### 5.5 Barriola–Vilenkin monopole

The Barriola–Vilenkin monopole [21] is given by the metric
with a constant . There is a deficit solid angle and a singularity at ; the plane
, has the geometry of a cone. (Similarly, for one gets a surplus solid
angle.) The Einstein tensor has non-vanishing components .
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 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 ).
Restricting metric (115) to the hyperplane and restricting metric (133) with to the
hyperplane 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 ) (compare Figures 18 and 19 with Figures 27
and 28).