The Superradiant Stability of Kerr-Newman Black Holes

The superradiant stability of Kerr-Newman black holes

Wen-Xiang Chen and Jing-Yi Zhang1∗ Department of Astronomy, School of Physics and Materials Science, GuangZhou University, Guangzhou 510006, China

Yi-Xiao Zhang School of Information and Optoelectronic Science and Engineering, South China Normal University

In this article, the superradiation stability of Kerr-Newman black holes is discussed by introducing the condition used in Kerr black holes y into them. Moreover, the motion equation of the minimal coupled scalar perturbation in a Kerr-Newman black hole is divided into angular and radial parts. We adopt the ﬁndings made by Erhart et al. on uncertainty principle in 2012, and discuss the bounds on y.

1. INTRODUCTION

The No Hair Theorem of black holes first proposed in 1971 by Wheeler and proved by Stephen Hawking[1], Brandon Carter, etc, in 1973. In 1970s, the development of black hole thermodynamics applies the basic laws of thermodynamics to the theory of black hole in the field of general relativity, and strongly implies the profound and basic relationship among general relativity, thermodynamics and quantum theory. The stability of black holes is an major topic in black hole physics. Regge and Wheeler[2] 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 < mΩH + qΦH ,ΩH = 2 2 (1) r+ + a where q and m are the charge and azimuthal quantum number of the incoming wave, ω denotes the wave frequency, 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[1–9], 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. Superradiant refers to the enhancement process of emission, flow density and intensity when a certain frequency wave or current density and intensity pass through the medium or the edge of the medium. In quantum mechanics, the superradiant effect can be traced back to Klein’s fallacy. Subsequently, more superradiation phenomena is discovered in classical physics and quantum mechanics. In 1971[1–9], Zel proved that a rotating conducting cylinder magnifies the scattered waves. In 1972[3–9], Misner first established a quantitative theory of the superradiance effect of black holes. Press and Teukolsky proposed a black hole bomb mechanism due to the existence of superradiation mode. If there is a mirror between the event horizon and space of the black hole, the amplified wave can scatter back and forth and grow exponentially, which leads to the superradiation instability in the black hole background. Specifically, black hole bomb is the name of physical effect. It is a physical phenomenon produced by the impact of a boson field amplified by superradiation scattering on a rotating black hole. The additional condition that the phenomenon must satisfy is that the field must have a static mass which is not equal to zero. In this phenomenon, the scattered wave will be reflected back and forth between the mass interference term and the black hole, and will be amplified at each reflection. As science and technology improved, voices began to point out in the 1980s that the law was not universal.Takeshi Ozawa, a professor at Nagoya University in Japan, put forward the “Ozawa Inequality” in 2003, suggesting that the “uncertainty principle” may have some flaws.To this end, his team made precise measurements of two values related to the ”rotation” tendency of the neutrons that make up the atoms, and succeeded in measuring the accuracy of the

∗Electronic address: [email protected] 2 two values above the so-called “limit”, making Ozawa apos;s inequality hold up, and also proving the contradiction with the “uncertainty principle”. A team led by Takeshi Ozawa of Nagoya University in Japan and Yuji Hasegawa of Vienna University of Technology in Austria have found flaws in the uncertainty principle, a fundamental law proposed about 80 years ago to explain quantum mechanics in the microscopic world. The discovery is the first of its kind in the world.The findings, which are said to be needed for the use of high-speed cryptographic communication technology and a change in textbooks, were published online in the British science journal Nature Physics on January 15, 2012[10]. They measured the spin angles of the neutrons with two different instruments and calculated them with smaller errors than those indicated by the Heisenberg uncertainty principle, thus disproving the measurement limits claimed by the Heisenberg uncertainty principle. In this article, we give an outline of the Kerr-Newman-black-hole-massive-scalar system and the angular part of the equation of motion in Section 2. We discuss some important asymptotic behaviors of the effective potential and its derivatives based on the Schrodinger like radial equation and the effective potential of scalar perturbation in kn background in Section 3. Finally, We obtain the parameter space region of super radiation stability by analyzing the effective potential in detail in Section 5.

