The metric (115) was briefly mentioned
as an example for a conical singularity by Sokolov and
Starobinsky [308
]. 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 [243].) For this reason,
the “almost exact lens map” of Virbhadra and Ellis [337] (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
).
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