A few years back, Achucarro, Gregory and Kuijken showed that there were solutions to the Abelian-Higgs equations in a black hole background in which the string threaded the black hole. They argued that at least topological scalar hair could reach to infinity in this system. We (Ashbourn-Chamblin, Chamblin, Emparan and myself) have been looking at cosmic strings in extremal black hole backgrounds. What is interesting is that extremal black holes have a Meissner effect in which flux lines are expelled from the hole. We show that this occurs also for cosmic strings, adding a new twist to the scalar hair discussion.

Below is a picture of a solution for E = 10.0, Beta = 0.5, Winding = 10.0 and Charge = 9.0. To get the solution we solve the Abelian-Higgs equations in a black hole background numerically both on and outside the horizon. We use similar techniques to AKG (see Phys. Rev. D52, 5729 (1995)).

An Abelian string threading a nonextremal black hole

Here is a picture of an extremal solution for E = 10.0, Beta = 0.5, Winding = 10.0 and Charge = 10.0.

An Abelian string being expelled from an extremal black hole

The transition between non-expelled and expelled cases is not gradual. We have done simulations for configurations with charge = 9.999999999999999 and found no expulsion. The extremal case seems to be the critical point, perhaps because of the known change in topology for extremal holes.

Below, we plot the energy in the gauge and scalar fields for a fixed (winding = 10.0) winding vortex with varying extremal hole size. Notice that for holes with mass less than 15 or so the field energy is less than that of the isolated string (the red line in the graph). This implies that it is energetically favorable for holes below this mass to attach themselves to strings. For holes larger than this mass, it is energetically favorable to be outside the string.

The following movies show the contours of the P and X fields in an x-z slice plotted from the vortex axis outwards.

P-field

X-field