2. THE SYSTEM OF KERR-NEWMAN BLACK HOLE

The metric of the 4-dimensional Kerr-Newman black hole under Boyer-Lindquist coordinates (t, r, θ, φ) is written as follow (with natural unit, G = ~ = c = 1)[11–24] 2 ∆ 2 ρ ds2 = − dt − a sin2 θdφ + dr2 ρ2 ∆ 2 sin θ 2 + ρ2dθ2 + r2 + a2 dφ − adt , (2) ρ2 where ∆ ≡ r2 − 2Mr + a2 + Q2 , ρ2 ≡ r2 + a2 cos2 θ, (3) a denotes the angular momentum per unit mass of certain Kerr-Newman black hole and Q, M denote its charge and mass. The inner and outer horizons of the Kerr-Newman black hole can be expressed as

p 2 2 2 r± = M ± M − a − Q , (4) and obviously

2 2 r+ + r− = 2M, r+r− = a + Q . (5) The background electromagnetic potential is written as follow

Qr aQr sin2 θ A = − , 0, 0, . (6) ν ρ2 ρ2 The following covariant Klein-Gordon equation

ν ν 2 (∇ − iqA )(∇ν − iqAν )Φ = µ Φ, (7) where ∇ν represents the covariant derivative under the Kerr-Newman background. We adopt the method of separation of variables to solve the above equation, and it is decomposed as

X imφ −iωt Φ(t, r, θ, φ) = Rlm(r)Slm(θ)e e . (8) lm where Rlm are the equations which satisfy the radial equation of motion (see Eq.(11) below). The angular function Slm denote the scalar spheroidal harmonics which satisfy the angular part of the equation of motion (see Eq.(9) below). l(= 0, 1, 2, ...) and m are integers, −l ≤ m ≤ l and ω denote the angular frequency of the scalar perturbation. The angular part of the equation of motion is an ordinary differential equation and it can be expressed as follows, 1 d dS (sin θ lm ) sin θ dθ dθ 2 2 2 2 2 m +[Klm + (µ − ω )a sin θ − ]Slm = 0, (9) sin2 θ 3 where Klm represent angular eigenvalues. The standard spheroidal differential equation above has been studied for a long time and of great significance in a great deal of physical problems. The spheroidal functions Slm are known as prolate (oblate) for (µ2 − ω2)a2 > 0(< 0), and only the prolate case is concerned in this article. We select the lower bound for this separation constant as follow,[25–35]

2 2 2 2 Klm ≥ m − a (µ − ω ). (10)

The radial part of the Klein-Gordon equation contented by Rlm is written as d dR ∆ (∆ lm ) + UR = 0, (11) dr dr lm where

U = [ω(a2 + r2) − am − qQr]2 2 2 2 +∆[2amω − Klm − µ (r + a )]. (12)

The study of the superradiant modes of KN black hole under the charged massive scalar perturbation require considering the asymptotic solutions of the radial equation near the horizon and at spatial inﬁnity for appropriate boundary conditions. Here we adopt tortoise coordinate to analyse the boundary conditions for the radial function. The tortoise coordinate r∗ is deﬁned by the following equation

dr r2 + a2 ∗ = . (13) dr ∆ The two boundary conditions we focused on are the purely ingoing wave next to the outer horizon and the exponentially decaying wave located at spatial inﬁnity. Thus the asymptotic solutions of the radial wave function under the above boundaries are selected as follows ( e−i(ω−ωc)r∗ , r∗ → −∞(r → r ) √ + Rlm(r) ∼ − µ2−ω2r e ∗ r , r → +∞(r → +∞). We can easily see that getting decaying modes at spatial inﬁnity requires following bound state condition

ω2 < µ2. (14)

The critical frequency ωc is deﬁned as

ωc = mΩH + qΦH , (15)

Qr+ where ΩH is angular velocity of the outer horizon and ΦH = 2 2 is the electric potential of whom. r++a

3. THE RADIAL EQUATION OF MOTION AND EFFECTIVE POTENTIAL

A new radial wave function is deﬁned as[11, 30–35]

