Hawking radiation by spherically-symmetric static black holes for all spins: I -- Teukolsky equations and potentials
Alexandre Arbey, Jérémy Auffinger, Marc Geiller, Etera R. Livine, Francesco Sartini
CCERN-TH-2021-007
Hawking radiation by spherically-symmetric static black holes for all spins:I - Teukolsky equations and potentials
Alexandre Arbey,
1, 2, 3, ∗ J´er´emy Auffinger, † Marc Geiller, ‡ Etera R. Livine, § and Francesco Sartini ¶ Univ Lyon, Univ Claude Bernard Lyon 1,CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland Institut Universitaire de France (IUF), 103 boulevard Saint-Michel, 75005 Paris, France Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1,CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France
In the context of the dynamics and stability of black holes in modified theories of gravity, we derivethe Teukolsky equations for massless fields of all spins in general spherically-symmetric and staticmetrics. We then compute the short-ranged potentials associated with the radial dynamics of spin 1and spin 1/2 fields, thereby completing the existing literature on spin 0 and 2. These potentials arecrucial for the computation of Hawking radiation and quasi-normal modes emitted by black holes.In addition to the Schwarzschild metric, we apply these results and give the explicit formulas forthe radial potentials in the case of charged (Reissner–Nordstr¨om) black holes, higher-dimensionalblack holes, and polymerized black holes arising from loop quantum gravity. The phenomenologicalapplications of these formulas will be the subject of a companion paper.
CONTENTS
Introduction 21. Metric and Newman–Penrose equations 32. Teukolsky equations for all spins 53. Short-ranged potentials 63.1. Spins 0 and 2 63.2. Spins 1 and 1/2 73.3. Summary 114. Some examples 114.1. tr -symmetric case 114.2. Charged black holes 124.3. Higher-dimensional black holes 134.4. Polymerized black holes 14Conclusion 15A. Details on the radial Teukolsky equations 15References 17 ∗ [email protected] † j.auffi[email protected] ‡ [email protected] § [email protected] ¶ [email protected] a r X i v : . [ g r- q c ] J a n INTRODUCTION
Black holes (BHs) are fascinating astrophysical objects. As the ultimate stage of the gravitational collapse of stars,they probe the limits of general relativity and our understanding of high energy and high density physics. With therecent rise of experimental gravitational wave detection, they have become the natural arena to seek and test modifiedtheories of gravity. For this reason, it is essential to analyze all facets of their phenomenology. At the theoretical level,the study of physical properties of black holes sets them at the interface between general relativity, thermodynamicsand quantum theory.Since Hawking discovered that black holes emit a quasi-thermal radiation [1], and therefore slowly evaporate away, avast literature has studied the characteristics of this Hawking radiation. Following Hawking’s seminal work, Teukolsky,Press, Page, Chandrasekhar, and Detweiler have worked out the equations governing the perturbations of rotatingand charged Kerr–Newman BHs for perturbations with spins 0, 1, 2 and 1 / e.g. massive gravity [22], cubic gravity [23], hairy BHs [24], Einstein–Gauss–BonnetBHs [25, 26], higher-dimensional BHs [14, 15], and Kerr–Newman massive scalar emission [27]. Beside the Hawkingradiation, another important near-horizon property of black hole, which is very sensitive to modifications of generalrelativity and can be measured from the outside, is the detail of quasi-normal modes. They constitute the ringdownsignal of a black hole relaxing towards its equilibrium state. This has become especially relevant in view of the recentgravitational wave detections from black hole mergers by LIGO/VIRGO (see [28] and references therein). Indeed, theincreasing sensitivity of the gravitational wave detectors promises an access to the fine structure of the quasi-normalmodes resulting from black hole mergers. Through this, we aim to push general relativity to its limits of validity.Indeed, there is (justified) hope that the measure of those quasi-normal mode gravitational waves will give access tothe precise characteristics of black hole horizons and thus to their correct metric description. Recent work has focusedfor example on charged Bardeen BHs [29], Gauss–Bonnet BHs [30, 31], Palatani gravity [32, 33], f ( R ) gravity [34],Kerr–de Sitter BHs [35], conformal gravity [36], higher derivative gravity [37], and so-called polymerized BHs withinloop quantum gravity [38–40].The computation of both Hawking radiation and quasi-normal modes is related to the response of black holesto perturbations. Thus, understanding the physically-measurable consequences of modified gravity on the Hawkingradiation and quasi-normal modes requires to work out the equations of motion of the various spin perturbationsto black hole metrics. This means generalizing the work of [2–12] to all black hole metrics predicted by the variousmodified gravity theories. In the present paper, we focus on spherically-symmetric static metrics of the form (1.1), andshow how the equations of motion can be written in a form similar to the Regge–Wheeler equation for SchwarzschildBHs, i.e. as a one-dimensional Schr¨odinger-like radial wave equations with a short-ranged potential. This potentialdepends on the spin of the perturbation field and we give its explicit expression for each spin 0, 1, 2 and 1 /
