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The stability of black holes is an major topic in black hole physics. Regge and Wheeler have proved that the spherically symmetric Schwarzschild black hole is stable under perturbation. The great impact of superradiance makes the stability of rotating black holes more complicated. Superradiative effects occur in both classical and quantum scattering processes. When a boson wave hits a rotating black hole, chances are that rotating black holes are stable like Schwarzschild black holes, if certain conditions are satisfied[1–9]
a ω ΩH is the angular velocity of black hole horizon and ΦH is the electromagnetic potential of the black hole horizon. If the frequency range of the wave lies in the superradiance condition, the wave reflected by the event horizon will be amplified, which means the wave extracts rotational energy from the rotating black hole when the incident wave is scattered. According to the black hole bomb mechanism proposed by Press and Teukolsky, if a mirror is placed between the event horizon and the outer space of the black hole, the amplified wave will reflect back and forth between the mirror and the black hole and grow exponentially, which leads to the super-radiative instability of the black hole. Hawking’s discovery suggests that black holes can radiate black body spectrums and take nothing away, which is a significant progress in the study of black hole thermodynamics. However, this pose a hard nut to crack for the conservation of information in the process of black hole evaporation as well, leading to the so-called “information loss paradox”, which violates the basic quantum unitary theory [3–6]. After Hawking’s major discoveries being published in the 1970s, more works are emerged in the intent of solving these two problems. In 2000, Parikh and Wilczek proposed a semi-classical method [7–10] to calculate emissivity. They viewed the Hawking radiation as a tunneling process and adopted the method of WKB approximation. The output particles would create a barrier in the event horizon. Later, Zhang and Zhao extended this method to some more general cases. They introduced a self-gravity factor of the particles into Hawking radiation when the corrected spectrum is given, and came to the conclusion that the accuracy of the spectrum is reduce while some information can be extracted from the black hole after introducing the self-gravity factor. Marco Angheben et al. proposed a new method to research on Hawking radiation to get the reasonable explanation for the information loss paradox and the loss of quantum unitary theory. Recently, Liu has proposed a new method about Hawking evaporation of Dirac particles under the method of the Damour-Ruffini [19]. In Kerr and R-N black holes, Starobin sky’s classical paper gives the qualitative theory of scalar field, electromagnetic field and gravitational wave superradiation for the first time. It concludes that rotating the energy layer of the black hole makes its negative energy flow into the internal energy layer, while the outward scattered waves in the black hole extract the energy of the black hole due to the conservation of energy and increase its amplitude, It provides a basis for the physical explanation of black hole’s superradiation. Due to the existence of the superradiation mode, Press and Teukolsky proposed the black hole bomb mechanism. If there is a mirror between the event horizon and the black hole space, the amplified waves will scatter back and forth and grow exponentially, leading to superradiative instability of the black hole background. Specifically, the black hole bomb is the name of the physical effect. It is a physical phenomenon caused by