Lower bound on the proper lengths of stationary bound-state charged massive scalar clouds

Abstract: It has recently been revealed that charged scalar clouds, spatially regular matter configurations which are made of linearized charged massive scalar fields, can be supported by spinning and charged Kerr-Newman black holes. Using analytical techniques, we establish a no-short hair theorem for these stationary bound-state field configurations. In particular, we prove that the effective proper lengths of the supported charged massive scalar clouds are bounded from below by the remarkably compact dimensionless relation $\ell/M>\ln(3+\sqrt{8})$, where $M$ is the mass of the central supporting black hole. Intriguingly, this lower bound is universal in the sense that it is valid for all Kerr-Newman black-hole spacetimes [that is, in the entire regime ${a/M\in(0,1],Q/M\in[0,1)}$ of the dimensionless spin and charge parameters that characterize the central supporting black holes] and for all values of the physical parameters (electric charge $q$, proper mass $\mu$, and angular harmonic indexes ${l,m}$) that characterize the supported stationary bound-state scalar fields.