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Superradiantly Stable Non-Extremal Reissner–Nordström Black Holes

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Superradiantly Stable Non-Extremal Reissner–Nordström Black Holes Eur. Phys. J. C (2016) 76:314 DOI 10.1140/epjc/s10052-016-4157-y Regular Article - Theoretical Physics Superradiantly stable non-extremal Reissner–Nordström black holes Jia-Hui Huang1,a, Zhan-Feng Mai2 1 Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China 2 Department of Physics, Center for Advanced Quantum Studies, Beijing Normal University, Beijing 100875, China Received: 5 March 2015 / Accepted: 25 May 2016 / Published online: 7 June 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract The superradiant stability is investigated for hole and grows exponentially. This leads to the superradiant non-extremal Reissner–Nordström black holes. We use an instability of the black hole. algebraic method to demonstrate that all non-extremal The superradiant mechanism has been studied by many Reissner–Nordström black holes are superradiantly stable authors for the (in)stability problem of black holes [14– against a charged massive scalar perturbation. This improves 26]. Recently, for a Kerr black hole under massive scalar the results obtained before for non-extremal Reissner– perturbation, Hod has proposed a stronger stability regime Nordström black holes. than before [27]. The extremal and non-extremal charged Reissner–Nordström (RN) black holes have been proved to The stability problem of black hole is an important topic be stable against a charged massive perturbation [28,29]. in black hole physics. Regge and Wheeler [1] proved that Similarly, the analog of the charged RN black hole in string the spherically symmetric Schwarzschild black hole is sta- theory has also been proved to be stable under a charged ble under perturbations. The stability problems of rotating massive scalar perturbation [30]. or charged black holes are complicated due to the signif- In fact, up to now, the non-extremal charged RN black hole icant effect of superradiance. The superradiance effect can is proved to be superradiantly stable when the mass M and occur in both classical and quantum scattering processes [2– charge Q of the black hole satisfy (Q/M)2 ≤ 8/9[28,29]. In 5]. When a charged bosonic wave is impinging on a charged this paper, we demonstrate that the all non-extremal charged rotating black hole, the wave reflected by the event horizon RN black hole is stable against a massive charged scalar will be amplified if the wave frequency ω lies in the following perturbation. We find that there is no trapping well outside superradiant regime: the black hole, which is separated from the horizon by a potential barrier. As a result, there is no bound state in the 0 <ω<m + e, (1) superradiant regime, that can lead to the instability of the charged RN black hole. where m and e are the azimuthal harmonic number and charge The metric of the RN black hole (in natural unit G = c = of the incoming charged wave, is the angular velocity of h¯ = 1) is black hole horizon, and = Q/rH is the electric potential of the black hole [6–12]. This means that when the incoming 2M Q2 1 wave is scattered, the wave extracts rotational energy from ds2 =− 1 − + dt2 + dr 2 r r 2 − 2M + Q2 the rotating black hole and electric energy from charged black 1 r r2 hole. According to the black hole bomb mechanism proposed +r 2(dθ 2 + sin2θdφ2), (2) by Press and Teukolsky [13], if there is a mirror between the black hole horizon and spatial infinity, the amplified wave can be reflected back and forth between the mirror and the black where M and Q are the mass and electric charge of the black hole. The dynamics of a charged massive scalar field pertur- bation is governed by the Klein–Gordon equation, After this paper was completed, Ref. [31] appeared which addresses the same issue with a different method and gets the same conclusion. ν ν 2 a e-mail: [email protected] (∇ − iqA )(∇ν − iqAν) − μ = 0, (3) 123 314 Page 2 of 4 Eur. Phys. J. C (2016) 76 :314 where q and μ are the charge and the mass of the scalar wave equation has the following asymptotic solutions: field. Aν =−δ0 Q/r is the vector potential that describes ν the spherically symmetric electric field. The solution of the −i(ω− qQ)y e r+ , y →−∞(r → r+) above equation with definite spherical harmonic eigenvalues R˜ ∼ √ (11) − μ2−ω2 y, →+∞( →+∞). can be written