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 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 $({r}_{\text{field}}\ensuremath{-}{r}_{+})/({r}_{+}\ensuremath{-}{r}_{\ensuremath{-}})>1/{s}^{2}$ 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 ${s\ensuremath{\equiv}J/{M}^{2},{r}_{\ifmmode\pm\else\textpm\fi{}}}$ 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.