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It has been argued that a black hole horizon can support the long rangefields of a Nielsen-Olesen string, and that one can think of such a vortex asblack hole ``hair''. In this paper, we examine the properties of an AbelianHiggs vortex in the presence of a charged black hole as we allow the hole toapproach extremality. Using both analytical and numerical techniques, we showthat the magnetic field lines (as well as the scalar field) of the vortex arecompletely expelled from the black hole in the extreme limit. This was to beexpected, since extreme black holes in Einstein-Maxwell theory are known toexhibit such a ``Meissner effect'' in general. This would seem to imply that avortex does not want to be attached to an extreme black hole. We calculate thetotal energy of the vortex fields in the presence of an extreme black hole.When the hole is small relative to the size of the vortex, it is energeticallyfavoured for the hole to remain inside the vortex region, contrary to theintuition that the hole should be expelled. However, as we allow the extremehorizon radius to become very large compared to the radius of the vortex, we dofind evidence of an instability. This proves that it is energeticallyunfavourable for a thin vortex to interact with a large extreme black hole.This would seem to dispel the notion that a black hole can support `long'abelian Higgs hair in the extreme limit. We show that these considerations donot go through in the near extreme limit. Finally, we discuss whether this hasimplications for strings that end at black holes.

Can extreme black holes have (long) Abelian Higgs hair?