The physical and mathematical properties of the composed Reissner-Nordström-black-hole-massive-scalar-field system are studied analytically in the dimensionless large-mass Mμ ≫ 1 regime here M, μ are respectively the mass of the central black hole and the proper mass of the scalar field. It is proved that, for a given value Q ¯ ≡ Q / M of the dimensionless charge parameter of the central black hole, the system is characterized by the presence of quasi-resonances, linearized perturbation modes with arbitrarily long lifetimes. In particular, using analytical techniques, we determine the black-hole-field critical mass spectrum M μ crit (Q ¯) which characterizes the long-lived resonances of the composed physical system.
Shahar Hod (Tue,) studied this question.