# Scalar clouds in charged stringy black hole-mirror system

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## Abstract

It was reported that massive scalar fields can form bound states around Kerr black holes (Herdeiro and Radu, Phys. Rev. Lett. 112:221101, 2014). These bound states are called scalar clouds; they have a real frequency \(\omega =m\Omega _\mathrm{H}\), where \(m\) is the azimuthal index and \(\Omega _\mathrm{H}\) is the horizon angular velocity of Kerr black hole. In this paper, we study scalar clouds in a spherically symmetric background, i.e. charged stringy black holes, with the mirror-like boundary condition. These bound states satisfy the superradiant critical frequency condition \(\omega =q\Phi _\mathrm{H}\) for a charged scalar field, where \(q\) is the charge of the scalar field, and \(\Phi _\mathrm{H}\) is the horizon’s electrostatic potential. We show that, for the specific set of black hole and scalar field parameters, the clouds are only possible for specific mirror locations \(r_\mathrm{m}\). It is shown that analytical results of the mirror location \(r_\mathrm{m}\) for the clouds perfectly coincide with numerical results in the \(qQ\ll 1\) regime. We also show that the scalar clouds are also possible when the mirror locations are close to the horizon. Finally, we provide an analytical calculation of the specific mirror locations \(r_\mathrm{m}\) for the scalar clouds in the \(qQ\gg 1\) regime.

## Keywords

Black Hole Scalar Field Kerr Black Hole Black Hole Charge Black Hole Background## 1 Introduction

It was firstly proposed by Hod that a scalar field can have real bound states in the near-extremal Kerr black hole [1, 2]. Soon later, it was reported in [3] that massive scalar fields can form bound states around Kerr black holes by using the numerical method to solve the scalar field equation in the background. These bound states are the stationary scalar configurations in the black hole backgrounds, which are regular at the horizon and outside. They are named scalar clouds. More importantly, it was shown that the backreaction of clouds can generate a new family of Kerr black holes with scalar hair [3, 4]. It is suggested that whenever clouds of a given matter field can be found around a black hole, in a linear analysis, there exists a fully non-linear solution of a new hairy black hole correspondingly. However, it requires that the field generating the clouds yields a time independent energy-momentum tensor. Generally, the field should be complex and have a factor \(\mathrm{e}^{-i\omega _\mathrm{c} t}\), where \(\omega _\mathrm{c}\) is the superradiance critical frequency. For instance, real scalar fields can give rise to clouds but not hairy black holes [5]. So, it seems that the studies of scalar clouds at the linear level are very important for us to find hairy black holes at the non-linear level. This subject has attracted a lot of attention recently [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

Generally speaking, the existence of stationary bound states of matter fields in black hole backgrounds requires two necessary conditions. The first is that the matter fields should undergo the classical superradiant phenomenon [17, 18] in the black hole background. This condition can be satisfied by the bosonic fields in the rotating black holes or the charged scalar fields in the charged black holes [19]. When the frequencies of these matter fields \(\omega \) are smaller than the superradiant critical frequency \(\omega _\mathrm{c}\), there are time growing quasi-bound states. When \(\omega >\omega _\mathrm{c}\), the fields are time decaying. So, the scalar clouds exist at the boundary between these two regimes, i.e. the frequencies of the fields are taken as the superradiant critical frequency \(\omega _\mathrm{c}\). For the rotating black holes, the critical frequency \(\omega _\mathrm{c}\) is \(m\Omega _\mathrm{H}\), where \(m\) is the azimuthal index and \(\Omega _\mathrm{H}\) is the horizon angular velocity. For the charged black holes, \(\omega =q\Phi _\mathrm{H}\), where \(q\) is the charge of the scalar field, and \(\Phi _\mathrm{H}\) is the horizon’s electrostatic potential. The second one is there should be a potential well outside the black hole horizon in which the bound states can be trapped. This potential well may be provided by the mass term of the field, i.e. \(\omega <\mu \), where \(\mu \) is the mass of the scalar field. However, sometimes the artificial boundary conditions can play the same role.

In this paper, we will study the scalar clouds in a spherically symmetric and charged background. Specifically, we will consider the charged scalar field in the backgrounds of charged stringy black holes. At first sight, it seems that the massive scalar field can form the clouds in this background. However, it is proved that the massive charged scalar field is stable in this background and there is no superradiant instability [20]. To generate the superradiant instability [21, 22], the mirror-like boundary condition should be imposed according to the black hole bomb mechanism [23, 24]. Analytical and numerical studies of this subject can be found in [25] and [26]. Correspondingly, the scalar clouds are only possible with the mirror-like boundary condition. Using numerical methods, we will study the dynamics of the massless charged scalar field satisfying the frequency condition \(\omega =q\Phi _\mathrm{H}\) and the mirror-like boundary condition. We will show that, for the specific set of black hole and scalar field parameters, the clouds are only possible for the specific mirror locations \(r_\mathrm{m}\). It will be shown that the analytical results of the mirror location \(r_\mathrm{m}\) for the clouds perfectly coincide with the numerical results. In addition, we will show that the scalar clouds are also possible when the mirror locations are close to the horizon. Finally, we will provide an analytical calculation of the specific mirror locations \(r_\mathrm{m}\) for the scalar clouds in the \(qQ\gg 1\) regime.

