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 $0<\omega <\min \{{\omega }_{\mathrm{c}}\equiv m{\mathrm{\Omega }}_{\mathrm{H}}+q{\mathrm{\Phi }}_{\mathrm{H}},\mu \}$, where $\{{\mathrm{\Omega }}_{\mathrm{H}},{\mathrm{\Phi }}_{\mathrm{H}}\}$ are, respectively, the angular velocity and electric potential of the Kerr-Newman black hole, and $\{m,q,\mu \}$ 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 ${\omega }_{\mathrm{c}}$ 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 $q/\mu$. In particular, for given parameters $\{M,Q,J\}$ of the central Kerr-Newman black hole, we determine analytically the optimal charge-to-mass ratio $q/\mu$ 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.