the influence of the boson field amplified by superradiation scattering on the rotating black hole. This phenomenon must meet an additional condition, that is, the field can have any static mass except zero. Scattered waves are reflected and amplified back 3 and forth between the mass interference term and the black hole. We know that all known particles belong to two categories: fermions (half-integer spin particles) and bosons (integer spin particles). Quarks and leptons are fermions, and the carrier particles of various forces are bosons. The main difference between these two particles is that fermions follow the Pauli exclusion principle, which stipulates that two identical fermions cannot be in the same state at the same point in time and space. At the same time, due to Pauli’s incompatibility principle, it is generally believed that the scattering of fermions cannot be superradiative. However, since the different conditions of various fermion fields follow different evolution equations, it is difficult to obtain the rigorous and general mathematical proof of the above statement. Studies have shown that the scattering of the free Dirac spin 1/2 field on the electrostatic barrier and the Kerr-Newman (charged, rotating) black hole cannot exhibit superradiation. The exact solution of the Dirac equation of general relativity is studied, which describes the dynamics of massive charged particles with half-integer spins in the curved space-time geometry of a charged, rotating Kerr-Newman- (anti)de Sitter black hole . We first use the generalized Kinnersley null tetrad in Newman-Penrose formalism to derive the Dirac equation in the background of the Kerr-Newman-de Sitter (KNdS) black hole. Subsequently, in this coordinate system and the KNdS black hole spacetime, we proved that the Dirac equation can be decomposed into the radial and angular parts of ordinary differential equations. Under the specific transformation of the independent 1 variable and the dependent variable, we prove that a large number of charged spin 2 fermions in the background KNdS black hole constitute a highly nontrivial generalization of the Heun equation. Because it has five regular finite singularities. Using independent variables similar to Regge-Wheeler, we convert the radial equation in the KNdS background into a differential equation similar to Schr¨odinger, and study its asymptotic behavior near the event and the universe horizon. For the case of the massive fermions in the background of the Kerr-Newman (KN) black hole, we first prove that the radial and angular equations separated by the Dirac equation are simplified to the generalized Heun differential equation (GHE). The local solution of this GHE is derived and can be described by a holomorphic function. The power series coefficients of the holomorphic function are determined by a four-term recurrence relationship. In addition, we use asymptotic analysis to derive the solution of the massive fermions far away from the KN black hole and the solution close to the event horizon. Traditionally, the phenomenon of superradiation is analogized by gravitation, and superradiation is expected to realize the rotating space-time of toy models. A widely considered configuration is based on vortices, such as drain bathtubs; recently, the scattering of such vortices by surface gravity waves in water has been used to obtain the first experimental evidence of superradiation scattering. In a recent article[41], it proposes to use the information flow from the simulation model to the gravity situation, which is to understand the superradiation phenomenon from the Bose-Einstein condensation. We have already said that the opposite arrow of this analogy can also