The massive Dirac equation in the Kerr-Newman-de Sitter and Kerr-Newman black hole spacetimes

Exact solutions of the Dirac general relativistic equation that describe the dynamics of a massive, electrically charged particle with half-integer spin in the curved spacetime geometry of an electrically charged, rotating Kerr-Newman-(anti) de Sitter black hole are investigated. We first, derive the Dirac equation in the Kerr-Newman-de Sitter (KNdS) black hole background using a generalised Kinnersley null tetrad in the Newman-Penrose formalism. Subsequently in this frame and in the KNdS black hole spacetime, we prove the separation of the Dirac equation into ordinary differential equations for the radial and angular parts. Under specific transformations of the independent and dependent variables we prove that the transformed radial equation for a massive charged spin 12 fermion in the background KNdS black hole constitutes a highly non-trivial generalisation of Heun’s equation since it possess five regular finite singular points. Using a Regge-Wheeler-like independent variable we transform the radial equation in the KNdS background into a Schrödinger like differential equation and investigate its asymptotic behaviour near the event and cosmological horizons. For the case of a massive fermion in the background of a Kerr-Newman (KN) black hole we first prove that the radial and angular equations that result from the separation of Dirac’s equation reduce to the generalised Heun differential equation (GHE). The local solutions of such GHE are derived and can be described by holomorphic functions whose power series coefficients are determined by a four-term recurrence relation. In addition using asymptotic analysis we derive the solutions for the massive fermion far away from the KN black hole and the solutions near the event horizon . The determination of the separation constant as an eigenvalue problem in the KN background is investigated. Using the aforementioned four-term recursion formula we prove that in the non-extreme KN geometry there are no bound states with ω2


