Resonance spectra of caged black holes

Recent numerical studies of the coupled Einstein-Klein-Gordon system in a cavity have provided compelling evidence that {\it confined} scalar fields generically collapse to form black holes. Motivated by this intriguing discovery, we here use analytical tools in order to study the characteristic resonance spectra of the confined fields. These discrete resonant frequencies are expected to dominate the late-time dynamics of the coupled black-hole-field-cage system. We consider caged Reissner-Nordstr\"om black holes whose confining mirrors are placed in the near-horizon region $x_{\text{m}}\equiv (r_{\text{m}}-r_+)/r_+\ll\tau\equiv (r_+-r_-)/r_+$ (here $r_{\text{m}}$ is the radius of the confining mirror and $r_{\pm}$ are the radii of the black-hole horizons). We obtain a simple analytical expression for the fundamental quasinormal resonances of the coupled black-hole-field-cage system: $\omega_n=-i2\pi T_{\text{BH}}\cdot n[1+O(x^n_{\text{m}}/\tau^n)]$, where $T_{\text{BH}}$ is the temperature of the caged black hole and $n=1,2,3,...$ is the resonance parameter.