We consider the spacetime metric

with constants and . As usual, the azimuthal coordinate is defined modulo . For and , metric (133) is the Minkowski metric in cylindrical coordinates. For any other values of and , the metric is still (locally) flat but not globally isometric to Minkowski spacetime; there is a singularity along the -axis. For and , the plane , has the geometry of a cone with a deficit angle (see Figure 23); for there is a surplus angle. Note that restricting the metric (133) with to the hyperplane gives the same result as restricting the metric (115) of the Barriola–Vilenkin monopole to the hyperplane .The metric (133) describes the spacetime around a straight spinning string. The constant is related to the string’s mass-per-length , in Planck units, via

whereas the constant is a measure for the string’s spin. Equation (135) shows that we have to restrict to the deficit-angle case to have non-negative. One may treat the string as a line singularity, i.e., consider the metric (133) for all . (This “wire approximation”, where the energy-momentum tensor of the string is concentrated on a 2-dimensional submanifold, is mathematically delicate; see [134].) For a string of finite radius one has to match the metric (133) at to an interior solution, thereby getting a metric that is regular on all of .Historical notes.

With , the metric (133) and its geodesics were first studied by Marder [214, 215]. He also discussed
the matching to an interior solution, without, however, associating it with strings (which were no issue at
that time). The same metric was investigated by Sokolov and Starobinsky [307] as an example
for a conic singularity. Later Vilenkin [331, 332] showed that within the linearized Einstein
theory the metric (133) with describes the spacetime outside a straight non-spinning
string. Hiscock [158], Gott [143], and Linet [208] realized that the same is true in the full
(non-linear) Einstein theory. Basic features of lensing by a non-spinning string were found by
Vilenkin [332] and Gott [143]. The matching to an interior solution for a spinning string, , was
worked out by Jensen and Soleng [169]. Already earlier, the restriction of the metric (133)
with to the hyperplane was studied as the spacetime of a spinning particle in
2 + 1 dimensions by Deser, Jackiw, and ’t Hooft [77]. The geodesics in this (2 + 1)-dimensional
metric were first investigated by Clément [60] (cf. Krori, Goswami, and Das [191] for the
(3 + 1)-dimensional case). For geodesics in string metrics one may also consult Galtsov and Masar [130].
The metric (133) can be generalized to the case of several parallel strings (see Letelier [206] for the
non-spinning case, and Krori, Goswami, and Das [191] for the spinning case). Clarke, Ellis and
Vickers [58] found obstructions against embedding a string model close to metric (133) into an
almost-Robertson–Walker spacetime. This is a caveat, indicating that the lensing properties of “real”
cosmic strings might be significantly different from the lensing properties of the metric (133).

Redshift and Fermat geometry.

The string metric (133) is stationary, so the results of Section 4.2 apply. Comparison of metric (133) with
metric (61) shows that the redshift potential vanishes, . Hence, observers on -lines see each other
without redshift. The Fermat metric and Fermat one-form read

Light cone.

For the metric (133), the lightlike geodesics can be explicitly written in terms of elementary functions.
One just has to apply the coordinate transformation to the lightlike
geodesics in Minkowski spacetime. As indicated above, the new coordinates are not globally
well-behaved on the entire spacetime. However, they can be chosen as continuous and single-valued
functions of the affine parameter along all lightlike geodesics through some chosen event,
with taking values in . In this way we get the following representation of the lightlike
geodesics that issue from the observation event into the past:

Lensing by a non-transparent string.

With the lightlike geodesics known in terms of elementary functions, positions and properties of images can
be explicitly determined without approximation. We place the observation event at , ,
, , and we consider a light source whose worldline is a -line at , ,
with . From Equations (138, 139, 140) we find that the images are in one-to-one
correspondence with integers such that

Lensing by a transparent string.

In comparison to a non-transparent string, a transparent string produces additional images.
These additional images correspond to light rays that pass through the string. We consider
the case and , which is illustrated by Figures 24 and 25. The general
features do not depend on the form of the interior metric, as long as it monotonously interpolates
between a regular axis and the boundary of the string. In the non-transparent case, there is a
single-imaging region and a double-imaging region. In the transparent case, the double-imaging region
becomes a triple-imaging region. The additional image corresponds to a light ray that passes
through the interior of the string and then smoothly slips over one of the cusp ridges. The
point where this light ray meets the worldline of the light source is on the sheet of the light
cone between the two cusp ridges in Figure 25, i.e., on the sheet that does not exist in the
non-transparent case of Figure 24. From the picture it is obvious that the additional image
shows the light source at a younger age than the other two images (so it is a “tertiary image”).
A light source whose worldline meets the caustic of the observer’s past light cone is on the
borderline between single-imaging and triple-imaging. In this case the tertiary image coincides
with the secondary image and it is particularly bright (even infinitely bright according to the
ray-optical treatment; recall Section 2.6). Under a small perturbation of the worldline the bright
image either splits into two or vanishes, so one is left either with three images or with one
image.

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