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: r(field) > r(null). 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} of the central black hole, we find the dimensionless ratio q/mu of the field parameters which minimizes the effective lengths (radii) of the exterior stationary charged scalar configurations (here {M, Q, J} are respectively the mass, charge, and angular momentum of the black hole, and {mu, q} are respectively the mass and charge coupling constant of the linearized scalar field). This allows us to prove explicitly that (nonspherically 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 r(field) -> r(H), can be supported in the exterior spacetime regions of extremal Kerr-Newman black holes (here r(field) is the peak location of the stationary scalar configuration and r(H) is the black-hole horizon radius). Furthermore, we prove that these remarkably compact stationary field configurations exist in the entirerange s equivalent to J/M-2 is an element of(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/mu=(1-2s(2))/(1-s(2)). (C) 2015 The Author. Published by Elsevier B.V.