Introduction
The problem of massive perturbations in the strong gravity background of a black hole is fascinating and fundamentally significant problem, as has been demonstrated recently for the case of scalar perturbations [1][2][3] which has acquired extra impetus after the discovery of gravitational waves in nature [4,5].
In particular in [1] the Klein-Gordon-Fock equation for a massive charged particle in the background of the Kerr-Newman-(anti) de Sitter black hole was separated. The resulting radial and angular equations, for particular values of the inverse Compton wavelength in terms of the cosmological constant Λ, were reduced to a Heun form. Then both Heun equations were solved in terms of an infinite series of hypergeometric functions using the idea of augmented convergence . In this setup, the solution converges inside the ellipse with foci at two of the finite regular singularities and passing through the third finite regular singular point with the possible exception of the line connecting the two foci. This method of constructing a Heun function 1 offers a perspective section 3 we prove by applying appropriate transformations of the dependent and independent variables that the separated massive radial equation for the spin half charged fermion in KNdS spacetime is a highly non-trivial generalisation of the Heun equation since it possess five regular finite singularities-see equation (62). Furthermore in 3.1 we derive the asymptotic forms of the radial equation in the KNdS black hole background. Using a Regge-Wheeler-like coordinate, equations (65), (68) we first transform the massive radial equation in the presence of the cosmological constant into equation (66). We then derive the asymptotics of (66) near the event horizon and near the cosmological horizon, equations (69) and (70) respectively. In section 4 we derive in the zero cosmological constant limit the separated radial and angular ODEs. In sections 4.2 and 4.3 we transform both radial and angular ODEs into the generalised Heun form, equations (104), (105)and (116) respectively. In section 5 we discuss the analytic local solutions of the GHE. As is shown there, the coefficients in the series expansion around the simple singularities obey a four-term recursion relation. From the various local solutions one can derive various important identities. In section 6 making use of the fact that the radial GHE in KN spacetime has an irregular singularity at ¥ we derive the asymptotic solutions near infinity i.e. far away from the black hole. In section 6.1 following the work in [28] we investigate the global solution associated with the connection problem between the regular singularity at 1 and the irregular one at ¥ for the GHE which after appropriate transformations of the dependent and the independent variables transforms to a connection relation in which the connection coefficients can be computed. Such connection coefficients involve besides the index parameters and the parameter α of the GHE the expansion coefficients in the transformed local solution. Again these coefficients are proved to obey a four-term recursion relation. This is of importance for constructing a global solution relating the event horizon with ¥ in the case of the KN black hole. In section 7 we derive the near horizon limit solution for the KN black hole using the local solution around the regular singularity associated with the event horizon. The theme of section 9 is the determination of the separation constant λ in KN-spacetime that also appears nontrivially in the radial equation (116). Following the approach in [22] we write the angular equation as an eigenvalue matrix equation. Using functional analysis techniques and linear operator perturbation theory [31] we prove that the λ-eigenvalues depend holomorphically on the two physical parameters ν, ξ defined in section 9 and their partial derivatives with respect to them are bounded by trigonometric functions and therefore by 1 (see equation (181)) 4 . The eigenvalues obey a particular partial differential equation (207). In section 9.1 we prove using Charpit's method that this differential equation reduces to the third Painlevé transcendent P III , a nonlinear ordinary differential equation-see equation (218) 5 . In section 9.2 we derive special closed form solutions for the P III the angular eigenvalues obey, for particular values of the parameters in terms of Bessel functions. In these special cases, the eigenvalues of the angular KN differential equation are expressible in closed analytic form in terms of Bessel functions. In section 9.3 we investigate a novel approach in which we derive the asymptotic solutions of Painlevé P III in terms of Jacobian elliptic functions. Therefore, the third Painlevé transcendent is asymptotically related to the Jacobian elliptic function. This is analogous to the scattering theory of ordinary quantum mechanics in which the Bessel functions have an asymptotic expansion in terms of trigonometric functions. The angular eigenvalues themselves 6 in this asymptotic elliptic limit of the transcendent P III are expressed in terms of a reduced form of the Jacobian elliptic functions sn,dn, and cn. In section 10.1 we prove that no fermionic bound states with 2 2 w m < , where ω is the energy of the fermion and μ its mass, exist in the nonextreme Kerr-Newman geometry.
We achieve that by using the fundamental four-term recursion relation the coefficients in the power series expansion of a closed form analytic solution of the radial GHE in the KN geometry satisfy.
In appendix A we discuss the connection problem for the regular singularities of the GHE. In appendix B we explore the representation of the Painlevé P III nonlinear ordinary differential equation (ODE) as the compatibility condition for a Lax pair of first order linear systems as was introduced in the works [35] and [36] 7 . This isomonodromic deformation method in the theory of Painlevé appears suitable for both integrating the Painlevé transcendents as well as for studying their asymptotic behaviour.
Taking into account the contribution from the cosmological constant , L the generalisation of the Kerr-Newman solution is described by the Kerr-Newman de Sitter (KNdS) metric element which in Boyer-Lindquist (BL) coordinates is given by [38][39][40][41] (in units where G=1 and c = 1): 4 An essential ingredient in obtaining these estimates is the computation of the operator norm of a matrix. 5 The mathematical importance of the six Painlevé transcendents stems from the fact that they belong to the class of ordinary differential equations of the form F y x , , y x y x d d d d 2 2 = ( ) where F is a rational function of y x d d and y, and an analytic function of x, which have the property that their solutions are free from movablecritical points [32,33]. 6 An additional physical significance of the knowledge of the angular eigenvalues for a rotating charged black hole is that they can be used in where a M e , , , denote the Kerr parameter, mass and electric charge of the black hole, respectively. The KN(a)dS metric is the most general exact stationary black hole solution of the Einstein-Maxwell system of differential equations. This is accompanied by a non-zero electromagnetic field F A d , = where the vector potential is [41,42]:

