Can extreme black holes have (long) Abelian Higgs hair? art. no. 124014

It has been argued that a black hole horizon;can support the long-range fie lds of a Nielsen-Olesen string and that one can think of such a vortex as b lack hole ''hair.'' In this paper, we examine the properties of an Abelian Higgs vortex in the presence of a charged black hole as we allow the hole t o approach extremality. Using both analytical and numerical techniques, we show that the magnetic field lines (as well as the scalar field) of the vor tex are completely expelled from the black hole in the extreme limit. This was to be expected, since extreme black holes in Einstein-Maxwell theory ar e known to exhibit such a "Meissner effect'' in general. This would seem to imply that a vortex does not want to be attached to an extreme black hole. We calculate the total energy of the vortex fields in the presence of an e xtreme black hole. When the hole is small relative to the size of the vorte x, it is energetically favored for the hole to remain inside the vortex reg ion, contrary to the intuition that the hole should be expelled. However, a s we allow the extreme horizon radius to become very large compared to the radius of the vortex, we do find evidence of an instability. This proves th at it is energetically unfavorable for a thin vortex to interact with a lar ge extreme black hole. This would seem to dispel the notion that a black ho le can support ''long'' Abelian Higgs hair in the extreme limit. We show th at these considerations do not go through in the near-extreme limit. Finall y, we discuss the implications for strings that end at black holes, as in t he processes where a string snaps by nucleating black holes. [S0556-2821(98 )08320-9].