2. Thisderivation already exists in the literature for fields of spins 2 and 0 (see e.g. respectively [39] and [41]), but here weextend it to spins 1 and 1 /
2. We also give a general derivation of the intermediate Teukolsky equation for spins 0, 1,2, 1 / / /
1. METRIC AND NEWMAN–PENROSE EQUATIONS
We consider spherically-symmetric static metrics, which constitute a subset of Petrov type D metrics. In four-dimensional Boyer–Lindquist coordinates, the general form of such metrics isd s = − G ( r )d t + 1 F ( r ) d r + H ( r )dΩ , (1.1)where dΩ = d θ + sin θ d ϕ is the solid angle in spherical coordinates. Within this family of metrics, we further focuson solutions to the Einstein equations which are asymptotically flat. This means that at spatial infinity the functions F , G , and H must satisfy the asymptotic conditions F ( r ) −→ r → + ∞ , G ( r ) −→ r → + ∞ , H ( r ) ∼ r → + ∞ r . (1.2)Many usual metrics fall into this category, such as charged BHs, higher-dimensional BHs or effective BH metricsinspired by (loop) quantum gravity. One particular case that will be especially relevant is G ( r ) = F ( r ) ≡ h ( r ) , H ( r ) = r , (1.3)to which we refer as tr -symmetric (for time-radius symmetric). For instance, charged and higher-dimensional BHsare tr -symmetric.We now have to describe the dynamics of matter fields in these types of spacetimes. This can be done either bystudying the equations of motion written in terms of the metric, or by using the Newman–Penrose formalism. Inthe following, we will use the most direct method to obtain the results. Starting with the spin 0 case, we consider amassive scalar field φ . In this case, it is easier to write the Proca equation in curved spacetime (cid:0) (cid:3) + m φ (cid:1) φ = 1 √− g ∂ a (cid:0) g ab √− g ∂ b φ (cid:1) + m φ φ = 0 , √− g = (cid:114) GF H sin θ , (1.4)where m φ is the mass of the field. For the other types of matter fields, the multiplicity of the vector, spinor ortensor components makes it difficult to obtain a single equation of motion when working directly with the metric.A simple and efficient way to bypass this difficulty is to exploit the Newman–Penrose formalism [12, 43], whichrelies on a reformulation of the equations of motion using a null tetrad field. A choice of null tetrad such that g ab = − l a n b − n a l b + m a ¯ m b + ¯ m a m b is given by l a = (cid:32) G , (cid:114)
FG , , (cid:33) , m a = (cid:18) , , √ H , i √ H sin θ (cid:19) ,n a = (cid:32) , − √ F G , , (cid:33) , ¯ m a = (cid:18) , , √ H , − i √ H sin θ (cid:19) , (1.5)where m and ¯ m are complex conjugate. This tetrad satisfies l · n = − m · ¯ m = 1, while all other scalar productsvanish. Introducing e ai = ( e a , e a , e a , e a ) = ( l a , n a , m a , ¯ m a ), we define the λ -coefficients as λ ijk ≡ (cid:0) e ai e bk − e ak e bi (cid:1) ∂ a e jb . (1.6)These coefficients enter the definition of the so-called Ricci spin (or rotation) coefficients γ ijk ≡
12 ( λ ijk + λ kij − λ jki ) , (1.7)and some specific linear combinations of these Ricci coefficients are then denoted by κ ≡ γ , ρ ≡ γ , (cid:15) ≡ ( γ + γ ) / ,σ ≡ γ , µ ≡ γ , γ ≡ ( γ + γ ) / ,λ ≡ γ , τ ≡ γ , α ≡ ( γ + γ ) / ,ν ≡ γ , π ≡ γ , β ≡ ( γ + γ ) / . (1.8)For the family of metrics (1.1), the only non-vanishing components are real and given by ρ = − H (cid:48) H (cid:114) FG , µ = − H (cid:48) H √ F G , γ = G (cid:48) (cid:114) FG , β = − α = cot θ √ H , (1.9)where X (cid:48) ≡ ∂ r X denotes the derivative in the radial direction. In the tr -symmetric case, these spin coefficients arethe same as in [14]. We define the covariant derivatives along the four directions of the tetrad (1.5) as D ≡ l a ∇ a , ∆ ≡ n a ∇ a , δ ≡ m a ∇ a , ¯ δ ≡ ¯ m a ∇ a . (1.10)These derivatives satisfy the general commutation relation (cid:0) D − ( p + 1) (cid:15) + qρ + ¯ (cid:15) − ¯ ρ (cid:1) ( δ − pβ + qτ ) = (cid:0) δ − ( p + 1) β + qτ + ¯ π − ¯ α (cid:1) ( D − p(cid:15) + qρ ) , (1.11)where p and q are arbitrary constants. This identity, which is valid for type D metrics (see equation (2.11) of [2]),ispivotal in what follows. In particular, for the family of spherically-symmetric static metrics (1.1) that we focus on, itreduces to ( D + qρ − ρ )( δ + pα ) = ( δ + pα )( D + qρ ) . (1.12)We are now equipped with the necessary material to write down the Newman–Penrose equations of motion for fieldsof various spins. Massless spin . For a massless gauge boson, satisfying the Einstein–Maxwell field equations d F = 0 andd ∗ F = 0, the general form of the Newman–Penrose equations is [2, 12, 14] Dφ − ¯ δφ + (2 α − π ) φ + κφ − ρφ = 0 , (1.13a) Dφ − ¯ δφ + (2 (cid:15) − ρ ) φ + λφ − πφ = 0 , (1.13b)∆ φ − δφ − (2 γ − µ ) φ − σφ + 2 τ φ = 0 , (1.13c)∆ φ − δφ − (2 β − τ ) φ − νφ + 2 µφ = 0 , (1.13d)where the three Maxwell scalars are φ ≡ F ab l a m b , φ ≡ F ab ( l a n b + ¯ m a m b ) , φ ≡ F ab ¯ m a n b . (1.14)The cancellation of many of the Ricci coefficients for the family of metrics (1.1) allows to write the first and thirdequations as a coupled system involving φ and φ only, i.e. (2 α − ¯ δ ) φ + ( D − ρ ) φ = 0 , (1.15a)(∆ − γ + µ ) φ − δφ = 0 . (1.15b)These coupled first order equations can then be turn into a pair of decoupled second order differential equations. Oneapplies δ to the first equation, and applies D − ρ to the second one. Adding the two resulting equations, and usingthe identity (1.12) with p = 0 and q = −
