Barriola–Vilenkin monopole

5.5 Barriola–Vilenkin monopole

The Barriola–Vilenkin monopole [21

] is given by the metric

with a constant $k < 1$ . There is a deficit solid angle and a singularity at $r = 0$ ; the plane $t = constant$ , $𝜗 = π ∕2$ has the geometry of a cone. (Similarly, for $k > 1$ one gets a surplus solid angle.) The Einstein tensor has non-vanishing components $G = − G = (1 − k2)∕r2 tt rr$ .

The metric (115 ) was briefly mentioned as an example for a conical singularity by Sokolov and Starobinsky [307 ]. Barriola and Vilenkin [21 ] realized that this metric can be used as a model for monopoles that might exist in the universe, resulting from breaking a global $𝒪 (3)$ symmetry. They also discussed the question of whether such monopoles could be detected by their lensing properties which were characterized on the basis of some approximative assumptions (cf. [82 ]). However, such approximative assumptions are actually not necessary. The metric (115 ) has the nice property that the geodesics can be written explicitly in terms of elementary functions. This allows to write down explicit expressions for image positions and observables such as angular diameter distances, luminosity distances, image distortion, etc. (see [271 ]). Note that because of the deficit angle the metric (115 ) is not asymptotically flat in the usual sense. (It is “quasi-asymptotically flat” in the sense of [244 ].) For this reason, the “almost exact lens map” of Virbhadra and Ellis [336 ] (see Section 4.3), is not applicable to this case, at least not without modification.

The metric (115 ) is closely related to the metric of a static string (see metric (133 ) with $a = 0$ ). Restricting metric (115 ) to the hyperplane $𝜗 = π ∕2$ and restricting metric (133 ) with $a = 0$ to the hyperplane $z = constant$ gives the same (2 + 1)-dimensional metric. Thus, studying light rays in the equatorial plane of a Barriola–Vilenkin monopole is the same as studying light rays in a plane perpendicular to a static string. Hence, the multiple imaging properties of a Barriola–Vilenkin monopole can be deduced from the detailed discussion of the string example in Section 5.10. In particular Figures 24 and 25 show the light cone of a non-transparent and of a transparent monopole if we interpret the missing spatial dimension as circular rather than linear. This makes an important difference. While in the string case the cone of Figures 24 has a 2-dimensional set of transverse self-intersection points, the corresponding cone for the monopole has a 1-dimensional radial caustic. The difference is difficult to visualize in spacetime pictures; it is therefore recommendable to use a purely spatial visualization in terms of instantaneous wave fronts (intersections of the light cone with hypersurfaces $t = constant$ ) (compare Figures 18 and 19 with Figures 27 and 28 ).

Figure 18: Instantaneous wave fronts in the spacetime of a non-transparent Barriola–Vilenkin monopole with $k = 0.8$ . The picture shows in 3-dimensional space the intersections of the past light cone of some event with four hypersurfaces $t = constant$ , at values $t1 > t2 > t3 > t4$ . Only one half of each instantaneous wave front and of the monopole is shown. When the wave front passes the monopole, a hole is pierced into it, then a tangential caustic develops. The caustic of each instantaneous wave front is a point, the caustic of the entire light cone is a spacelike curve in spacetime which projects to part of the axis in 3-space.

Figure 19: Instantaneous wave fronts in the spacetime of a transparent Barriola–Vilenkin monopole with $k = 0.8$ . In addition to the tangential caustic of Figure 18

, a radial caustic is formed. For each instantaneous wave front, the radial caustic is a cusp ridge. The radial caustic of the entire light cone is a lightlike 2-surface in spacetime which projects to a rotationally symmetric 2-surface in 3-space.