Abstract
It is reported that massive scalar fields can form bound states around Kerr black holes [C. Herdeiro, and E. Radu, Phys. Rev. Lett. 112, 221101 (2014)]. These bound states are called scalar clouds, which have a real frequency $\omega=m\Omega_H$, where $m$ is the azimuthal index and $\Omega_H$ 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 $\omega=q\Phi_H$ for the charged scalar field, where $q$ is the charge of scalar field, and $\Phi_H$ is the horizon electrostatic potential. We show that, for the specific set of black hole and scalar field parameters, the clouds are only possible for the specific mirror locations $r_m$. It is shown that the analytical results of mirror location $r_m$ for the clouds are perfectly coincide with the numerical results. We also show that the scalar clouds are also possible when the mirror locations are close to the horizon. At last, we provide an analytical calculation of the specific mirror locations $r_m$ for the scalar clouds in the $qQ\gg 1$ regime.
Highlights
Speaking, the existence of stationary bound states of matter fields in black hole backgrounds requires two necessary conditions
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 ωc
The first line indicates that the scalar field is regular near the horizon and the second line implies that the system is placed in a perfectly reflecting cavity
Summary
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. 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 ωc. To generate the superradiant instability [21,22], the mirror-like boundary condition should be imposed according to the black hole bomb mechanism [23,24]. We will study the dynamics of the massless charged scalar field satisfying the frequency condition ω = q H and the mirror-like boundary condition.
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