as e y r −iωt lm(t, r,θ,φ)= Rlm(r)Ylm(θ, φ)e , (4) It is obvious that when ω2 <μ2 where Ylm is the spherical harmonic function, l is the spheri- (12) cal harmonic index, m is the azimuthal harmonic index with −l m l, and ω is the energy of the mode. The radial there is a bound state of the scalar field. Klein–Gordon equation obeyed by Rlm (we denote Rlm by In the following, we prove that there is no trapping well R in the following) is given by outside the black hole horizon when the parameters of the scalar field and the black hole satisfy the bound state con- d dR dition (12) and the superradiance condition of the RN black + UR= 0, (5) dr dr hole, <ω< = / . where = r 2 − 2Mr + Q2 and 0 q H qQ r+ (13) 1 We define a new radial function φ by φ = 2 R, then the U = (ωr 2 − qQr)2 − [μ2r 2 + l(l + 1)]. (6) radial Eq. (5) can be rewritten as The inner and outer horizons of the black hole are d2φ + (ω2 − V )φ = 0, (14) dr 2 r± = M ± M2 − Q2, (7) where and it is obvious that 2 2 2 U + M − Q r+ + r− = 2M, r+r− = Q . V = ω2 − . (15) 2 In order to study the superradiance stability of the black hole against the massive charged perturbation, the asymp- In order to see if there exists a trapping potential outside totic solutions of the radial wave equation near the horizon the horizon, we should analyze the shape of the effective and at infinity will be considered with proper boundary con- potential V . From the following asymptotic behavior of the ditions. We proceed by defining the tortoise coordinate y by potential V : 2 the equation dy = r and a new radial function R˜ = rR. dr 2Mμ2 + 2Qqω − 4Mω2 The radial wave Eq. (5) can be written as V (r →+∞) → μ2 + r 1 d2 R˜ + o , (16) + U˜ R˜ = 0, (8) r 2 dy2 V (r → r+) →−∞, (17) where we know there is at least one maximum for V outside the event horizon. It is easy to see that the asymptotic behavior ˜ U d U = − . (9) of the derivative of V is r 4 r 3 dr r 2 2Mμ2 + 2Qqω − 4Mω2 1 It is easy to obtain the asymptotic behavior of the new poten- V =− + o . 2 3 (18) tial U˜ as r r (ωr+ − qQ)2 lim U˜ = , lim U˜ = ω2 − μ2. (10) We can prove the coefficient 2Mμ2 + 2Qqω − 4Mω2 > → 2 →∞ r r+ r+ r 0 when ω satisfies the superradiance and the bound state The chosen boundary conditions are ingoing waves at the conditions. Define a quadratic function f for ω horizon (y →−∞) and bound states (exponentially decay- ing modes) at spatial infinity (y →+∞). Then the radial f (ω) =−4Mω2 + 2Qqω + 2Mμ2. (19) 123 Eur. Phys. J. C (2016) 76 :314 Page 3 of 4 314 It is obvious that there are two zero points for f with opposite where sign, and the positive one is a =−4Mω2 + 2qQω + 2Mμ2, (26) 2 2 2 2 2 2 Qq + Q q + 8M μ c = 12r− −4r−ω + 6r−ωqQ ω+ = . (20) 4M 2 2 2 2 −[2q Q + (3r+r− + r−)μ ] − 3(r+ − r−)l(l + 1), (ω) > ω To verify f 0 when satisfies the superradiance and (27) the bound state conditions, we just need to prove ω<ω+. 2 2 1 3 e = 2r−(r+ − r−)(ωr− − qQ) + (r+ − r−) . (28) Case I: ω<μ≤ qQ/r+ 2 > With the obvious relation r+ M, we can get From the asymptotic behaviors of the effective potential at 2 2 the inner and outer horizons and infinity, we know that there qQ q2 Q2 μ2 μr+ μ r+ μ2 ω+ = + + > + + are at least two roots for V (z) = 0 when z > 0. If a trapping 2 2 4M 16M 2 4r+ 16r+ 2 potential existed, there would be at least four positive roots = μ>ω. (21) for V (z) = 0. Next, we will demonstrate that it is impos- sible for the equation V (z) = 0 to have four positive roots Case II: ω<qQ/r+ <μ when the superradiance condition (13) and the bound state We can also easily get condition (12) are satisfied. Because we are just concerned with the roots of V (z) = qQ q2 Q2 μ2 qQ q2 Q2 q2 Q2 ω+ = + + > + + 0, the numerator of Eq. (25) will be considered only. We 4M 16M2 2 4r+ 16r 2 2r 2 + + denote the roots of V (z) = 0by{z1, z2, z3, z4} and z1, z2 = / >ω. qQ r+ (22) are the two known positive roots (r− < z1 < r+, r+ < z2 < +∞). According to the Vieta theorem, we have the following So when ω satisfies the superradiance and the bound state relations for the roots: conditions, f (ω) > 0. It implies that c z1z2 + z1z3 + z1z4 + z2z3 + z2z4 + z3z4 = , (29) ( →∞) → −. a V r 0 (23) e z z z z = . (30) 1 2 3 4 a This means there is no potential well when r →+∞.In the following, we will show that there is only one maximum The coefficient a(= f (ω)) has been proved to be positive outside the event horizon for V , no trapping potential exists before.
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