Null tetrad and the Dirac equation in Kerr-Newman-de Sitter black hole spacetime
The Kerr-Newman-de Sitter geometry (as was the case with the Kerr and Kerr-Newman geometries) can be described in terms of a local Newman-Penrose null tetrad frame that is adapted to the principal null geodesics, i.e. the tetrad coincides with the two principal null directions of the Weyl tensor C mnre . In this generalised Kinnersley frame, the null tetrad is constructed directly from the tangent vectors of the principal null geodesics: The Ricci rotation coefficients a b c g ( )( )( ) are expressed through the λ-coefficients as follows: we obtain the following 2-spinor form of the Dirac equation ṁ˙, q is the charge or the coupling constant of the massive Dirac particle to the vector field and * m is the particle mass. Thus equivalently the 2-spinor form of general relativistic Dirac's equation is the following: where assuming that the azimuthal and time-dependence of the fields will be of the form e i m t f w -( ) we calculate the directional derivatives to be

Separation of the Dirac equation in the Kerr-Newman-de Sitter spacetime
Applying the separation ansatz: we obtain the following ordinary differential equations for the radial and angular polar parts: Taking the 0 L = limit of (39) yields the differential equation: where K r r a ma eqr r eqr . Applying to (41) the independent variable transformation: r r z r r , 4 2 so that the quartic polynomial that determines the horizons of the Kerr-Newman-de Sitter black hole factorises as follows 8 : with the notation:     8 The quantity r KN D in terms of the radii of the event and Cauchy horizons r r , + -and the cosmological horizon r L + for positive cosmological constant is written as: r r r r r r r r In equation (46), we use the notation R z R r z r r . Let us calculate the exponents of the five singular points in (46). For the singularity at z=0 the indicial equation becomes Likewise the exponents at the singular points z z z z z 1, , 3 4 = = = read as follows: On the other hand the exponents at the fifth singular point z z 5 = are computed to be 0, 2 { }, i.e. they are both integers. Now by applying the F-homotopic transformation of the dependent variable: We observe that equation (62) possess five regular finite singularities and therefore constitutes a highly nontrivial generalisation of the Heun differential equation-the latter has three finite regular singular points.
As we shall see in section 4 for zero cosmological constant the corresponding radial equation-see equation (116)-also leads to a generalisation of Heun's equation, in particular it has the specific mathematical structure of a GHE, however with fewer finite singularities than equation (62).

Asymptotic forms of the radial equation in KNdS spacetime
The investigation of the asymptotic forms of the radial equation (41) and using the Regge-Wheeler-like (or "tortoise") coordinate 3.1.1. The near event horizon limit r  r + In the near event horizon limit r  r + (r*  -¥), equation (

The general relativistic Dirac equation in the Kerr-Newman spacetime
In the case of the Kerr-Newman spacetime (KN) the Kinnersley null tetrad is a special case of (7) for 0 L = and takes the form: In the KN spacetime the non-vanishing λ-symbols are computed by setting 0 L = in the expressions (14), a procedure that yields: while the Dirac equations in the curved background of the KN black hole spacetime have the same general form as in the more general KNdS case: however with different Ricci coefficients and differential operators for the directional derivatives calculated from those of the KNdS case by setting 0 L = . In equations (77), (78): We have also made use of the tetrad formalism to define associated local spinor components for the original 2-spinor components y y , The spin coefficients are defined generically as

Separation of the general relativistic Dirac equation in the Kerr-Newman space-time
Applying the separation ansatz: we obtain the following ordinary radial and angular differential equations The radial equation for the R -( ) mode can be written as follows [16]:

Transforming the angular KN equation into a generalised Heun form
Using the variable x cos q ≔ the previous angular differential equation (40) is written: , 97 . Equation (97) (100) is a particular case of the following generic form: n a a n n a a n = ¢ + ¢ + ¢ = +

Transforming the radial KN equation into a generalised Heun form
We will apply the following transformation of variables to the radial equation where R z R r z where 1  is the coefficient of the term The fact that the radial equation in the KN spacetime has the mathematical structure of a GHE was also noted in [18], on the basis of a computation using a Carter tetrad.

