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

## Abstract

It has been shown that the mass of a charged scalar field in the background of a charged stringy black hole is never able to generate a potential well outside the event horizon to trap the superradiant modes. This is to say that the charged stringy black hole is stable against massive charged scalar perturbations. In this paper we will study the superradiant instability of the massless scalar field in the background of charged stringy black hole due to a mirror-like boundary condition. The analytical expression of the frequencies of unstable superradiant modes is derived by using the asymptotic matching method. It is also pointed out that the black hole mirror system becomes extremely unstable for a large charge \(q\) of the scalar field and a small mirror radius \(r_m\).

## Keywords

Black Hole Scalar Field Event Horizon Massless Scalar Field Superradiant Instability## 1 Introduction

Long ago, there was proposal of building a black hole bomb [1] by using the classical superradiance phenomenon [2, 3, 4, 5, 6, 7]. It seems that the mechanism of a black hole bomb is very simple. When an impinging bosonic wave with the frequency satisfying the superradiant condition is scattered by the event horizon of the rotating black hole, the amplitude of this bosonic wave will be enlarged. If one places a mirror outside of the hole, the enlarged wave will be reflected into the hole once again. Then this wave will be bounced back and forth between the event horizon and the mirror. Meanwhile, the energy of this wave can become sufficiently big in this black hole mirror system until the mirror is destroyed.

The black hole bomb mechanism firstly proposed by Press and Teukolsky [1] was studied by Cardoso et al. in [8] recently. It is found that there exists a minimum mirror’s radius to make the black hole mirror system unstable. See also Refs. [9, 10, 11, 12, 13, 14] for recent studies on this topic. The black hole bomb mechanism can be generalized to other cases. The first case is to study the massive bosonic field in rotating black holes, for example in [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], where the mass term can play the role of the reflecting mirror. In this case, the wave will be trapped in the potential well outside of the hole and the amplitude will grow exponentially, which triggers the instability of the system. The second case is to study the bosonic field perturbation in a black hole background with the Dirichlet boundary condition at asymptotic infinity. These background spacetimes include black holes in AdS spacetime [27, 28, 29, 30, 31, 32, 33], black holes in a Gödel universe [34, 35], and black holes in a linear dilaton background [36, 37]. In all these spacetimes, the Dirichlet boundary condition provides the reflecting mirror, which results in the instabilities of the systems.

For a charged scalar wave in the background of the spherical symmetric charged black hole, if the frequency of this impinging wave satisfies the superradiant condition, the wave will also undergo the superradiant process when scattered by the horizon [38]. 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. Soon after, Degollado et al. [41, 42] found that the same system can be made unstable by adding a mirror-like boundary condition like the case of the Kerr black hole. However, whether all of the charged black holes have similar properties to the RN black hole is still an interesting question that deserves further studies.

In [43], we have shown that the mass term of the scalar field in the charged stringy black hole is never able to generate a potential well outside the event horizon to trap the superradiant modes. This is to say that the charged stringy black hole is stable against massive charged scalar perturbations. In this paper, we will further study the superradiant instability of the massless scalar field in the background of the charged stringy black hole due to a mirror-like boundary condition.

It has been shown by analyzing the behavior of the effective potential that for both the nonextremal black holes and the extremal black holes there is no potential well which is separated from the horizon by a potential barrier. Thus, the superradiant modes of the charged scalar field cannot be trapped and lead to the instabilities of the black holes. This indicates that the extremal and the nonextremal charged black holes in string theory are stable against charged scalar field perturbations [43].

Now we will employ the matched asymptotic expansion method [48, 49] to compute the unstable modes of a charged scalar field in this black hole mirror system. We shall assume that the Compton wavelength of the scalar particles is much larger than the typical size of the black hole, i.e. \(1/\omega \gg M\). With this assumption, we can divide the space outside the event horizon into two regions, namely, a near-region, \(r-r_+\ll 1/\omega \), and a far-region, \(r-r_+\gg M\). The approximated solution can be obtained by matching the near-region solution and the far-region solution in the overlapping region \(M\ll r-r_+\ll 1/\omega \). Finally, we can impose the mirror’s boundary condition to obtain the analytical expression of the unstable modes in this system.

Finally, one should note that, for the RN black hole in a cavity [13, 42], the charged scalar field has a rapid growth of superradiant instability. The expression of the imaginary part of BQN frequencies is very similar to the result given in [42]. For the present case, one can also observe that \(\delta \) grows with the charge \(q\) of the scalar field. This implies the instability becomes stronger as \(q\) increases. So one can expect that, for large \(q\) and small \(r_m\), the instability time scale of this charged spherical symmetric black hole mirror system will become very short. This result is different from the rotating black hole mirror system. For the rotating black hole [8], the superradiant condition is given by \(\omega <m\Phi _H\), where \(m\) and \(\Phi _H\) are the azimuthal number and the angular velocity of the horizon, respectively. The value of \(m\) cannot be taken arbitrarily large because of the limit condition \(m\le l\) with \(l\) being the spherical harmonic index.

In summary, we have studied the instability of the massless charged scalar field in the stringy black hole mirror system. By imposing the mirror boundary condition, we have analytically calculated the expression of the BQN frequencies. Based on this result, we also point out that the black hole mirror system becomes extremely unstable for the large charge \(q\) of the scalar field and the small mirror radius \(r_m\). In [13], it is deduced by Hod using the analytical method that, for the RN black hole, the instability time scale can be made arbitrarily short in a special limit. So, the analytical computation and the numerical simulation are still required to verify the conclusion.

## Notes

### Acknowledgments

This work was supported by NSFC, China (Grant No. 11205048).

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