2, gives a differential equation involving φ only: (cid:16) ( D − ρ )(∆ − γ + µ ) − δ (¯ δ − α ) (cid:17) φ = 0 . (1.16)This is the equation of motion for a massless spin 1 field. Massless spin . For purely gravitational perturbations, which are equivalent to a massless spin 2 graviton field,the general form of the Newman–Penrose equations is [2, 12]( D − ρ − (cid:15) ) ψ − (¯ δ − α + π ) ψ + 3˜ κψ ◦ = 0 , (1.17a)(∆ − γ + µ ) ψ − ( δ − τ − β ) ψ − σψ ◦ = 0 , (1.17b)( D − ρ − ¯ ρ − (cid:15) + ¯ (cid:15) )˜ σψ ◦ − ( δ − τ + ¯ π − ¯ α − β )˜ κψ ◦ − ψ ψ ◦ = 0 , (1.17c)where the ψ i are the perturbed components of the Weyl tensor ( e.g. ψ ≡ − C abcd l a m b l c m d ), ψ ◦ is the only non-vanishing background component, and the tilde on a spin coefficient indicates a perturbed quantity. We now specializeto the family of metrics (1.1). If we remove the vanishing unperturbed spin coefficients, apply the operator δ − β tothe first equation, and the operator D − ρ to the second one, add the two and make use of identity (1.12) with p = 2and q = −
4, we obtain an equation involving solely ψ , with the ˜ σψ ◦ and ˜ κψ ◦ contributions replaced by ψ ψ ◦ thanksto the third equation. The resulting equation reads (cid:16) ( D − ρ )(∆ − γ + µ ) − ( δ + 2 α )(¯ δ − α ) − ψ ◦ (cid:17) ψ = 0 , (1.18)where the background ψ ◦ is given by the Ricci identity as [12] ψ ◦ = Dµ − δπ − ¯ ρµ − σλ − π ¯ π + ( (cid:15) + ¯ (cid:15) ) µ + (¯ α − β ) π + νκ ⇒ ψ ◦ = Dµ − ρµ . (1.19)Equation (1.18) is the equation of motion for a massless spin 2 field. Massless spin / . The Newman–Penrose equations for the massless Dirac spin 1 / δ − α + π ) χ − ( D − ρ + (cid:15) ) χ = 0 , (1.20a)(∆ − γ + µ ) χ − ( δ + β − τ ) χ = 0 , (1.20b)where χ i are the two components of the spinor. We now specialize to the metrics (1.1). We remove the vanishing spincoefficients, apply the operator δ − α to the first equation, apply the operator D − ρ to the second one, subtract thetwo and make use of identity (1.12) with p = − q = −
1. This produces a decoupled differential equation for χ only: (cid:16) ( D − ρ )(∆ − γ + µ ) − ( δ − α )(¯ δ − α ) (cid:17) χ = 0 . (1.21)This is the equation of motion for a massless spin 1 / Massless spin / . Finally, the general form of the Newman–Penrose equations for a Rarita–Schwinger masslessspin 3 / D − (cid:15) − ρ ) H − (¯ δ − α + π ) H − ψ ◦ ψ = 0 , (1.22a)( δ − β − τ ) H − (∆ − γ + µ ) H − ψ ◦ ψ = 0 , (1.22b)where H = ( δ − β − ¯ α + ¯ π ) ψ − ( D − (cid:15) + ¯ (cid:15) − ¯ ρ ) ψ is a combination of the spinor components, and ψ ◦ isthe same background component as in (1.19). Specializing to the metric ansatz (1.1), we remove the vanishing spincoefficients, apply the operator δ + α to the first equation, apply the operator D − ρ to the second one, subtract thetwo and use identity (1.12) with p = 1 and q = −
3. This leads to an equation on H ≡ H only, which reads (cid:16) ( D − ρ )(∆ − γ + µ ) − ( δ + α )(¯ δ − α ) − ψ ◦ (cid:17) H = 0 , (1.23)where we have also used ( D − ρ ) ψ ◦ = 0 and ( δ − τ ) ψ ◦ = 0, which follows from the Bianchi identities [2]. Equation(1.23) is the equation of motion for a massless spin 3 /
2. TEUKOLSKY EQUATIONS FOR ALL SPINS
In this section we now derive an equivalent of the radial Teukolsky equation for all spins in the general spherically-symmetric and static metric (1.1). The first step of this calculation consists in developing explicitly all the terms inequations (1.4), (1.16), (1.18), (1.21), (1.23). Then, based on the spherical and time symmetries of the metric (1.1),we choose (cid:0) φ, φ , ψ , χ , H (cid:1) = Φ s ( r ) S s(cid:96),m ( θ, ϕ ) e − iωt , (2.1)as an ansatz for the wavefunctions. Here S s(cid:96),m are the spin- s weighted spherical harmonics for angular modes (cid:96), m ,satisfying the equation (cid:18) θ ∂ θ (sin θ ∂ θ ) + csc θ ∂ ϕ + 2 is cot θ sin θ ∂ ϕ + s − s cot θ + λ s(cid:96) (cid:19) S s(cid:96),m = 0 , (2.2)where the separation constant is λ s(cid:96) ≡ (cid:96) ( (cid:96) + 1) − s ( s + 1). In the spin 0 case, S (cid:96),m = Y (cid:96),m are just the sphericalharmonics. As we are here considering metrics with spherical and not axial symmetry, the dependency on the angularmomentum projection m factorizes as S s(cid:96),m ( θ, ϕ ) = S s(cid:96) ( θ ) e imϕ . Expanding with (2.1) the equations of motion obtainedabove for all spins will now allow us to decouple the angular and radial equations, just like in the Schwarzschild andKerr cases [2, 4]. Furthermore, the time symmetry replaces time derivatives by the energy ω of the field.For the sake of clarity, we give the details of the calculations in appendix A. The final result takes a remarkablysimple form, and we obtain the one-dimensional radial Teukolsky equations (A.2), (A.4), (A.6), (A.8), and (A.10), forspins 0, 1, 2, 1 /
2, and 3 / A s (cid:0) B s Φ (cid:48) s (cid:1) (cid:48) + (cid:32) ω + iωs (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) + C s (cid:33) Φ s = 0 , (2.3)where the radial functions A s ( r ), B s ( r ), and C s ( r ) can be read in appendix A for the various values of the spin s , andwhere once again a prime denotes the radial derivative. The consistency of this equation can be checked by choosing a tr -symmetric metric with (1.3). Inserting this in (2.3) reproduces the Teukolsky master equation for all spins derivedin [14], which is 1∆ s (cid:0) ∆ s +1 Φ (cid:48) s (cid:1) (cid:48) + (cid:18) ω r h + 2 iωsr − isωr h (cid:48) h + s (∆ (cid:48)(cid:48) − − λ s(cid:96) (cid:19) Φ s = 0 , (2.4)where in [14] the notation is ∆( r ) ≡ r h ( r ).
