A no-short scalar hair theorem for rotating Kerr black holes

If a black hole has hair, how short can this hair be? A partial answer to this intriguing question was recently provided by the 'no-short hair' theorem which asserts that the external fields of a spherically-symmetric electrically neutral hairy black-hole configuration must extend beyond the null circular geodesic which characterizes the corresponding black-hole spacetime. One naturally wonders whether the no-short hair inequality $r_{\text{hair}}>r_{\text{null}}$ is a generic property of all electrically neutral hairy black-hole spacetimes? In this paper we provide evidence that the answer to this interesting question may be positive. In particular, we prove that the recently discovered cloudy Kerr black-hole spacetimes -- non-spherically symmetric non-static black holes which support linearized massive scalar fields in their exterior regions -- also respect this no-short hair lower bound. Specifically, we analytically derive the lower bound $r_{\text{field}}/r_+>r_+/r_-$ on the effective lengths of the external bound-state massive scalar clouds (here $r_{\text{field}}$ is the peak location of the stationary bound-state scalar fields and $r_{\pm}$ are the horizon radii of the black hole). Remarkably, this lower bound is universal in the sense that it is independent of the physical parameters (proper mass and angular harmonic indices) of the exterior scalar fields. Our results suggest that the lower bound $r_{\text{hair}}>r_{\text{null}}$ may be a general property of asymptotically flat electrically neutral hairy black-hole configurations.