Analytic treatment of the system of a Kerr-Newman black hole and a charged massive scalar field

Charged rotating Kerr-Newman black holes are known to be superradiantly unstable to perturbations of charged massive bosonic fields whose proper frequencies lie in the bounded regime $0<\ensuremath{\omega}<\phantom{\rule{0ex}{0ex}}\mathrm{min}{{\ensuremath{\omega}}_{\mathrm{c}}\ensuremath{\equiv}m{\mathrm{\ensuremath{\Omega}}}_{\mathrm{H}}+q{\mathrm{\ensuremath{\Phi}}}_{\mathrm{H}},\ensuremath{\mu}}$, where ${{\mathrm{\ensuremath{\Omega}}}_{\mathrm{H}},{\mathrm{\ensuremath{\Phi}}}_{\mathrm{H}}}$ are, respectively, the angular velocity and electric potential of the Kerr-Newman black hole, and ${m,q,\ensuremath{\mu}}$ are, respectively, the azimuthal harmonic index, the charge-coupling constant, and the proper mass of the field. In this paper we study analytically the complex resonance spectrum which characterizes the dynamics of linearized charged massive scalar fields in a near-extremal Kerr-Newman black hole spacetime. Interestingly, it is shown that near the critical frequency ${\ensuremath{\omega}}_{\mathrm{c}}$ for superradiant amplification and in the eikonal large-mass regime, the superradiant instability growth rates of the explosive scalar fields are characterized by a nontrivial (nonmonotonic) dependence on the dimensionless charge-to-mass ratio $q/\ensuremath{\mu}$. In particular, for given parameters ${M,Q,J}$ of the central Kerr-Newman black hole, we determine analytically the optimal charge-to-mass ratio $q/\ensuremath{\mu}$ of the explosive scalar field which maximizes the growth rate of the superradiant instabilities in the composed Kerr-Newman-black-hole-charged-massive-scalar-field system.

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