be interesting. Here, we will use this point of view to reconsider the stability of quantized vortices in BECs, and study the special instabilities that occur in non-uniform flow BECs similar to hydrodynamic parallel shear flows. In this article,we can see that when the derivatives of V (imaginary part) equal to 0, and the second derivatives of V (imaginary part) is negative the superradiation condition is established. The establish of superradiation condition proves that there is a potential barrier near the horizon. The phenomenon of superradiation is a stable superradiation phenomenon when superradiation conditions are established, and when there is a potential barrier near the field of action.We conclude here that the Kerr-Newman ds black hole can produce superradiation phenomena. 4 II. DIRAC EQUATIONS IN A KERR-NEWMAN DS SPACE-TIME The metric of the 4-dimensional Kerr-Newman ds black hole under Boyer-Lindquist coordinates (t,r,θ,φ) is written as follow (with natural unit, G = ~ = c = 1)[11–24]: KN 2 2 2 ∆r 2 2 ρ 2 ρ 2 ds = 2 2 (dt a sin θdφ) KN dr dθ Ξ ρ − − ∆r − ∆θ ∆ sin2 θ θ (adt (r2 + a2)dφ)2 (1) − Ξ2ρ2 − a2Λ a2Λ ∆ := 1 + cos2 θ, Ξ := 1 + , (2) θ 3 3 Λ ∆KN := 1 r2 r2 + a2 2Mr + e2, (3) r − 3 − ρ2 = r2 + a2 cos2 θ, (4) 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 where the vector potential is: er A = (dt a sin2 θdφ). (5) −Ξ(r2 + a2 cos2 θ) − The generalised Kinnersley null tetrad in the Kerr-Newman-de Sitter spacetime is given by 2 2 2 2 KN µ (r + a )Ξ aΞ µ Ξ(r + a ) ∆r aΞ l = KN , 1, 0, KN , n = 2 , 2 , 0, 2 ∆r ∆r 2ρ − 2ρ 2ρ µ 1 iΞ m = iaΞ sin θ, 0, ∆θ, (r + ia cos θ)√2∆θ sin θ µ 1 iΞ m = − iaΞ sin θ, 0, ∆θ, (6) (r ia cos θ)√2∆ − sin θ − θ Dirac equations in the Newman-Penrose formalism for the Kerr-Newman de Sitter black hole spacetime: ˙ µ (1) µ (0) (0) (D′ γ + µ + iqn Aµ)P + (δ τ + β + iqm Aµ)P = iµ Q , (7) − − − ∗ ˙ µ (0) µ (1) (1) ( D + ̺ ε iql Aµ)P + ( δ′ + α π iqm Aµ)P = iµ Q (8) − − − − − − − ∗ i(mφ ωt) where assuming that the azimuthal and time-dependence of the fields will be of the form e − we calculate the directional derivatives to be µ ∂ iΞ D = l ∂µ = + KN K 0, (9) ∂r ∆r ≡ D KN KN µ ∆r ∂ iΞK ∆r † D′ ∆= n ∂µ = 2 KN 2 0 (10) ≡ − 2ρ ∂r − ∆r ≡− 2ρ D µ √∆θ ∂ ΞH √∆θ δ = m ∂µ = + 0, (11) √2ρ ∂θ ∆θ ≡ √2ρ L µ √∆θ ∂ ΞH √∆θ δ′ δ∗ = m ∂µ = 0†, (12) ≡ √2ρ∗ ∂θ − ∆θ ≡ √2ρ∗ L 5 where m K := ma ω(r2 + a2), H := ωa sin θ . (13) − − sin θ III. THE SUPERRADIATION EFFECT AND UNCERTAINTY PRINCIPLE 1 Superradiance in Flat Spacetime:Let us consider the Dirac equation [36]for a spin- 2 massless fermion Ψ, minimally coupled to the same EM potential Aµ as in Eq.. µ γ Ψ;µ =0 , (14) where γµ are the four Dirac matrices satisfying the anticommutation relation γµ,γν =2gµν. The solution to takes { } iωt the form Ψ = e− χ(x), where χ is a two-spinor given by f1(x) χ = . (15) f2(x) Using the representation i 0 0 i γ0 = , γ1 = , (16) 0 i i 0 − − the functions f1 and f2 satisfy the system of equations: df /dx i(ω eA )f =0, df /dx i(ω eA )f =0. (17) 1 − − 0 2 2 − − 0 1 One set of solutions can be once more formed by the ‘in’ modes, representing a flux of particles coming from x → −∞ being partially reflected (with reflection amplitude 2) and partially transmitted at the barrier |R| in in iωx iωx iωx iωx f ,f = e e− , e + e− as x (18) 1 2 I − R I R → −∞ eikx, eikx as x + (19) T T → ∞ µ 0 µ On the other hand, the conserved current associated with the Dirac equation is given by j = eΨ†γ γ Ψ and, − by equating the latter at x and x + , we find some general relations between the reflection and the → −∞ → ∞ transmission coefficients, in particular, 2 = 2 2 . (20) |R| |I| − |T | Therefore, 2 2 for any frequency, showing that there is no superradiance for fermions. The same kind