Analytic solutions of the generalised Heun equation
where a 0, 1  Î ⧹{ } and 0 a ¹ and , j j m b are complex parameters. This equation with three regular singular points and one irregular singular point at a 0, 1, and ¥ has been discussed in [28]. In this form the exponents at the singularities z a 0, 1, = are respectively 0, , 0, , 0, Because of the symmetry of (120) in the parameters , , ) . 11 We note that when 0 a = and 0 , ¥ is also a regular singularity and (120) reduces to Heun's equation.
Substituting (122) into (120) we obtain recursion relations for the coefficients T k : T  T  a  T  a  T  1  1   1  2  2 which are summarised in the four-term recurrence relation for the T k The automorphism group of the generalised Heun equation has been investigated in [28] by studying the index transformations:  Table 1, appendix A. Using these results for the automorphism group one can define for each j 0, 1, 2 = a set of two Floquet solutions y y , with T a , , , ) .

Asymptotic solutions at infinity r  ¥
We can also obtain the far horizon limit of our closed form analytic radial solutions as follows. The radial GHE (116) is a differential equation with an irregular singularity at infinity. Following the discovery of Thomé that such a differential equation can be satisfied in the neighbourhood of an irregular singularity by a series of the form [29] Y e a 140 6.1. The connection problem for a regular and the irregular singular point at ¥ for the radial generalised Heun equation More exactly then in general we know for second order equations such as (120) that there exist uniquely determined solutions y z 1 ( ) and y z 2 ( ) defined in [29] From [28] we know that (120) and (108) have one solution y(z) that is holomorphic near 1 with y z form (120)), which can be analytically continued to a neighbourhood in 1, .
[ The corresponding connection problem [30] is to decompose the solution y(z) in the form: y z y z y z z z , 1, , sufficiently large . 148 Following [30] we shall transform the connection problem between 1 and ¥ by defining z t ). Indeed, we shall apply the combined transformation Starting from (120), one proves that the new dependent variable v(t) satisfies the differential equation: 0, 160 The following theorem, which was proven in [30], supplies a limit formula for the connection coefficient 2 g : Theorem 2. Let 2 g be the connection coefficient for the connection problem (144)-(148) . Assume the coefficient 1 m in (120) is not a positive integer e.g. that 1, 2, 3, Moreover the above sequence with limit 2 g has an asymptotic expansion, involving powers of k 1 .
As it is explained in [30] all connection coefficients between 1 and ¥ can be computed once a formula for 2 g is known. The necessity of the conditions on the GHE parameters in the above theorem is also discussed in [30].
For fixed values of ξ and ν, the differential operator A generated by the left hand side of (179) is a self-adjoint operator acting on the Hilbert L 0, , 2 sin 2 2 p q (( ) ) of square integrable vector functions with respect to the weight function 2 sin q.
The operator A A , x n = ( )as well as its eigenvalues m, , j j l l x n = ( )depend holomorphically on , x n. The partial derivatives of the differential operator A are given by: Using analytic perturbation theory, in particular theorem 3.6, VII, section 3 in [31], we obtain for the eigensystem (179) the following estimates: In (181) the function   · denotes the operator or (spectral) norm of a matrix: ) denote the space of continuous linear operators X Y  , where X and Y are normed ) is a vector space with a norm defined by [47]: 13 ) is complete when Y is complete. For a matrix A the operator norm is related to the spectral radius: Thus for a real 2×2 matrix A we compute: Alternatively we can express the angular equations in the following manner: . We will first determine the eigenvalues of (185) for a=0 (non-rotating black hole). For this we will need the following lemmas concerning Jacobi polynomials.
Lemma 6 x P x P x P x 1 2 2 1 2 1 186 Proof. We will prove it using the relation of Jacobi polynomials to hypergeometric functions. Indeed we have: 13 The norm is well defined in the sense that if  is an operator, then for all non-zero x X Î , and the supremum   of such upper bounds b exists. In fact, a linear map belongs to B X Y , ( ) if, and only if,  n a b n n a b n a n n a b n + + + + + + = - Proof. Using the recurrence relation for the Gaußhypergeometric function: Proof. Using (187) and the following recurrence relation for the Gaußhypergeometric function: the lemma is proved. + The result follows directly from lemmas 7 and 8: Thus, we can write (197) as follows: and substituting for u(x) in (196) yields If we compare the last equation with the equation for the Jacobi polynomials: x y x x y x y and using the derivative of the Jacobi polynomial: in equation (199) we obtain: Then using lemmas 6-8 and proposition 9 we obtain the result: As we shall see in section 9.1 the eigenvalues of the angular equation in KN spacetime for different generic values of the parameters , n x obey a partial differential equation whose solution using the method of Charpit is reduced to the solution of the nonlinear ODE of Painlevé P III . The theory of solutions of the Painlevé P III constitutes a very active field of research in mathematical analysis [48,49]. In section 9.2 we obtain new exact solutions of the angular Painlevé P III in terms of Bessels functions.
In section 9.3 we present a new asymptotic analysis for the Painlevé P III which describes the leading order behaviour of eigenvalues of the angular equation in KN spacetime for generic values of the physical parameters ν and ξ. In particular we shall derive in the large limit of the independent variable a closed form solution for the eigenvalues of the angular equation in terms of Jacobian elliptic functions 14 .
A very important property of non-linear differential equations such as P III is that they can be considered in the framework of the inverse problem and can be represented as compatibility conditions for a system of auxiliary linear problems. A particular linear representation of the nonlinear ODE P III by a Lax pair was provided in [36] and is discussed in appendix B (see equations (279)-(280)). In the same appendix we briefly discuss how this coupled linear system can be solved by the Inverse Monodromy Tranform and obtain rational solutions for the Painlevé P III . 9.1. Reducing the partial differential equation satisfied by the eigenvalues to P III In appendix B.2, it is proved that the equation the eigenvalues obey for different parameters , n x is: Introducing as in [44] new coordinates: 14 In [50], section 7.5, the authors discuss the asymptotics of solutions for the first two Painlevé transcendents P , P I II . As is mentioned there, Boutroux initiated such a research programme by studying the asymptotics of solutions of the first Painlevé differential equation  2 3 Ã( ) denotes the Weierstraß elliptic function [51].
and also w t v , , l n x = ( ) ( )we compute: In the new coordinates equation (207) becomes: In order to solve equation (211) we will use the method of Charpit 15 . For a pde in the form: )is a general solution of (212), we have the following system of differentials: Applying this method to (211) we obtain: The procedure yields: Applying the transformations: Thus the general solution of (223) is written as follows: where J x  ( ) and Y x  ( ) are Bessel functions. Thus P III possesses solutions that are expressed in terms of Bessel functions. As is pointed out in [49] the freedom in the choice of sign of the square roots of γ and d means that the one-parameter family condition may be regarded as four individual conditions. Indeed, the one-parameter family condition for 1, g d = -= taking into account the third constraint in (222) yields 2 0 b a + + = .

