Extremal Kerr–Newman black holes with extremely short charged scalar hair

The recently proved ‘no short hair’ theorem asserts that, if a spherically-symmetric static black hole has hair, then this hair (the external fields) must extend beyond the null circular geodesic (the “photonsphere”) of the corresponding black-hole spacetime: rfield>rnullrfield>rnull. In this paper we provide compelling evidence that the bound can be violated by non  -spherically symmetric hairy black-hole configurations. To that end, we analytically explore the physical properties of cloudy Kerr–Newman black-hole spacetimes – charged rotating black holes which support linearized stationary charged scalar configurations in their exterior regions. In particular, for given parameters {M,Q,J}{M,Q,J} of the central black hole, we find the dimensionless ratio q/μq/μ of the field parameters which minimizes   the effective lengths (radii) of the exterior stationary charged scalar configurations (here {M,Q,J}{M,Q,J} are respectively the mass, charge, and angular momentum of the black hole, and {μ,q}{μ,q} are respectively the mass and charge coupling constant of the linearized scalar field). This allows us to prove explicitly that (non-spherically symmetric non-static) composed Kerr–Newman-charged-scalar-field configurations can violate the no-short-hair lower bound. In particular, it is shown that extremely compact stationary charged scalar ‘clouds’, made of linearized charged massive scalar fields with the property rfield→rHrfield→rH, can be supported in the exterior spacetime regions of extremal Kerr–Newman black holes (here rfieldrfield is the peak location of the stationary scalar configuration and rHrH is the black-hole horizon radius). Furthermore, we prove that these remarkably compact stationary field configurations exist in the entire   range s≡J/M2∈(0,1)s≡J/M2∈(0,1) of the dimensionless black-hole angular momentum. In particular, in the large-mass limit they are characterized by the simple dimensionless ratio q/μ=(1−2s2)/(1−s2)q/μ=(1−2s2)/(1−s2).