of |R| ≤ |I| relation can be found for massive fields. The reflection coefficient and transmission coefficient depend on the specific shape of the potential A0. However one can easily show that the Wronskian df˜ df˜ W = f˜ 2 f˜ 1 , (21) 1 dx − 2 dx 6 between two independent solutions, f˜1 and f˜2, of is conserved. From the equation on the other hand, if f is a solution 2 2 ω eV 2 then its complex conjugate f ∗ is another linearly independent solution. We find = − .Thus,for |R| |I| − ω |T | 0 <ω IV. SEPARATION OF THE DIRAC EQUATION IN THE KERR-NEWMAN-DE SITTER SPACETIME The separation parts: iωt imφ (+) ( ) iωt imφ ( ) (+) (0)˙ e e S (θ)R (r) (1)˙ e e S (θ)R (r) Q = − − , Q = − − (22) KN − √2(r + ia cos θ) ∆r e iωteimφS( )(θ)R( )(r) e iωteimφSp(+)(θ)R(+)(r) P (0) = − − − , P (1) = − . (23) √2(r ia cos θ) ∆KN − r The following ordinary differential equations for the radial and angular polarp parts: ( ) 2 2 ( ) KN dR − (r) Ξi(ma ω(r + a )) ( ) iqerR − (r) (+) ∆r + − KN R − (r) KN = (λ + iµr)R (r), (24) q dr ∆r − ∆r (+) 2 2 (+) (+) KN dR (r) iΞ(ma ω(r + a ))R (r) ieqrR (r) ( ) ∆ − + = (λ iµr)R − (r), (25) r KN KN q dr − ∆r ∆r − ∆ dS(+)(θ) Ξ p m p1 d θ + ωa sin θ + S(+)(θ)+ ( ∆ sin θ)S(+)(θ) √ dθ √ sin θ 2 sin θ dθ θ ∆θ ∆θ h− i p ( ) = ( λ + µa cos θ)S − (θ), (26) − ( ) ∆θ dS − (θ) m Ξ ( ) 1 d ( ) + ωa sin θ S − (θ)+ ( ∆ sin θ)S − (θ) √ dθ sin θ √ 2 sin θ d sin θ θ ∆θ − ∆θ p = (λ + µa cos θ)S(+)(θ), (27) where λ is a separation constant and µ = µ/√2. ∗ 7 V. SPHERICAL QUANTUM SOLUTION IN VACUUM STATE The general relativity theory’s field equation is well known as[39], 1 8πG R g R = T (28) µv − 2 µv − c4 µv when Tµv = 0, the Ricci tensor is written as Rµv =0 (29) which indicates that the Ricci tensor is in vacuum state. The proper time of is The general form of metric for spherical coordinates is expressed as follows, 1 dτ 2 = A(t, r)dt2 B(t, r)dr2 + r2dθ2 + r2 sin2θdφ2 (30) − c2 From Eq(28), we can obtain the Ricci-tensor equations. A A B A A 2 B¨ B˙ 2 A˙B˙ R = ′′ + ′ ′ ′ + ′ + = 0 (31) tt −2B 4B2 − Br 4AB 2B − 4B2 − 4AB A A 2 A B B B¨ A˙B˙ B˙ 2 R = ′′ ′ ′ ′ ′ + + = 0 (32) rr 2A − 4A2 − 4AB − Br − 2A 4A2 4AB 1 rB rA R = 1+ ′ + ′ = 0 (33) θθ − B − 2B2 2AB 2 Rφφ = Rθθ sin θ =0 (34) Rtθ = Rtφ = Rrθ = Rrφ = Rθφ = 0 (35) B˙ R = =0 (36) tr −Br ∂ 1 ∂ Substituting ′ = , = into Eq, we conclude that ∂r · c ∂t B˙ =0 (37) R R 1 A B (AB) tt + rr = ′ + ′ = ′ = 0 (38) A B −Br A B − rAB2 Hence, the result is obtained. 1 A = (39) B 1 rB′ rA′ r ′ Rθθ = 1+ 2 + = 1+ = 0 (40) − B − 2B 2AB − B 8 By solving Eq, we can obtain that r 1 C = r + C =1+ (41) B → B r According to a well-known conclusion, when tortoise coordinates tend to the boundary of event horizon, the inde- y pendent variable r approaches radial negative infinity. In order to preset a boundary condition, we select C = ye− , When r tends to infinity, the relation expressions between A, Σ , dr2 and y are as follows[40] 1 y y A = =1 Σ, Σ= e− (42) B − r y dτ 2 = 1 dt2 (43) − r X In this time, if particles’ mass are mi, the excessive energy will be e. It is seen that the effect of presetting boundary is similar to that of the cosmological constant of ds space-time 2 2 2 2 E = Mc = m1c + m2c + ... + mnc + e (44) VI. THE RADIAL EQUATION OF MOTION AND EFFECTIVE POTENTIAL When[17] r r z = − − , (45) r+ r − − in terms of the original variables(Asymptotic solutions at infinity r in the Kerr-Newman-de Sitter spacetime): → ∞ 2 2 (C1+C2+C3)(2i√ω µ (r+ r−)) i ω2 µ2(r r−)z r r− ∓ − − √ + − e± − − r r− +− R(r) 2 2 (C1+C2+C3)(2i√ω µ (r+ r−)) ∼ (i ω2 µ2(r r−)z) r r− ± − − √ + − e− − − r r− +− We see that in the event horizon (r tends to be negative infinity) the wave function under the Kerr-Newmann ds black hole tends to the wave function under the