3. SHORT-RANGED POTENTIALS
The next step towards an applicable formulation of the equations of motion, for the computation of both quasinormal modes and Hawking radiation, is to write the Teukolsky equations (2.3) in the form of a Schr¨odinger waveequation with short-ranged potentials. Even if the equations can in principle be solved in the form (2.3), precise andstable numerical computations require to work with potentials which fall off at least as 1 /r at infinity. Furthermore,working with real-valued potentials also constitutes an appreciable bonus. We therefore need to get rid of the firstorder radial derivatives and of the complex isω terms in equation (2.3), which have a 1 /r behaviour at infinity.For all of this section, it will be convenient to define a generalized tortoise coordinate r ∗ as [39, 41]d r ∗ d r = 1 √ F G . (3.1)In what follows we will give the expressions of the potentials with both the r ∗ and r coordinates, because the firstone is more concise and the second one is better suited for numerical calculations. Furthermore, we will also considerthe general redefinition of the wave function as Ψ s ≡ Φ s (cid:115) B s √ F G , (3.2)where all quantities are functions of r and we keep track of the spin s . Finally, for each spin our goal will be to finda wave function Z s satisfying the general Schr¨odinger-like equation ∂ ∗ Z s + (cid:16) ω − V s (cid:0) r ( r ∗ ) (cid:1)(cid:17) Z s = 0 , (3.3)with spin-dependent potentials V s , and where ∂ ∗ denotes the derivative with respect to the tortoise coordinate r ∗ .Spins 0 and 2 are already treated in the literature, while spins 1/2 and 1 require more work, and in particular the useof the Chandrasekhar transformation. We now study in detail these aspects. For the massive spin 0 field, there is no complex term in (A.2), and all the terms are already decreasing fasterthan 1 /r at infinity because of the fall-offs (1.2). Applying the transformations (3.1) and (3.2), we obtain simply aSchr¨odinger wave equation for Z ≡ Ψ , with a potential given by [41] V (cid:0) r ( r ∗ ) (cid:1) = − Gm φ + Gλ (cid:96) H + 12 (cid:114) F GH (cid:32)(cid:114)
F GH H (cid:48) (cid:33) (cid:48) = − Gm φ + Gλ (cid:96) H + ∂ ∗ √ H √ H . (3.4)This is the short-ranged potential for the massive spin 0 field in the metric (1.1).For the massless spin 2 field, reference [39] follows [12]. They consider clever combinations of the metric componentsand the vanishing of the Ricci tensor components at first order in the perturbation to obtain directly a decoupledradial equation of the Schr¨odinger-like form, with the potential V (cid:0) r ( r ∗ ) (cid:1) = G ( λ (cid:96) + 4) H + F GH (cid:48) H − (cid:114) F GH (cid:32)(cid:114)
F GH H (cid:48) (cid:33) (cid:48) = G ( λ (cid:96) + 4) H + ( ∂ ∗ H ) H − ∂ ∗ √ H √ H . (3.5)This is the short-ranged potential for the massless spin 2 field in the metric (1.1).
We now complete the above results, which are already present in the literature, by deriving the short-rangedpotentials for spins 1 and 1/2. These represent the main results of this article. In order to do so, we follow themethod which has been used by Chandrasekhar and Detweiler to find the short-ranged potentials for the Kerr metric[8–11], and perform a Chandrasekhar transformation of the radial Teukolsky equations (A.4) and (A.8).Let us briefly look at the massless spin 1 and spin 1/2 fields separately before going back to general expressions forspin s . For the massless spin 1 field, applying the transformations (3.1) and (3.2) to (A.4) gives (cid:32) ω + iω (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) + F G (cid:48)(cid:48) − F GH (cid:48)(cid:48) H − F G (cid:48) G + F GH (cid:48) H + F (cid:48) G (cid:48) − F (cid:48) GH (cid:48) H + F G (cid:48) H (cid:48) H − G ( λ (cid:96) + 2) H (cid:33) Ψ + ∂ ∗ Ψ = 0 . (3.6)For the massless spin 1/2 field, (A.8) becomes (cid:32) ω + iω (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) + F G (cid:48)(cid:48) − F GH (cid:48)(cid:48) H − F G (cid:48) G + 3 F GH (cid:48) H + F (cid:48) G (cid:48) − F (cid:48) GH (cid:48) H − G ( λ / (cid:96) + 1) H (cid:33) Ψ / + ∂ ∗ Ψ / = 0 . (3.7)The general form of these equations is (cid:32) ω + iωs (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) + D s (cid:33) Ψ s + ∂ ∗ Ψ s = 0 . (3.8)The only way to suppress the complex term without re-introducing first order derivatives is to change the unknownfunction Ψ s by a linear combination of itself and its first order derivative. In this context, this is called the Chan-drasekhar transformation. In order to achieve this, we first define the intermediate function Y s byΨ s = α s Y s . (3.9)This function is such that equation (3.8) can be written in the formΛ Y s + P s Λ − Y s − Q s Y s = ∂ ∗ Y s + ω Y s + P s ( ∂ ∗ Y s + iωY s ) − Q s Y s = 0 , (3.10)with two functions P s and Q s , and the operatorsΛ ± ≡ ∂ ∗ ± iσ , Λ ≡ Λ ± Λ ∓ = ∂ ∗ + σ , (3.11)with σ ≡ − ω . When written using (3.9), equation (3.8) becomes ∂ ∗ Y s + ω Y s + iωs (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) Y s + D s Y s + 1 α s (cid:0) ∂ ∗ α s ∂ ∗ Y s + Y s ∂ ∗ α s (cid:1) = 0 . (3.12)Comparing this result with (3.10) then reveals that the two new functions are defined by the requirements Q s = − D s − ∂ ∗ α s α s , P s = 2 ∂ ∗ α s α s = s (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) = s ∂ ∗ ln (cid:18) HG (cid:19) . (3.13)One can then show that this gives Q = G ( λ (cid:96) + 2) H , Q / = G ( λ / (cid:96) + 1) H , α s = (cid:18) HG (cid:19) s/ , (3.14)where the expressions for Q and Q / can explicitly be checked using (3.6) and (3.7). Note that Q s takes a remarkablysimple form, as displayed here, in the case of spin 1/2 and 1. Unfortunately, this is not true for spin 2 and 3/2, inwhich case the explicit expression is actually much more complicated. The solution for α s , however, is valid for allspins.In order to continue with a lighter notation, from now on we remove the explicit spin label s from all the variousfunctions involved. We simply need to keep in mind that all the functions encountered below depend on the spin s .Now, let us further decompose Y as a linear combination of the function Z satisfying the Schr¨odinger wave equation(3.3), by writing Y ≡ f Λ + Λ + Z + W Λ + Z , (3.15)where on the right-hand side we have two unknown functions f and W . The Schr¨odinger equation (3.3) takes theform Λ Z = V Z , where V is the short-ranged potential that we are trying to determine for spin 1 and 1 /
