Stability of the extremal Reissner–Nordström black hole to charged scalar perturbations

Abstract

The fundamental role played by black holes in many areas of physics makes it highly important to explore the nature of their stability. The stability of charged Reissner-Nordström black holes to neutral (gravitational and electromagnetic) perturbations was established almost four decades ago. However, the stability of these charged black holes under charged perturbations has remained an open question due to the complexity introduced by the well-known phenomena of superradiant scattering: A charged scalar field impinging on a charged Reissner-Nordström black hole can be amplified as it scatters off the hole. If the incident field has a non-zero rest mass, then the mass term effectively works as a mirror, preventing the energy extracted from the hole from escaping to infinity. One may suspect that the superradiant amplification of charged fields by the charged black hole may lead to an instability of the Reissner-Nordström spacetime (in as much the same way that rotating Kerr black holes are unstable under rotating scalar perturbations). However, in this Letter we show that, for charged Reissner-Nordström black holes in the regime (Q/M ) 2 ≤ 8/9, the two conditions which are required in order to trigger a possible superradiant instability [namely: (1) the existence of a trapping potential well outside the black hole, and (2) It is well-known that rotating Kerr black holes suffer from superradiant instability which may be triggered by massive scalar perturbations [1][2][3]. Superradiant scattering is a well-known phenomena in quantum systems [4,5] as well as in classical ones [6,7]. Considering a wave of the form e imφ e −iωt incident upon a rotating object whose angular velocity is Ω, one finds that if the frequency ω of the incident wave satisfies the relationthen the scattered wave is amplified. A bosonic field impinging upon a rotating Kerr black hole can be amplified if the superradiance condition (1) is satisfied, where in this case Ω is the angular velocity of the black-hole horizon. Press and Teukolsky suggested to use this mechanism to build a black-hole bomb [8]: If one surrounds the black hole by a reflecting mirror, the wave will bounce back and forth between the black hole and the mirror amplifying itself each time. It was later realized [9][10][11][12][13][14][15][16][17] that a natural mirror exists in a system composed of a rotating black hole and a massive scalar field: In this case the mass term effectively works as a mirror for modes in the regime ω < µ ≡ MG/hc, where M is the mass of the field. The gravitational force binds the massive field and keeps it from escaping to infinity. At the event horizon some of the field goes down the black hole, and if the frequency of the wave is in the superradiance regime (1) then the field is amplified. In this way the field is amplified at the horizon while being bound away from infinity. As a consequence, the rotational energy extracted from the black hole by the incident bosonic field grows exponentially over time [9][10][11][12][13][14][15][16][17].A sim...

Superradiantly stable non-extremal Reissner–Nordström black holes

“…In fact, up to now, the non-extremal charged RN black hole is proved to be superradiantly stable when the mass M and charge Q of the black hole satisfy (Q/M) 2 ≤ 8/9 [28,29]. In this paper, we demonstrate that the all non-extremal charged RN black hole is stable against a massive charged scalar perturbation.…”

“…Recently, for a Kerr black hole under massive scalar perturbation, Hod has proposed a stronger stability regime than before [27]. The extremal and non-extremal charged Reissner-Nordström (RN) black holes have been proved to be stable against a charged massive perturbation [28,29]. Similarly, the analog of the charged RN black hole in string theory has also been proved to be stable under a charged massive scalar perturbation [30].…”

“…The extremal and non-extremal charged Reissner-Nordström (RN) black holes have been proved to be stable against a charged massive perturbation [28,29]. Similarly, the analog of the charged RN black hole in string theory has also been proved to be stable under a charged massive scalar perturbation [30].In fact, up to now, the non-extremal charged RN black hole is proved to be superradiantly stable when the mass M and charge Q of the black hole satisfy (Q/M) 2 ≤ 8/9 [28,29]. In this paper, we demonstrate that the all non-extremal charged RN black hole is stable against a massive charged scalar perturbation.…”

Quasi-bound state resonances of charged massive scalar fields in the near-extremal Reissner–Nordström black-hole spacetime

“…a e-mail: shaharhod@gmail.com black-hole spacetime have demonstrated explicitly that exponentially decaying quasi-bound state charged matter configurations, which are characterized by extremely long lifetimes, 2 can be supported in the external spacetime region of the central charged black hole. These external quasi-bound field configurations are characterized by the physically motivated boundary condition of ingoing waves at the outer black-hole horizon and exponentially decaying (bounded) radial eigenfunctions at spatial infinity [37][38][39][40][41][42].…”

Superradiant instability of charged scalar field in stringy black hole mirror system

“…But it is pointed out in [18] that there is no unstable mode of a scalar field in a Reissner-Nordström (RN) black hole. More recently, it was proved by Hod in [39,40] that, for the Reissner-Nordström (RN) black holes, the existence of a trapping potential well outside the black hole and superradiant amplification of the trapped modes cannot be satisfied simultaneously. This means that the RN black holes are stable under the perturbations of massive charged scalar fields.…”