1 ψlm ≡ ∆ 2 Rlm. (16) in order to substitute the radial equation of motion (11) for a Schrodinger-like wave equation

d2Ψ lm + (ω2 − V )Ψ = 0, (17) dr2 lm where U + M 2 − a2 − Q2 ω2 − V = , (18) ∆2 in which V denotes the effective potential. Taking the superradiant condition (1), i.e. ω < ωc, and bound state condition (14) into consideration, the Kerr- Newman black hole and charged massive scalar perturbation system are superradiantly stable when the trapping 4 potential well outside the outer horizon of the Kerr-Newman 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 behaviors of the effective potential V around the inner and outer horizons and at spatial infinity can be expressed as

2(2Mω2 − qQω − Mµ2) 1 V (r → +∞) → µ2 − + O( ), (19) r r2

V (r → r+) → −∞,V (r → r−) → −∞. (20) √ If a Kerr black hole satisfy the condition of µ = yω, it will be superradiantly stable when µ < 2mΩH . In this article, we introduce the above condition into Kerr-Newman black holes. Therefore, the formula of the asymptotic behaviors is written as

2[M(2 − y2)ω2 − qQω] 1 V (r → +∞) → y2ω2 − + O( ), (21) r r2

V (r → r+) → −∞,V (r → r−) → −∞. (22)

It is concluded from the equations above that the effective potential approximates a constant at infinity in space, and the extreme between its inner and outer horizons cannot be less than one.The asymptotic behaviour of the derivative of the effective potential V at spatial infinity can be expressed as

2[M(2 − y2)ω2 − qQω] 1 V 0(r → +∞) → + O( ), (23) r2 r3 The derivative of the eﬀective potential has to be negative in order to satisfy the no trapping well condition,

2M(2 − y2)ω2 − 2Qqω < 0. (24)

4. THE SUPERRADIATION EFFECT AND UNCERTAINTY PRINCIPLE

We ﬁnd the Klein-Gordon equation[36]

;µ Φ;µ = 0 , (25)

µ where we deﬁned Φ;µ ≡ (∂µ − ieAµ)Φ and e is the charge of the scalar ﬁeld.We get A = {A0(x), 0},and eA0(x)can be equal to µ(where µ is the mass).

0 as x → −∞ A → . (26) 0 V as x → +∞

With Φ = e−iωtf(x), which is determined by the ordinary diﬀerential equation

d2f + (ω − eA )2 f = 0 . (27) dx2 0 We see that particles coming from −∞ and scattering oﬀ the potential with reﬂection and transmission amplitudes R and T respectively. With these boundary conditions, the solution to behaves asymptotically as

iωx −iωx fin(x) = Ie + Re , x → −∞, (28)

ikx fin(x) = T e , x → +∞ (29) where k = ±(ω − eV ). To deﬁne the sign of ω and k we must look at the wave’s group velocity. We require ∂ω/∂k > 0, so that they travel from the left to the right in the x–direction and we take ω > 0. 5

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 , (30) 1 dx 2 dx ˜ ˜ between two independent solutions, f1 and f2, 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|R| = |I| − ω |T | .Thus,for 0 < ω < eV ,it is possible to have superradiant amplification of the reflected current, i.e, |R| > |I|. There are other potentials that can be completely resolved, which can also show superradiation explicitly. We can pre-set the boundary conditions eA0(x) = yω(which can be µ = yω)(with natural unit, G = ~ = c = 1), and 2 ω−eV 2 we see that when y is relatively large(according to the properties of the boson, y can be very large),|R| ≥ − ω |T | may not hold.If the boundary conditions of the incident boson are set in advance, the two sides of the probability flow density equation are not equal because of the boundary conditions. Valagiannopoulos’s paper[39] 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.The principle of joint uncertainty shows that the joint measurement of position and momentum is impossible, that is, the simultaneous measurement of position and momentum can only be an approximate joint measurement, and the error follows the 2 2 ω−eV 2 2 ω−eV 2 inequality ∆x∆p ≥ 1/2(in natural unit system).We find|R| = |I| − ω |T | ,and we know that|R| ≥ − ω |T | is a necessary condition for the inequality ∆x∆p ≥ 1/2 to be established.We can pre-set the boundary conditions eA0(x) = yω(which can be µ = yω), and we see that when y is relatively large(according to the properties of the 2 ω−eV 2 boson, y can be very large),|R| ≥ − ω |T | may not hold.In the end,we can get ∆x∆p ≥ 1/2 may not hold.

