Rotating black hole hair

A Kerr black hole sporting cosmic string hair is studied in the context of the abelian Higgs model vortex. It is shown that a such a system displays much richer phenomenology than its static Schwarzschild or Reissner--Nordstrom cousins, for example, the rotation generates a near horizon electric field. In the case of an extremal rotating black hole, two phases of the Higgs hair are possible: Large black holes exhibit standard hair, with the vortex piercing the event horizon. Small black holes on the other hand, exhibit a flux-expelled solution, with the gauge and scalar field remaining identically in their false vacuum state on the event horizon. This solution however is extremely sensitive to confirm numerically, and we conjecture that it is unstable due to a supperradiant mechanism similar to the Kerr-adS instability. Finally, we compute the gravitational back reaction of the vortex, which turns out to be far more nuanced than a simple conical deficit. While the string produces a conical effect, it is conical with respect to a local co-rotating frame, not with respect to the static frame at infinity. As a consequence, we find that the ergosphere is shifted, and geodesics around the black hole are perturbed.