How short can stationary charged scalar hair be?

Abstract: It is by now well established that charged rotating Kerr-Newman black holes can support bound-state charged matter configurations which are made of minimally coupled massive scalar fields. We here prove that the externally supported stationary charged scalar configurations {\it cannot} be arbitrarily compact. In particular, for linearized charged massive scalar fields supported by charged rotating near-extremal Kerr-Newman black holes, we derive the remarkably compact lower bound $(r_{\text{field}}-r_+)/(r_+-r_-)>1/s^2$ on the effective lengths of the external charged scalar `clouds' [here $r_{\text{field}}$ is the radial peak location of the stationary scalar configuration, and ${s\equiv J/M^2, r_{\pm}}$ are respectively the dimensionless angular momentum and the horizon radii of the central supporting Kerr-Newman black hole]. Remarkably, this lower bound is universal in the sense that it is independent of the physical parameters (proper mass, electric charge, and angular momentum) of the supported charged scalar fields.