2. Actingon (3.15) with Λ − and using Λ + = Λ − + 2 iσ then leads toΛ − Y = (cid:0) ∂ ∗ ( f V ) + W V (cid:1) Z + (cid:0) f V + ∂ ∗ ( W + 2 iσf ) (cid:1) Λ + Z ≡ − βα Z + R Λ + Z , (3.16)where on the right-hand side we have introduced two unknown functions β and R . Acting once again with Λ − onboth sides gives Λ − Λ − Y = (cid:18) iσ βα − ∂ ∗ (cid:18) βα (cid:19) + RV (cid:19) Z + (cid:18) ∂ ∗ R − βα (cid:19) Λ + Z . (3.17)Next, we can use Λ + = Λ − + 2 iσ once again to rewrite equation (3.10) in the formΛ − Λ − Y = − ( P + 2 iσ )Λ − Y + QY = (cid:18) βα ( P + 2 iσ ) + Qf V (cid:19) Z + (cid:0) Q ( W + 2 iσf ) − ( P + 2 iσ ) R (cid:1) Λ + Z , (3.18)where P is given in equation (3.13). Matching the Z and Λ + Z terms of these two different expansions for Λ − Λ − Y now tells us that we must have RV − Qf V = ∂ ∗ βα , ∂ ∗ ( α R ) = β + α (cid:0) Q ( W + 2 iσf ) − iσR (cid:1) , (3.19)in addition to which we should remember that, because of (3.16), we also have the definitions − βα = ∂ ∗ ( f V ) + W V , R = f V + ∂ ∗ ( W + 2 iσf ) . (3.20)Now, one can check by a direct substitution that the four previous equations lead to the conservation equation ∂ ∗ (cid:0) α Rf V + β ( W + 2 iσf ) (cid:1) = 0 , (3.21)which is a generalization of Chandrasekhar’s result [8–11]. We call this constant K , and we will see later on that itsimplifies the calculations neatly. We also define the quantity T ≡ W + 2 iσ . Using the identity (3.21) to remove anunwanted derivative of the potential V (which would have caused further difficulties), we finally obtain that (3.19)and (3.20) reduce to the following system of four equations: RV − Qf V = ∂ ∗ βα , (3.22a) ∂ ∗ ( α R ) = β + α ( QT − iσR ) , (3.22b) R ( R − ∂ ∗ T ) + βTα = Kα , (3.22c) R = f V + ∂ ∗ T , (3.22d)where (3.22c) has been obtained by combining (3.20) and (3.21). This is the system that we have to solve in order toprove that a solution Z satisfying the Schr¨odinger wave equation in the potential V does indeed exist. This systemfollows from the form of the Chandrasekhar transformation, and is valid for all spins. Chandrasekhar and Detweilerhave solved it for the Kerr metric and for spins 0, 1, 2 and 1 /
2. We will now solve it in the general case of the metric(1.1) for spin 1 following [8], and for spin 1 / Spin In the case of spin 1 we look for a simple solution, i.e. we suppose that the unknown quantities are linear in σ andof the form A = A + 2 iσA , and that the desired potential V is of course independent of σ (together with Q whichis the initial potential without the iω part). Looking at the system (3.22) tells us that the only σ term will comefrom R , meaning that we need to actually choose R = 0. Then, if we do not wish to carry out the integrations, wefurther assume that ∂ ∗ T = 0. The only remaining term in iσ then come from f and ∂ ∗ β , so we take f = 0 and β to be constant. Indeed as in [8], both R and f are also independent of σ with these hypotheses. We therefore onlyneed to decompose T ≡ T + 2 iσT , K ≡ K + 2 iσK , β ≡ β + 2 iσβ . (3.23)With all these assumptions, the system (3.22) simply becomes RV − Qf V = ∂ ∗ β α , (3.24a) ∂ ∗ ( α R ) = β + 2 iσβ + α (cid:0) Q ( T + 2 iσT ) − iσR (cid:1) , (3.24b) R ( R − ∂ ∗ T ) + 1 α (cid:0) β T + 2 iσ ( β T + β T ) − σ β T (cid:1) = 1 α ( K + 2 iσK ) , (3.24c) R = f V + ∂ ∗ T . (3.24d)Identifying the no- σ and σ terms in (3.24b) then gives us the two equations ∂ ∗ ( α R ) = β + α QT , R = β α + QT , (3.25)while doing the same in (3.24c) leads to R ( R − ∂ ∗ T ) + 1 α (cid:0) β T − σ β T (cid:1) = K α , β T + β T = K . (3.26)We can see from the last equation that the numerical value of the constant T can be absorbed in the other unknownquantities, so we set T = 1 and define κ ≡ K + 4 σ β . In order to rewrite the system in an elegant way, we nowdefine the function F ≡ α Q = (cid:96) ( (cid:96) + 1) . (3.27)With this, the second equation in (3.25) gives α R = β + F , (3.28)which can then be injected in the first equation of (3.25) to find ∂ ∗ F = β + T F . (3.29)We can now use (3.26) to eliminate β from all other equations, and (3.28) to eliminate R . We can then write theprevious equation as T = 1 F − β ( ∂ ∗ F − K ) , (3.30) We remember that in all the functions appearing in this system we have kept the spin label s implicit for conciseness. α ( F + β ) − ( F + β ) ∂ ∗ T + T ( K − β T ) = κ . (3.31)Substituting the expression (3.30) for T , we finally obtain an identity on F which reads [8] F ( ∂ ∗ F ) + ( β − F ) ∂ ∗ F + F α − (cid:18) β α + κ (cid:19) F + (2 κβ − K ) F + (cid:18) β α − κβ (cid:19) = 0 . (3.32)The goal is now to find a set of constants β , κ , and K compatible with this identity. Since F is a constant givensimply by F = (cid:96) ( (cid:96) + 1), this is actually straightforward. We deduce that β = ± (cid:96) ( (cid:96) + 1) and K = 0, while κ isunconstrained. We then choose K = 0 and obtain κ = 4 σ β . Finally, since (3.30) does not constrain T , we choose T = 0, which implies β = 0 thanks to (3.26). We have therefore found a consistent set of constants satisfying theassumptions, and all the remaining functions can be analytically computed. At the end, equations (3.24a) and (3.24d)lead to the very simple result V (cid:0) r ( r ∗ ) (cid:1) = Q = (cid:96) ( (cid:96) + 1) GH . (3.33)This is the short-ranged potential for a massless spin 1 field in the metric (1.1).
