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

Abstract

It has been argued that a black hole horizon can support the long range fields of a Nielsen-Olesen string, and that one can think of such a vortex as black 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 to approach extremality. Using both analytical and numerical techniques, we show that the magnetic field lines (as well as the scalar field) of the vortex are completely expelled from the black hole in the extreme limit. This was to be expected, since extreme black holes in Einstein-Maxwell theory are 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 extreme black hole. When the hole is small relative to the size of the vortex, it is energetically favoured for the hole to remain inside the vortex region, contrary to the intuition that the hole should be expelled. However, as 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 that it is energetically unfavourable 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 do not go through in the near extreme limit. Finally, we discuss whether this has implications for strings that end at black holes.

21 pages REVTeX plus 9 figures. Additional figures and mpeg movies available at http://www.damtp.cam.ac.uk/user/ats25/strhole.html We have made several cosmetic changes, and we have revised and extended the discussion of strings which end on extreme horizons

References in corpus (2)

• Superconducting p-branes and Extremal Black Holes
• Selection rules for splitting strings

Cited by in corpus (22)

• Breaking an Abelian gauge symmetry near a black hole horizon
• Black holes on thick branes
• Superconducting p-branes and Extremal Black Holes
• Vortices and extreme black holes: the question of flux expulsion
• Decay of Dirac Massive Hair in the Background of Spherical Black Hole
• Rotating black hole hair
• Dilaton Black Holes on Thick Branes
• Abelian Higgs Hair for a Static Charged Black String
• Abelian Higgs Hair for Rotating and Charged Black Holes
• Abelian Higgs Hair for Electrically Charged Dilaton Black Holes
• Abelian-Higgs hair on stationary axisymmetric black hole in Einstein-Maxwell-axion-dilaton gravity
• Reissner-Nordstrom Black Holes and Thick Domain Walls
• The Question of Abelian Higgs Hair Expulsion from Extremal Dilaton Black Holes
• Euclidean Dilaton Black Hole Vortex and Dirac Fermions
• Hairy rotating black string in the Einstein-Maxwell-Higgs system
• Vortices and black holes in dilatonic gravity
• Thick Domain Walls and Charged Dilaton Black Holes
• Cosmological Black Holes on Branes
• Thick Domain Walls in AdS Black Hole Spacetimes
• Superconducting Hair on Charged Black String Background
• Zooming in on the horizon when in its Meissner state
• Dark matter cosmic string in the gravitational field of a black hole