= -+ ( )
We now prove the following proposition: This is an elliptic integral equation and its inversion will involve an elliptic function for discrete roots of the quartic equation If all the roots of the quartic are real and distinct and arranged in an > > > then we prove the following result: Proposition 13. Equation (232) can be solved in terms of the Jacobi sinus amplitudinus function Applying the transformation: where the modulus k 2 is given by: ò - and inverting Following [54] and the results in proposition 12, we can define an energy-like quantity for the generic P III : ) and the Jacobi modulus satisfies the equation: For the specific angular P III differential equation ( = . Solving equation (215) for w results in the following equation that relates the eigenvalues of the KN angular equation to the large-t limit of the Painlevé P III in terms of Jacobian elliptic functions (in the degenerate limit in terms of hyperbolic functions) 16 : where a a 1 1 , 258 We will have a polynomial solution if the local solution near r + for the radial GHE in KN spacetime terminates at some positive integer N: demands the equality of a real number with a complex number which is absurd 19 . Likewise, we have checked that none of the conditions in (264) can be satisfied if one of the two factors 18 Also C C C .  Our exact mathematical method that was culminated in the theorem 14, corroborates in the most emphatic way the results in [21,24], obtained by different means, for the absence of physical fermionic bound states in the non-extreme KN geometry .