Spin / In order to study the case of spin 1/2, we first note that the definition of F gives the simple result F = α Q = ( λ / (cid:96) + 1) (cid:114) FG . (3.34)In spite of this simple form, using the same hypothesis as in the previous subsection, leading to (3.32), we find thatthe latter has no solution. We therefore need to make fewer assumptions than above. We will in fact follow [11], andgo back to the system of equations (3.22). In this system, integrations can be avoided by assuming ∂ ∗ T = 0 , ∂ ∗ ( α R ) = 0 , (3.35)which in turn implies that ˜ R ≡ α R is a constant. Thus we have the system V = ∂ ∗ β ˜ R + ( λ / (cid:96) + 1) α , (3.36a)0 = β + ( λ / (cid:96) + 1) Tα − iσ ˜ R , (3.36b)˜ R α + βTα = Kα , (3.36c)˜ Rα = f V , (3.36d)where, in order to obtain (3.36a), we have used (3.36d). We see that (3.36a) already gives us the potential as afunction of β and ˜ R . The goal is therefore to determine these functions. For this, we set T = 2 iσ by analogy withthe final result of [11] and the result of the spin 1 calculation. Equation (3.36b) then becomes β = 2 iσ (cid:32) ˜ R − ( λ / (cid:96) + 1) α (cid:33) , (3.37)and (3.36c) gives ˜ R + 4 σ ( λ / (cid:96) + 1) = α (4 σ ˜ R + K ) . (3.38)1In this equation, everything is a constant except α , we can therefore identify separately K = − σ ˜ R , ˜ R = ± iσ (cid:113) ( λ / (cid:96) + 1) . (3.39)We have therefore found a set of constants and a function β satisfying the system (3.36). We can finally use (3.36a)to write the potential as V / (cid:0) r ( r ∗ ) (cid:1) = (cid:0) (cid:96) ( (cid:96) + 1) + 1 / (cid:1) GH ± (cid:112) (cid:96) ( (cid:96) + 1) + 1 / ∂ ∗ (cid:32)(cid:114) GH (cid:33) = (cid:0) (cid:96) ( (cid:96) + 1) + 1 / (cid:1) GH ± (cid:112) (cid:96) ( (cid:96) + 1) + 1 / √ F G (cid:32)(cid:114) GH (cid:33) (cid:48) . (3.40)This is the short-ranged potential for the massless spin 1/2 field in the metric (1.1). In this section we have obtained the short-ranged massless potentials for all spins in elegant forms using the tortoisecoordinate r ∗ . This is summarized as V = ν GH + ∂ ∗ √ H √ H , (3.41a) V = ν GH , (3.41b) V = ν GH + ( ∂ ∗ H ) H − ∂ ∗ √ H √ H , (3.41c) V / = ν / GH ± √ ν / ∂ ∗ (cid:32)(cid:114) GH (cid:33) , (3.41d)where we have defined for conciseness ν ≡ (cid:96) ( (cid:96) + 1) ≡ ν , ν ≡ (cid:96) ( (cid:96) + 1) − ν / ≡ (cid:96) ( (cid:96) + 1) + 1 /
4. Theseresults are surprisingly compact, and extend the existing literature to the case of spin 1 and 1/2 massless fields in themetric (1.1). We can now focus on specific examples of metrics, and see what the resulting potentials look like whencompared to the Schwarzschild ones.
4. SOME EXAMPLES
There are numerous physically-motivated spherically-symmetric and static metrics of the form (1.1). These examplescome from both classical general relativity and modified theories of gravity with e.g. quantum gravity corrections.In this section we will study three examples of potentials, for charged BHs, higher-dimensional BHs, and finallyso-called polymerized BHs with corrections from loop quantum gravity. We will have a more extensive discussion ofthe applications of the formalism presented here in the companion paper [42].There is already important physical information which can be extracted directly from the form of the potentialsgiven below. For example, BHs which have a higher potential barrier than in the Schwarzschild case will have a lowerHawking emission rate. While this cannot directly be seen from the plots below because they are rescaled, it caneasily be seen by looking at the analytic form of the potentials. tr -symmetric case Charged and higher-dimensional BHs fall within the family of tr -symmetric metrics (1.3). We therefore first givegeneral results about this case. Using a tr -symmetric ansatz, which depends only on a single function h ( r ), the2massless potentials (3.41) become V = h (cid:18) (cid:96) ( (cid:96) + 1) r + 1 r h (cid:48) (cid:19) , (4.1a) V = h (cid:96) ( (cid:96) + 1) r , (4.1b) V = h (cid:18) (cid:96) ( (cid:96) + 1) r − r h (cid:48) + 2( h − r (cid:19) , (4.1c) V / = h (cid:96) ( (cid:96) + 1) + 1 / r ± h / (cid:112) (cid:96) ( (cid:96) + 1) + 1 / r h (cid:48) ∓ h / (cid:112) (cid:96) ( (cid:96) + 1) + 1 / r . (4.1d)The first three potentials, which are bosonic, can be written as a single master potential V s = h (cid:18) (cid:96) ( (cid:96) + 1) r + 1 − sr h (cid:48) + s ( s − h − r (cid:19) . (4.2)We note that the last term in this master potential is absent from equation (6) of [45], but is coherent with our resultsand with that of [39]. For the Schwarzschild metric, we recall that F = G = h = 1 − r S r , H = r , (4.3)where r S = 2 M ≡ r H is the Schwarzschild radius of the horizon. After the Schwarzschild solution (4.3), the simplest tr -symmetric physically-relevant BH which is solution of classicalgeneral relativity equations is the charged BH with F = G = h = 1 − r S r + r r , H = r , (4.4)where r ≡ Q and Q < M is the charge of the BH (since we are working with natural units 4 πε = 1, the finestructure constant is α em = 1). The exterior horizon is given by r H ≡ r + = r S (cid:113) − r /r . (4.5)For neutral particles ( i.e. with no additional coupling between the charge of the BH and that of the particle), thepotentials