r M (cid:17) + σ (cid:105)(cid:111) , (36)where σ ( ω ) is the phase shift.In order to study the radiation emitted by the black hole, one has to write down the wavefunction components near the exterior horizon, r → r + . Using (32), for z →
1, the (radial)components of Ψ defined in (12), (15) and (18) can be writte asΨ out ∼ e − iωt ( r − r + ) ir + − r − (2 ωMr + − ma ) (37)By definition, the component ψ out near the event horizon should asymptotically have theform [23] Ψ out ∼ ( r − r h ) i κh ( ω − ω h ) (38)and the scattering probability at the exterior event horizon surface is given byΓ = (cid:12)(cid:12)(cid:12)(cid:12) Ψ out ( r > r + )Ψ out ( r < r + ) (cid:12)(cid:12)(cid:12)(cid:12) = exp (cid:20) − πκ h ( ω − ω h ) (cid:21) . (39)In our case, using the explicit expressions κ h = r + − r − M r + = √ M − a M ( M + √ M − a ) , ω h = ma M r + , (40)we get the Bose–Einstein distribution for the emitted particles N = Γ1 − Γ = 1 λe ωT − , with T = κ h π = √ M − a πM ( M + √ M − a ) (41)and λ = exp (cid:20) − π ω h κ h (cid:21) = exp (cid:20) − πmar + − r − (cid:21) . One may notice that the expression of the temperature (41) agrees with the one obtainedfollowing the usual thermodynamical procedure. Thus, by using the formula of the entropy12 = π (cid:2) r + a (cid:3) , with r + = M + √ M − a , and a = J/M , we express the mass in terms ofthe entropy as M = r + a r + = (cid:20) S π + πJ S (cid:21) / and compute the temperature on the event horizon as the following derivative T = ∂M∂S = S − π J √ πS (cid:20) SS + 4 π J (cid:21) / = r − a πr + ( r + a ) = √ M − a πM r + . (42)The corresponding heat capacity at constant angular momentum, i.e. C J = T (cid:18) ∂S∂T (cid:19) J = − S ( S − π J ) S − π J S − π J = 2 π ( r − a )( r + a ) a + 6 r a − r , (43)is positive for the following range of the parameter a/M : (cid:104) √ − (cid:105) / < aM < , for which the thermal system is stable on the event horizon.For a slowly rotating black hole with a/M < (cid:2) √ − (cid:3) / , the heat capacity becomesnegative, corresponding to a thermodynamically unstable phase.A particular value of a/M where the Kerr–Newman black hole undergoes a phase transi-tion and the heat capacity has an infinite discontinuity was found many years ago by Davies[24].Secondly, the angular equations coming from the system (20), i.e. T (cid:48)(cid:48) A + cot θT (cid:48) A + (cid:20) − cot θ − ∓ H (cid:48) − H + λ (cid:21) T A = 0 , (44)where prime means the derivative with respect to θ , for ξ = cos θ , is the spheroidal Teukolskyequation. However, for y = cos θ , the solutions are given by the Heun Confluent functionsas T = e ωa cos θ (cid:18) cos θ (cid:19) γ (cid:40) C (cid:18) sin θ (cid:19) β HeunC [ α, β, γ, δ, η, y ]+ C (cid:18) sin θ (cid:19) − β HeunC [ α, − β, γ, δ, η, y ] (cid:41) (45)and similarly for T , with the real parameters α = 4 ωa , β = m + 12 , γ = m − , δ = − ωa ,η = (1 − m ) ωa − λ + m . (46)13s expected, for given parameters of the black hole ( M, a ), the Dirac solutions are enu-merated by the halfinteger positive multipole number m ± /
2. Since β is not integer, thetwo functions in (45) form linearly independent solutions of the confluent Heun differentialequation.Similar expressions have been obtained for the solutions of the Klein-Gordon equationdescribing a charged massive scalar field in the Kerr-Newman spacetime [23], [42].Up to a normalization constant A , the first component of Ψ defined in (12), (15) and(18) has the following behavior for large r valuesΨ ≈ Ar exp (cid:110) i (cid:104) ωr + 2 ωM log (cid:16) r M (cid:17) + σ (cid:105)(cid:111) e i ( mϕ − ωt ) T ( θ ) , (47)while the other components can be easily built using the relations (18).Let us notice that, by introducing the new coordinate r ∗ = r + 2 M log (cid:0) r M (cid:1) , the radialpart of the above component has the form obtained by Starobinsky, for the Klein–Gordonequation in the Kerr metric, [39], namely R ∼ r (cid:2) Ae iωr ∗ + Be − iωr ∗ (cid:3) , where A and B are for the incident and reflected wave coefficients, respectively. The extreme Kerr metric can be easily written from (1), by setting the Kerr parameter a equal to M , so that there is a single (degenerate) horizon at r = M with zero Hawkingtemperature and horizon angular velocity Ω H = 1 / (2 M ). Thus, for the massless case, theradial equation (31) has the same form, but with ∆ = ( r − M ) and K = mM − ω ( r + M ) r − M .
The solutions are given by the Heun Double Confluent functions [18], [19] as being R ∼ (cid:26) C exp (cid:20) − iω ( r − M ) − ikMr − M (cid:21) HeunD [ α, β, γ, δ, ζ ]+ C exp (cid:20) iω ( r − M ) + ikMr − M (cid:21) HeunD [ − α, β, γ, δ, ζ ] (cid:27) (48)with k = 2 M (cid:16) ω − m M (cid:17) , ζ = r − M + i (cid:113) kMω r − M − i (cid:113) kMω (49)14nd the parameters α = − i √ ωkM , β = − ωM √ ωkM + 16 ω M − λ ,γ = 2 α , δ = − ωM √ ωkM − ω M + 4 λ . (50)Usually, the double confluent Heun functions are obtained from the confluent ones,through an additional confluence process [18], [19].One may notice that, for ω m = m/ (2 M ) and r = M , one has to deal with the irregularsingularities, at ζ = ±
1. For ω < m/ (2 M ), the variable in (49) is real. Since the pioneering works of Teukolsky [2] and Chandrasekhar [5] the study of the solutionsof the massive Dirac equation in the background of an electrically charged black hole has along history.The method used in the present paper, while based on Cartan’s formalism with anorthonormal base, is an alternative to the Newman-Penrose (NP) formalism [12], which isusually employed for solving Dirac equation describing fermions in the vicinity of differenttypes of black holes.The solutions to the radial Teukolsky equations (31), with two regular singularities at r = r ± and an irregular singularity at r = ∞ , have been written in the form of series ofhypergeometric functions [4]. Similar expressions as the ones in (32) have been found forthe exact solutions of the Teukolsky master equation for electromagnetic perturbations ofthe Kerr metric [40] and in the study of bosons in a Kerr–Sen black hole [41].By imposing the necessary condition for a polynomial form of the Heun confluent func-tions [18], [19], we get the resonant frequencies, which are of a crucial importance for gettinginformation on the black holes interacting with different quantum fields [23].By identifying the out modes near the r + horizon, one is able to compute the scatteringprobability (39) and the Bose–Einstein distribution of the emitted particles. For a = 0, weidentify the expected Hawking black body radiation and the Hawking temperature T h =1 / (8 πM ). 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