5. ALGEBRAIC ANALYSIS OF THE SUPERRADIANT STABILITY OF KERR-NEWMAN BLACK HOLES

A new radial coordinate is deﬁned as z, z = r−r−. The explicit expression of the derivative of the eﬀective potential V in radial coordinates z and r is expressed as[37–39]

Ar4 + Br3 + Cr2 + Dr + E V 0(r) = −∆3 A z4 + B z3 + C z2 + D z + E = V 0(z) = 1 1 1 1 1 . (31) −∆3 The two sets of coeﬃcients satisfy the following relation, where

A1 = A, (32)

B1 = B + 4r−A1, (33) 2 C1 = C + 3r−B1 − 6r−A1, (34) 2 2 D1 = D + 4r−A1 − 3r−B1 + 2r−C1, (35) 4 3 2 E1 = E − r−A1 + r−B1 − r−C1 + r−D1. (36)

The complete expressions of coeﬃcients A1, E1 and C1 are as follows 6

2 2 A1 = 2qQω + 2Mµ − 4Mω , (37) 2 2 2 3 2 E1 = −4a M + 4a m M + 4M − 4MQ +2amqQ(a2 + Q2) + 4a2M(−a2 − Q2)µ2 2 2 2 2 2 2 +2a M(a + Q )µ + 4M(−a − Q )λlm 2 2 2 2 2 +2M(a + Q )λlm + 4a r− − 4a m r− 2 2 2 2 2 2 +4amMqQr− + 2a q Q r− + 4a M µ r− 2 2 2 2 −2a Q µ r− + 4M (−1 + λlm)r− 2 2 2 +2a λlmr− + 2Q [2 + Q (q − µ)(q + µ) 2 2 2 +λlm]r− − 6amqQr− − 6M(−Q µ 2 2 2 3 2 2 2 3 +λlm)r− − 4M µ r− − 2Q (q + µ )r− 3 2 4 2 4 2 2 +2λlmr− + 2Mµ r− + ω (4a M + 4a Q r− 2 3 4 3 +4Q r− − 4Mr−) + ω[−8a mM −8amM(−a2 − Q2) − 4amM(a2 + Q2) 2 2 2 2 −2a qQ(a + Q ) − 4a MqQr− 2 2 2 3 2 −8am(M + Q )r− + 12amMr− − 6qQ r− 3 4 +4MqQr− + 2qQr−]. (38)

2 2 2 C1 = −3(r+ − r−)λlm + 12r−(Q − r− − r+r−)ω 2 2 +6[am(r− + r+) − (Q − 3r− − r−r+)qQ]ω 2 2 3 2 2 −3(Q r− − Q r+ − r− + r+r−)µ 2 2 −6amqQ − 6q Q r−, (39) In this article, we represent the numerator of the derivative of the eﬀective potential V 0(z). This quartic polynomial of z enable us to study the existence of the trapping well outside the horizon by analyzing the property of the roots of the equation. We adopt z1, z2, z3 and z4 to denote the four roots of f1(z) = 0. The relationships among them adhere to the Vieta theorem.

E1 z1z2z3z4 = , (40) A1 C1 z1z2 + z1z3 + z1z4 + z2z3 + z2z4 + z3z4 = . (41) A1 When z > 0, one can infer from the asymptotic behaviors of the eﬀective potential at the inner and outer horizons 0 and spatial inﬁnity that the number of positive roots of quation V (z) = 0(or f1(z) = 0) cannot less than two. Hence these two positive roots is written as z1, z2. Studies have shown that for any ω

E1 > 0. (42) and under the condition of

E1 > 0,C1 < 0, (43) f1(z) = 0, i.e., z3, z4 are both negative. The expression of C1 is written as 2 2 2 C1 = −3(r+ − r−)Klm + 12r−(Q − r− − r+r−)ω 2 2 +6[am(r− + r+) − (Q − 3r− − r−r+)qQ]ω 2 2 3 2 2 −3(Q r− − Q r+ − r− + r+r−)µ 2 2 −6amqQ − 6q Q r−. (44)

Here, the eigenvalue of the spheroidal angular equation Klm follows the lower bound, 2 2 2 2 Klm ≥ m − a (µ − ω ). (45) 7

When µ = yω, under the consideration of the inequality (14), the further expression of the above inequality is

2 2 2 2 2 C1 <3[2(a + 2Q )r− − (a + 4r−)2M− 2 2 2 r−y (r+ + r−)]ω 3 (46) + 6[−qQ + qr−(2r− + 2M)Q + 2amM]ω 2 2 2p 2 2 2 − 6q r−Q − 6amqQ − 6m M − a − Q .