This paper is organized as follows. In Sect. 2, we will present the background geometry of a charged string black hole and the dynamic equation of the scalar field. In particularly, we will give the superradiant condition and the boundary condition of this black hole–mirror system. In Sect. 3, we describe the numerical procedure to solve the radial equation under a certain boundary condition. In this section, the numerical results are illustrated. Some general discussions of the numerical results are also presented. In Sect. 4, an analytical calculation of the mirror radius \(r_\mathrm{m}\) for scalar clouds in the \(qQ\gg 1\) regime is presented. The conclusion is in Sect. 5.

## 2 Description of the system

## 3 Numerical procedure and results

The numerical methods employed in this problem are based on the shooting method, which is also called the direct integration (DI) method [31, 32, 33, 34]. It is shown that the DI method is specially suited to find a stationary field configuration with the mirror-like boundary condition.

Firstly, near the event horizon \(r=2M\), we require that the radial function is regular and expand the radial function \(R(r)\) as a generalized power series in terms of \((r-r_+)\) as in the first line of Eq. (9). Because the radial equation is linear, we can take \(R_0=1\) without loss of generality. Substituting the expansion of the radial wave function into the radial equation (5), we can solve the coefficient \(R_k\) order by order in terms of \((r-r_+)\). We have only considered six terms in the expansion. The \(R_k\)s can be expressed in terms of the parameters \((M, Q, q, l)\), which are not exhibited here.

Then we can integrate the radial equation (5) from \(r=r_+(1+\epsilon )\) and stop the integration at the radius of the mirror. In this procedure, we have taken the small \(\epsilon \) as \(10^{-6}\). The procedure can be repeated by varying the input parameters \((M, Q, q, l)\) until the mirror-like boundary condition \(R(r_\mathrm{m})=0\) is reached with the desired precision. We can use a numerical root finder to search the location of the mirror that supports the stationary scalar configuration. We have found that, for the given input parameters \((M, Q, q, l)\), scalar clouds exist for a discrete set of \(r_\mathrm{m}\), which is labeled by the quantum number \(n\) of nodes of the radial function \(R(r)\).

## 4 Scalar clouds in \(qQ\gg 1\) regime

In the above numerical calculations, we find that the radial equation becomes hard to integrate when the scalar charge \(q\) is large. So it is important to make an analytical study of the stationary charged scalar clouds in the \(qQ\gg 1\) regime. In this section, we will give an analytical expression of the special mirror radius \(r_\mathrm{m}\) in the \(qQ\gg 1\) limit, for which the charged scalar field can be confined to form a stationary cloud configuration.

## 5 Conclusion

In summary, in this paper, we have studied the massless scalar clouds in the charged stringy black holes with the mirror-like boundary conditions. The scalar clouds are stationary bound states satisfying the superradiant critical frequency \(\omega =q\Phi _\mathrm{H}\). The scalar clouds in rotating black holes [3, 8] can be heuristically interpreted in terms of a mechanical equilibrium between the black hole-cloud gravitational attraction and angular momentum driven repulsion. For the charged black hole cases, the charged clouds cannot be formed, because gravitational attraction and electromagnetic repulsion cannot reach equilibrium [7]. An additional mirror should be placed at a special location to reflect the charged scalar wave.

We show that, for the specific set of black hole and scalar field parameters, the clouds are only possible for the specific mirror location \(r_\mathrm{m}\). For example, for the fixed parameters of black hole and scalar field \(M, Q, q, l\), the discrete set of the mirror location \(r_\mathrm{m}\) is characterized by the node number \(n\) of the radial wave function. It is shown that the analytical results of the mirror location \(r_\mathrm{m}\) for the clouds perfectly coincide with the numerical results in the region of \(qQ\ll 1\). However, the agreement becomes less impressive for \(qQ=O(1)\) values. In addition, we also show that the massless scalar clouds are also possible when the mirror locations are very close to the horizon. Finally, we present an analytical calculation of the specific mirror locations \(r_\mathrm{m}\) for the scalar clouds in the \(qQ\gg 1\) regime.

## Notes

### Acknowledgments

The authors would like to thank Dr. Hongbao Zhang for useful discussion on the numerical methods. This work was supported by NSFC, China (Grant No. 11205048).

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