Kerr-Newmann black hole. Under the Kerr-Newmann ds black hole, the special solution relationship between the wave function at the edge of the cosmological constant horizon and the wave function at the edge of the event horizon is R(Ξ)=iR(r+). Both R and R+ are not related to θ. Therefore, by simplifying the expressions of f1, f2 and g2 and making θ = π/2, the following results is obtained iK r M ∆(∂ + + − )R √2qR = √2(irµ)R (46) r ∆ ∆ + − − The expression of R+ is obtained by solving Eq. and making r approach negative infinity 1 1 R Ce− 2 √2iRe− (47) + → − 1 when C = 0 is √2iRe− Valagiannopoulos’s paper[40] attempts to transfer the classical electrodynamic concept to the quantum realm, with an emphasis on quantum scattering. The meaning of the preset boundary condition y is shown here. 9 In order to substitute the radial equation of motion for a Schrodinger-like wave equation d2Ψout (r > r ) ω + + (ω2 V )Ψout (r > r )=0, (48) dr2 − ω + where 2 2 i(ω ωc) ω V = i(r r )− ( 1+ − )(ω ω ), (49) − − + − K − c in which V denotes the effective potential. Taking the superradiant condition, i.e. ω<ωc, and bound state condition into consideration, the KN black hole and charged massive scalar perturbation system are superradiantly stable when the trapping potential well outside the outer horizon of the KN black hole does not exist. As a result, the shape of the effective potential V is analyzed next in order to inquiry into the nonexistence of a trapping well.The asymptotic behaviour of the derivative of the effective potential V at spatial infinity can be expressed as 2 2 2 4 a +r (mΩH +ω( 1+y)) ( +) − V ′(realpart)= (r r1)3(r1+r2) − − 2(mΩH +ω( 1+y)) V ′(imaginarypart)= −3 (r r+) 2 2 − 2 (50) 12 a +r1 (mΩH +ω( 1+y)) ( ) − V ′′(realpart)= (r r )4(r +r−) − + + 6(mΩH +ω( 1+y)) V ′′(imaginarypart)= (r r −)4 − − + When Λ = 0, the space-time is close to the boundary of event horizon, the phase difference of space-time curving coordinates approaches π/2, and the kinetic energy of incident particles needs to be multiplied by i. The new equation is expressed as follow: d2Ψout (r > r ) ω + + i(ω2 V )Ψout (r > r )=0, (51) dr2 − ω + We differentiate again, and obtain that: 2 2 2 4 a +r (mΩH +ω( 1+y)) ( +) − V ′(imaginarypart)= (r r )3(r +r−) − − + + 2(mΩH +ω( 1+y)) V ′(realpart)= −3 (r r+) − 2 2 2 (52) 12 a +r1 (mΩH +ω( 1+y)) ( ) − V ′′(imaginarypart)= (r r )4(r +r−) − + + 6(mΩH +ω( 1+y)) V ′′(realpart)= (r r −)4 − − + If we preset the boundary condition as qΦH = yω, the imaginary part of V will be displayed due to the rotation of the black hole and the charge near the horizon[25–36]. It is seen that when the derivatives of V (imaginary part) equal to 0, and the second derivatives of V (imaginary part) is negative the superradiation condition is established. The establish of superradiation condition proves that there is a potential barrier near the horizon. As we can see, when the wave functions are combined into one, the Schr¨odinger equation holds, creating another reasonable wave function, where the imaginary part of V becomes the real part. If there is a potential barrier beyond the horizon, there could be a superradiation effect. The Kerr-Newman-de Sitter geometry (as in the case of Kerr and Kerr-Newman geometry) can be described in terms of the local Newman-Penrose zero-tetrad framework applicable to the main zero geodesic, that is, the tetrad and Weyl tensor Two main zero directions Cµνρε. In this generalized Kinnersley framework, the null tetrad is constructed directly from the tangent vector of the main null geodesic: 2 2 2 2 ˙ dt Ξ (r + a ) ˙ ˙ aΞ t := = KN E, r˙ = ΞE, θ =0, φ = KN E, (53) dλ ∆r ± ∆r 10 where the dot denotes differentiation with respect to the affine parameter λ and E