The right side of the above inequality is denoted as f(ω) i.e a quadratic function with ω as independent variable

2 2 2 2 2 f(ω) =3[2(a + 2Q )r− − (a + 4r−)2M− 2 2 2 r−y (r+ + r−)]ω 3 (47) + 6[−qQ + qr−(2r− + 2M)Q + 2amM]ω 2 2 2p 2 2 2 − 6q r−Q − 6amqQ − 6m M − a − Q .

Let

2 2 2 2 2 2 2 a1 = 3[2(a + 2Q )r− − (a + 4r−)2M − r−y (r+ + r−)] (48)

3 b1 = 6[−qQ + qr−(2r− + 2M)Q + 2amM] (49)

2 2 2p 2 2 2 c1 = −6q r−Q − 6amqQ − 6m M − a − Q (50)

In the case of f(ω) < 0, the numerical value of m is a problem. We next discuss the b1ω + c1 part of f(ω). When b1ω + c1 < 0, a1 < 0 both conditions are satisﬁed, f(ω) < 0. b1ω + c1 < 0 can be regarded as a problem of M qQ quadratic equation less than 0. As a result, we suppose when ω < 2M , b1ω + c1 < 0. We obtain a suﬃcient condition of f(w) < 0 from the above equation

p 2 2 2 2 2 2 H(m) = − M − a − Q m + a(−qQ + 2Mω)m − qQ(qQr− + (Q − 2r−(M + r−))ω) (51) The quadratic equation of b1ω + c1, when p −a(−qQ + 2Mω) a2(−qQ + 2Mω)2 − 4qQ −a2 + M 2 − Q2(qQr2 + Q2 − 2r (M + r ))ω m > − − − − − (52) 2p−a2 + M 2 − Q2 2p−a2 + M 2 − Q2

qQ As we can see that b1ω + c1 < 0 when ω < 2M . In the parameter region where f(ω) < 0 is a suﬃcient condition for C1 < 0. Firstly, the quadratic coeﬃcient of f(ω) is negative. Therefore, the following inequality is derived

2 2 r− 2Q 2 2 y (r+ − r−) > 2r− + − a − 4r− (53) r+ r+

2(a2 + 2Q2) 2pM 2 − a2 − Q2(M − pM 2 − a2 − Q2)y2 r+ > 2 2 − 2 2 . (54) a + 4r− a + 4r−

2 When y > 2(A1 > 0) for E1 > 0, C1 < 0 at this time, then f(ω) < 0, and we can know that the equation 0 V1 (z) = 0 cannot have more than two positive roots. So the Kerr-Newman black hole is superradiantly stable at that time.

6. SUMMARY

In this article, we introduced µ = yω[38, 39] into Kerr-Newman black holes, and discussed the superradiant stability of Kerr-Newman black holes. We adopt the variable separation method to divide the motion equation of the minimally coupled scalar perturbation in Kerr-Newman black hole into two forms: angular and radial. According to 8

Erhart et al’s research on uncertainty principle in 2012, the range of y discussed in this paper is of certain significance. The research[40] in 2012 shows that the small mass asymptotically non-flat Kerr-Newman-anti-de Sitter black holes produce superradiation instability under the scalar perturbation of charged mass. In 2016 [41], by analyzing the complex resonance spectrum of the charged mass scalar field in the near extremum Kerr-Newman black holes space- time, the available dimensionless charge mass ratio is obtained q/µ to characterize the growth rate of super-radiative instability of scalar field, where q is the charge of scalar field. [40] studied the small Kerr-Newman-anti-de Sitter black hole, and [41] studied the case without cosmological constant. In our paper, we obtain a new condition for the stability of superradiance when the extreme value of the uncertainty principle is smaller. Acknowledgements: We would like to thank Jing-Yi Zhang for generous help. This work is partially supported by National Natural Science Foundation of China(No. 11873025).

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