take the form V = ν r + (1 − ν ) r S r + r ( ν − − r r + r r S r − r r , (4.6a) V = ν r − ν r S r + ν r r , (4.6b) V = ν + 2 r − ( ν + 3) r S r + ( ν + 4) r + r r − r r S r + 2 r r , (4.6c) V / = ν / r − ν / r S r + ν / r r ∓ √ ν / r (cid:115) − r S r + r r (cid:32) r − r S r + 4 r r (cid:33) . (4.6d)In figure 1 we show these potentials compared to the Schwarzschild ones for the minimum possible angular momenta (cid:96) = s and for r Q = r S / Q = 2 M/ We therefore conclude that the master equation of [45] is not valid for spin 2. r/r H . . . . . . . r H V s ( r ) spin 0 , ‘ = 0spin 1 , ‘ = 1spin 2 , ‘ = 2spin 1 / , ‘ = 1 / FIG. 1. Comparison of the potentials for a charged BH with r Q = r S / Another simple case of tr -symmetric metrics describes (4 + n )-dimensional BHs [14, 15]. In this case the geometryis specified by F = G = h ≡ − (cid:16) r H r (cid:17) n +1 , H = r , (4.7)where the horizon radius is r H = 1 √ πM ∗ (cid:18) MM ∗ (cid:19) / ( n +1) (cid:32) (cid:0) ( n + 3) / (cid:1) n + 2 (cid:33) / ( n +1) , (4.8)and M ∼ M n +2 ∗ R n defines the fundamental mass scale of the theory. In this geometry, the massless potentialsbecome V = ν r + r n +1H ( n + 1 − ν ) r n +3 − ( n + 1) r n +2H r n +4 , (4.9a) V = ν r − ν r n +1H r n +3 , (4.9b) V = ν + 2 r − (cid:0) ν + 2 + ( n + 1) (cid:1) r n +1H r n +3 + ( n + 1) r n +2H r n +4 , (4.9c) V / = ν / r − ν / r n +1H r n +3 ∓ √ ν / (cid:114) − (cid:16) r H r (cid:17) n +1 (cid:18) r − ( n + 3) r n +1H r n +3 (cid:19) . (4.9d)Note that these potentials describe the radiation truncated to the four-dimensional ( t, r, θ, ϕ ) subspace, and in par-ticular does not describe the radiation within the extra dimensions. In figure 2 we plot these potentials and comparethem to the Schwarzschild ones for n = 2, M = 10 M P and M ∗ = 10 TeV [15]. [14] assumes that these BHs satisfy (cid:96) P (cid:28) r H (cid:28) R where (cid:96) P is the Planck length and R is the typical size of the extra dimensions. r/r H . . . . . . . . r H V s ( r ) spin 0 , ‘ = 0spin 1 , ‘ = 1spin 2 , ‘ = 2spin 1 / , ‘ = 1 / FIG. 2. Comparison of the potentials for a higher-dimensional BH with n = 2, M = 10 M P and M ∗ = 10 TeV (solid lines)and for the Schwarzschild metric (dashed lines). The vertical black line denotes the BH horizon. Interesting metrics which are not tr -symmetric arise in loop quantum gravity, where effective semi-classical cor-rections due to effects of quantum gravity have been derived and give rise to so-called polymerized BHs. There aremany proposals for deriving such BH metrics [18, 19, 21, 46–49], as we will review in the companion paper [42]. Here,for the sake of the example and in order to compare with previous results obtained in [39, 41], we will focus on theparticular type of polymerized BHs with [50] F = ( r − r + )( r − r − ) r ( r + r ∗ ) ( r + a ) , G = ( r − r + )( r − r − )( r + r ∗ ) r + a , H = r + a r . (4.10)Here a is the area gap of loop quantum gravity, and the radii are given by r + = 2 m ≡ r H , r − = 2 mP , r ∗ = √ r + r − , (4.11)where P = ( √ (cid:15) − / ( √ (cid:15) + 1) is the so-called polymeric function, and the parameter m is related to theso-called ADM mass M by M = m (1 + P ) . These polymerized BH solutions therefore have two free parameters,which are a and (cid:15) . With these ingredients, the massless potentials become V = ( r − r + )( r − r − )( r + a ) (cid:16) ν r + (2 ν r ∗ + r + + r − ) r + ( ν − r ∗ r + 2 a ( ν + 5) r + 2 a (cid:0) ν r ∗ − r + + r − ) (cid:1) r + 2 a r ∗ ( ν + 5) r + a ( ν − r + a (2 ν r ∗ + r + + r − ) r + a ν r ∗ r (cid:17) , (4.12a) V = ν r ( r − r + )( r − r − )( r + r ∗ ) ( r + a ) , (4.12b) V = ( r − r + )( r − r − )( r + a ) (cid:16) ( ν + 1) r + (2 ν r ∗ + r + + r − ) r + ( ν + 2) r ∗ r + a (2 ν − r + 2 a (cid:0) ν r ∗ + 5( r + + r − ) (cid:1) r + a r ∗ ( ν − r + a ( ν + 1) r + a (2 ν r ∗ − r + − r − ) r + ν a r ∗ r + a (cid:17) , (4.12c) V / = ν / r ( r − r + )( r − r − )( r + r ∗ ) ( r + a ) ± √ ν / r (cid:112) ( r − r + )( r − r − )( r + a ) (cid:16) ( r + a ) (cid:104) r ( r + r ∗ )(2 r − r + − r − )+ 2 r ( r − r + )( r − r − ) + 2 r ( r − r + )( r − r − )( r + r ∗ ) (cid:105) − r ( r − r + )( r − r − )( r + r ∗ ) (cid:17) . (4.12d)5 r/r H . . . . . r H V s ( r ) spin 0 , ‘ = 0spin 1 , ‘ = 1spin 2 , ‘ = 2spin 1 / , ‘ = 1 / FIG. 3. Comparison of the potentials for a polymerized BH with (cid:15) = 0 . a = 10 − r (solid lines) and for the Schwarzschildmetric (dashed lines). The vertical black line denotes the BH horizon. In figure 3 we show these potentials compared to the Schwarzschild ones for (cid:15) = 0 . a = 10 − r . For thisparticular example, the spin 0 potentials almost coincide because of the cancellation of most of the corrections dueto the choice of the angular mode (cid:96) = 0. CONCLUSION
In this paper we have studied the dynamics of massless fields of all spins in the general spherically-symmetric andstatic black hole metrics (1.1), deriving a generic one-dimensional radial Teukolsky equation. For the spin 1 and spin1 / Appendix A: Details on the radial Teukolsky equations
In this appendix we give the detailed equations leading to the radial Teukolsky equations for all spins, which takethe general form (2.3). We recall that prime denotes a radial derivative.
