Revisiting Kerr–Newman black hole’s charged scalar cloud: flux balance

Abstract In the present study, we investigate the quasibound states and scalar cloud of relativistic charged scalar fields bound to a Kerr–Newman black hole. We present the exact eigensolutions of the governing Klein–Gordon equation with non-minimal coupling in the black hole background. By imposing boundary conditions on the quasibound states, we are able to find the exact complex quasibound state frequencies of the corresponding radial wave functions in terms of the confluent Heun polynomial. The quantization formula obtained here allows us to determine the exact resonance frequency of a charged scalar cloud, thereby revisiting and extending the earlier result derived through the asymptotic matching approach. Our analysis shows that a stationary configuration can exist only when the charge coupling satisfies qrQ q r Q 0. In this case, the scalar cloud remains stationary due to a flux balance: an outgoing flux from the Cauchy horizon compensates for the ingoing particle flux at the event horizon. This mechanism distinguishes scalar clouds from ordinary quasibound states, where such a compensating flux is absent.