Conclusions
In this work we have investigated the massive Dirac equation in the KNdS and the KN black hole backgrounds. First we derived and separated the Dirac equation in the Kerr-Newman-de Sitter (KNdS) black hole background using a generalised Kinnersley null tetrad in the Newman-Penrose formalism. By using appropriate transformations for the independent variable and appropriate index transformations for the dependent variable we proved the novel result that the resulting second order differential radial equation for a spin 1 2 massive charged fermion in KNdS background generalises in a highly non-trivial way the ordinary Heun equation. With the aid of a Regge-Wheeler-like coordinate and a suitable change of dependent variable we transformed the massive radial equation for the KNdS black hole into a Schrödinger-like differential equation. Subsequently we investigated the asymptotics of this novel equation and derived its near-event and near-cosmological horizon limits.
Taking the zero cosmological constant limit we investigated the massive Dirac equation in the Kerr-Newman spacetime. In the Kinnersley tetrad, by suitable transformations of the independent and dependent variables we transformed the separated angular and radial parts in the KN spacetime into generalised Heun equations (GHEs). Such GHEs are characterised by the fact that they possess three regular singularities and an irregular singularity at ¥. Heun's differential equation is a special case of such a GHE. We then derived local analytic solutions of these GHEs in which the series coefficients obey a four-term recursion relation. Moreover we investigated the radial solutions near the event horizon and far away from the black hole. The global behaviour of the solutions, along the lines of [30], was also studied with emphasis on the connection problem for a regular and an irregular singular point and the computation of the corresponding connection coefficients. We also performed a detailed analysis for the determination of the separation constant λ that appears in both radial and angular equations in the Kerr-Newman background. The procedure involves an eigenvalue matrix problem for the KN angular equations. The angular eigenvalues obey a pde which when integrated with Charpit's method leads to the Painlevé P III ordinary nonlinear differential equation. There is a vast interest in the solutions of the P III transcendent. For particular values of the parameters we derived closed form solutions for this angular P III nonlinear ODE in terms of Bessel functions. We then performed a novel asymptotic analysis of the specific angular P III transcendent in terms of Jacobian elliptic functions. This is analogous to the scattering theory of ordinary quantum mechanics in which the Bessel functions that solve the radial equation have as an asymptotic limit trigonometric functions. Our closed form solutions of both the radial and angular components of the Dirac equation in the KN curved background are of physical importance for the theory of general relativistic quantum scattering by rotating charged black holes as well as for the computation of emission rates from the KN black hole. It would also be very interesting to obtain further closed form solutions of the angular Painlevé P III transcendent, by working in the framework of the linearisation of P III by the Lax pair [36]. A background for the procedure is outlined in appendix B. Such an analysis will be a task for the future. Using the four-term recurrence relation the coefficients in the power series representation of the holomorphic local solutions of the radial GHE in the KN spacetime obey, we proved that bound fermionic states do not exist with 2 2 w m < .
Future research will also be conducted to obtain analytic solutions of the separated radial and angular ODEs  20 as well as the analytic computation of emission rates from a KNdS black hole. They would also shed light on the issue of bound states. We hope to engage in such exciting research endeavours and report progress in a future publication.
where U a c , , x ( )is the Kummer hypergeometric function of the second kind with asymptotic behaviour: B.1.1. The Picard-Fuchs differential equation that the elliptic integrals satisfy in the asymptotic limit of the generic P III with respect to III  . In order to study the dependence of the Abelian elliptic integrals on the energy-like quantity E III , to prove boundedness properties of E III and to determine further regions in the complex plane in which the elliptic asymptotic limit of P III is valid it is convenient to make the change of variables v e u = [54]. Then P III is transformed into the differential equation: as t  ¥ | | . The full analysis will be performed in a separate publication [56]. However, we will calculate in this appendix the E IIIdependence of the Abelian elliptic integrals defined in the elliptic limit of the Painlevé