Massive spin . The massive spin 0 field satisfies the equation of motion (1.4). Using the metric (1.1), thisbecomes − ∂ t φ + GH (cid:18) θ ∂ θ (sin θ ∂ θ ) + csc θ ∂ ϕ (cid:19) φ + √ F GH (cid:16) √ F GHφ (cid:48) (cid:17) (cid:48) + Gm φ φ = 0 . (A.1)Using the ansatz (2.1), the radial part decouples and becomes (cid:18) ω + Gm φ − Gλ (cid:96) H (cid:19) Φ + √ F GH (cid:16) √ F GH Φ (cid:48) (cid:17) (cid:48) = 0 . (A.2)This is the radial Teukolsky equation for a massive spin 0 field.6 Massless spin . For the massless spin 1 field, equation (1.16) takes the explicit form − ∂ t φ + (cid:114) FG (cid:18) G (cid:48) − GH (cid:48) H (cid:19) ∂ t φ + (cid:18) θ ∂ θ (sin θ ∂ θ ) + csc θ ∂ ϕ + 2 i cot θ sin θ ∂ ϕ − − cot θ (cid:19) φ + (cid:18) F G (cid:48)(cid:48) + F GH (cid:48)(cid:48) H − F G (cid:48) G + F GH (cid:48) H + F (cid:48) G (cid:48) F (cid:48) GH (cid:48) H + 7 F G (cid:48) H (cid:48) H (cid:19) φ + 1 H (cid:114) FG (cid:16) √ F GGH φ (cid:48) (cid:17) (cid:48) = 0 . (A.3)Using the ansatz (2.1), the radial part decouples and becomes (cid:32) ω + iω (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) + F G (cid:48)(cid:48) + F GH (cid:48)(cid:48) H − F G (cid:48) G + F GH (cid:48) H + F (cid:48) G (cid:48) F (cid:48) GH (cid:48) H + 7 F G (cid:48) H (cid:48) H − G ( λ (cid:96) + 2) H (cid:33) Φ + 1 H (cid:114) FG (cid:16) √ F GGH Φ (cid:48) (cid:17) (cid:48) = 0 . (A.4)This is the radial Teukolsky equation for a massless spin 1 field. Massless spin . For the massless spin 2 field, equation (1.18) takes the explicit form − ∂ t ψ + 2 (cid:114) FG (cid:18) G (cid:48) − GH (cid:48) H (cid:19) ∂ t ψ + (cid:18) θ ∂ θ (sin θ ∂ θ ) + csc θ ∂ ϕ + 4 i cot θ sin θ ∂ ϕ − − θ (cid:19) ψ + (cid:18) F G (cid:48)(cid:48) − F GH (cid:48)(cid:48) H − F G (cid:48) G + 3 F GH (cid:48) H + F (cid:48) G (cid:48) − F (cid:48) GH (cid:48) H + 9 F G (cid:48) H (cid:48) H (cid:19) ψ + 1 GH (cid:114) FG (cid:16) √ F GG H ψ (cid:48) (cid:17) (cid:48) = 0 . (A.5)Using the ansatz (2.1), the radial part decouples and becomes (cid:32) ω + 2 iω (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) + 2 F G (cid:48)(cid:48) − F GH (cid:48)(cid:48) H − F G (cid:48) G + 3 F GH (cid:48) H + F (cid:48) G (cid:48) − F (cid:48) GH (cid:48) H + 9 F G (cid:48) H (cid:48) H − G ( λ (cid:96) + 4) H (cid:33) Φ + 1 GH (cid:114) FG (cid:16) √ F GG H Φ (cid:48) (cid:17) (cid:48) = 0 . (A.6)This is the radial Teukolsky equation for a massless spin 2 field. Massless spin / . For the massless spin 1/2 field, equation (1.21) takes the explicit form − ∂ t χ + 12 (cid:114) FG (cid:18) G (cid:48) − GH (cid:48) H (cid:19) ∂ t χ + (cid:18) θ ∂ θ (sin θ ∂ θ ) + csc θ ∂ ϕ + i cot θ sin θ ∂ ϕ − −
14 cot θ (cid:19) χ + (cid:18) F G (cid:48)(cid:48)
F GH (cid:48)(cid:48) H − F G (cid:48) G + F (cid:48) G (cid:48) F (cid:48) GH (cid:48) H + 3 F G (cid:48) H (cid:48) H (cid:19) χ + 1 H (cid:114) FH (cid:16) √ F HGHχ (cid:48) (cid:17) (cid:48) = 0 . (A.7)Using the ansatz (2.1), the radial part decouples and becomes (cid:32) ω + iω (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) + F G (cid:48)(cid:48)
F GH (cid:48)(cid:48) H − F G (cid:48) G + 3 F G (cid:48) H (cid:48) H + F (cid:48) G (cid:48) F (cid:48) GH (cid:48) H − G ( λ / (cid:96) + 1) H (cid:33) Φ / + 1 H (cid:114) FH (cid:16) √ F HGH Φ (cid:48) / (cid:17) (cid:48) = 0 . (A.8)This is the radial Teukolsky equation for a massless spin 1/2 field.7 Massless spin / . For the massless spin 3/2 field, equation (1.23) takes the explicit form − ∂ t H + 32 (cid:114) FG (cid:18) G (cid:48) − GH (cid:48) H (cid:19) ∂ t H + (cid:18) θ ∂ θ (sin θ ∂ θ ) + csc θ ∂ ϕ + 3 i cot θ sin θ ∂ ϕ − −
94 cot θ (cid:19) H + (cid:18) F G (cid:48)(cid:48) − F G (cid:48) G + 3 F GH (cid:48) H + 3 F (cid:48) G (cid:48) F G (cid:48) H (cid:48) H (cid:19) H + 1 GH (cid:114) FH (cid:16) √ F HG H H (cid:48) (cid:17) (cid:48) = 0 . (A.9)Using the ansatz (2.1), the radial part decouples and becomes (cid:32) ω + iω (cid:114) FG (cid:18) GH (cid:48) H − G (cid:48) (cid:19) + 3 F G (cid:48)(cid:48) − F G (cid:48) G + 3 F GH (cid:48) H + 3 F (cid:48) G (cid:48) F G (cid:48) H (cid:48) H − G ( λ / (cid:96) + 3) H (cid:33) Φ / + 1 GH (cid:114) FH (cid:16) √ F HG H Φ (cid:48) / (cid:17) (cid:48) = 0 . (A.10)This is the radial Teukolsky equation for a massless spin 3/2 field. [1] S. W. Hawking, Particle creation by black holes, Communications in Mathematical Physics , 199 (1975).[2] S. A. Teukolsky, Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic,and Neutrino-Field Perturbations, Astrophysical Physics Journal , 635 (1973).[3] W. H. Press and S. A. 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