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 ! < mΩH + qΦH ;ΩH = 2 2 (1) r+ + a where q and m are the charge and azimuthal quantum number of the incoming wave, ! 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 (r − iqA )(rν − iqAν )Φ = µ Φ; (7) where rν 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 −i!t 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 ! 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 − ! )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 − !2)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 − ! ): (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 = [!(a2 + r2) − am − qQr]2 2 2 2 +∆[2am! − 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(!−!c)r∗ ; r∗ ! −∞(r ! r ) p + Rlm(r) ∼ − µ2−!2r e ∗ r ; r ! +1(r ! +1): We can easily see that getting decaying modes at spatial infinity requires following bound state condition !2 < µ2: (14) The critical frequency !c is defined as !c = 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 lm ≡ ∆ 2 Rlm: (16) in order to substitute the radial equation of motion (11) for a Schrodinger-like wave equation d2Ψ lm + (!2 − V )Ψ = 0; (17) dr2 lm where U + M 2 − a2 − Q2 !2 − V = ; (18) ∆2 in which V denotes the effective potential.