denotes a constant. Cooper’s pair:We will assume that the total spin and center of mass momentum ~~q of the pair are constant, so an orbital wave function of the pair is[42] i~q R~ ψ (~r1, ~r2)= ϕq(~ρ)e · (54) with center of mass and relative coordinates defined as R~ = (~r + ~r ) /2 and ρ = ~r ~r , respectively. As ~q 0 the 1 2 1 − 2 → relative coordinate is spherically symmetric and hence, ϕ(~ρ) is an eigenfunction of angular momentum with angular momentum quantum numbers 1 and m. If ~q = 0, 1 is not a good quantum number, but the component of angular 6 momentum along ~q and parity are. Assuming ~q =0, ψ can be expanded as i~k ~ri i~k ~r2 ψ (~r1, ~r2)= ϕq(~ρ)= ake · e− · (55) Xk ~ik ~r1 i~k ~r2 where the sum is restricted to states with εk > 0. If e · and e− · are thought of as plane wave states, then the pair wave function is a superposition of definite pairs where ~k is occupied. ± We see that the phases of the wave functions between two fermions with opposite spins coincide (phase difference π/2).We know that the Cooper pair is caused by the coupling between two fermions due to the weak gravitational force and the coincidence of the phases of the wave functions between the two fermions with opposite spins.Find the combined wave functions, they are normalized(A is a normalized constant): i√ω2 Vr i√ω2 Vr iπ/2 Ψ= A(e − + e − e ). (56) We can prove that Dirac particles in the KN black hole background can have a special solution through a certain operation, forming a Cooper pair, thus producing superradiation.When the space-time is close to the boundary of event horizon, the phase difference of space-time curving coordinates approaches π/2, and the kinetic energy of incident particles needs to be multiplied by i. The new equation is expressed as follow: d2Ψout (r > r ) ω + + (ω2 V )(eiπ/2Ψout ) (r > r )=0, (57) dr2 − ω + We differentiate again, and obtain that: 2 2 2 4 a +r (Ξ(mΩH +ω( 1+y)) ( Λ) − V ′(imaginarypart)= (r r1)3(r +r−) − − + 2Ξ(mΩH +ω( 1+y)) V ′(realpart)= −3 (r rΛ) − 2 2 2 (58) 12 a +r (Ξ(mΩH +ω( 1+y))) ( Λ) − V ′′(imaginarypart)= (r r )4(r +r−) − Λ + 6Ξ(mΩH +ω( 1+y)) V ′′(realpart)= (r r )−4 − − Λ If the imaginary part of V will be displayed due to the rotation of the black hole and the charge near the horizon[25– 36]. It is seen that when the derivatives of V (imaginary part) equal to 0, and the second derivatives of V (imaginary part) is negative the superradiation condition is established. The establish of superradiation condition proves that there is a potential barrier near the horizon. As we can see, when the wave functions are combined into one, the Schr¨odinger equation holds, creating another reasonable wave function, where the imaginary part of V becomes the real part. If there is a potential barrier beyond the horizon, there could be a superradiation effect. 11 VII. SUMMARY In this article, we introduced into Kerr-Newman black holes, and discussed the superradiance of Kerr-Newman ds black holes. In this paper, we deal with the approximate wave function of the Dirac particle out of the horizon of KN ds Black Hole, obtain V, and then derive V with real part and imaginary part. We’ll deal with the real part and the imaginary part. When the derivatives of V (imaginary part) equal to 0, and the second derivatives of V (imaginary part) is negative the superradiation condition is established. We generate another equation by setting the boundary. The V (imaginary part) of the previous equation becomes the V (real part) of the new equation. In this way, there is a potential barrier outside the event horizon, which meets the condition of superradiation which acts as a cosmological constant in ds space. 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