Abelian Higgs hair for rotating and charged black holes

We study the problem of vortex solutions in the background of rotating black holes in both asymptotically flat and asymptotically anti-de Sitter spacetimes. We demonstrate that the Abelian Higgs field equations in the background of four-dimensional Kerr, Kerr-AdS, and Reissner-Nordstrom-AdS black holes have vortex line solutions. These solutions, which have axial symmetry, are a generalization of the Nielsen-Olesen string. By numerically solving the field equations in each case, we find that these black holes can support an Abelian Higgs field as hair. This situation holds even in the extremal case, and no flux expulsion occurs. We also compute the effect of the self-gravity of the Abelian Higgs field and show that the vortex induces a deficit angle in the corresponding black hole metrics