Abstract
In previous work, black hole vortex solutions in Einstein gravity with AdS3 background were found where the scalar matter profile had a singularity at the origin r = 0. In this paper, we find numerically static vortex solutions where the scalar and gauge fields have a non-singular profile under Einstein gravity in an AdS3 background. Vortices with different winding numbers n, VEV v and cosmological constant Λ are obtained. These vortices have positive mass and are not BTZ black holes as they have no event horizon. The mass is determined in two ways: by subtracting the numerical values of two separate asymptotic metrics and via an integral that is purely over the matter fields. The mass of the vortex increases as the cosmological constant becomes more negative and this coincides with the core of the vortex becoming smaller (compressed). We then consider the vortex with gravity in asymptotically flat spacetime for different values of the coupling α = 1/(16πG). At the origin, the spacetime has its highest curvature and there is no singularity. It transitions to an asymptotic conical spacetime with angular deficit that increases significantly as α decreases. For comparison, we also consider the vortex without gravity in flat spacetime. For this case, one cannot obtain the mass by the first method (subtracting two metrics) but remarkably, via a limiting procedure, one can obtain an integral mass formula. In the absence of gauge fields, there is a well-known logarithmic divergence in the energy of the vortex. With gravity, we present this divergence in a new light. We show that the metric acquires a logarithmic term which is the 2 + 1 dimensional realization of the Newtonian gravitational potential when General Relativity is supplemented with a scalar field. This opens up novel possibilities which we discuss in the conclusion.
Highlights
We find numerically static vortex solutions where the scalar and gauge fields have a non-singular profile under Einstein gravity in an AdS3 background
This logarithmic term can be viewed as the 2 + 1 dimensional realization of the Newtonian gravitational potential with mass m when General Relativity is supplemented with a complex scalar field and symmetry breaking potential
Near the core of the vortex, the metric looks nothing like the BTZ black hole metric
Summary
One can solve analytically for the metric in vacuum by setting f = v and a = n identically in eq (2.11) (this eliminates matter from the Lagrangian density (2.5)). Substituting the above vacuum solution into eq (2.7) yields B0(r) = k0 (−Λr2 + C) where k0 is a positive integration constant which can be absorbed into a redefinition of time. Using (2.7) we obtain B(R) = k A(R) where the positive integration constant k can no longer be fully absorbed into a redefinition of time (since that has already been done once with the constant k0). In the presence of matter, at large radius R, we have that B(R) is proportional to A(R) but not equal to it
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