5.10 Straight spinning string
Cosmic strings (and other topological defects) are expected to exist in the universe, resulting from a
phase transition in the early universe (see, e.g.,  for a detailed account). So far, there is no direct
observational evidence for the existence of strings. In principle, they could be detected by their lensing effect
(see  for observations of a recent candidate and  for a discussion of the general perspective). Basic
lensing features for various string configurations are briefly summarized in . Here we consider the simple
case of a straight string that is isolated from all other masses. This is one of the most attractive
examples for investigating lensing from the spacetime perspective without approximations. In
particular, studying the light cones in this metric is an instructive exercise. The geodesic equation is
completely integrable, and the geodesics can even be written explicitly in terms of elementary
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 .) 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 .
Figure 23: On a cone with deficit angle , the point can be connected to every point
in the double-imaging region (shaded) by two geodesics and to a point in the single-imaging
region (non-shaded) by one geodesic.
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  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 , Gott , and Linet  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  and Gott . The matching to an interior solution for a spinning string, , was
worked out by Jensen and Soleng . 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 . The geodesics in this (2 + 1)-dimensional
metric were first investigated by Clément  (cf. Krori, Goswami, and Das  for the
(3 + 1)-dimensional case). For geodesics in string metrics one may also consult Galtsov and Masar .
The metric (133) can be generalized to the case of several parallel strings (see Letelier  for the
non-spinning case, and Krori, Goswami, and Das  for the spinning case). Clarke, Ellis and
Vickers  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.
As the Fermat one-form is closed, , the spatial paths of light rays are the geodesics of the Fermat
metric (cf. Equation (64)), i.e., they are not affected by the spin of the string. can be transformed
to zero by changing from to the new time function . Then the influence of the string’s spin on
the travel time (62) vanishes as well. However, the new time function is not globally well-behaved (if
), because is either discontinuous or multi-valued on any region that contains a full circle
around the -axis. As a consequence, can be transformed to zero on every region that does not
contain a full circle around the -axis, but not globally. This may be viewed as a gravitational analogue of
the Aharonov–Bohm effect (cf. ). The Fermat metric (136) describes the product of a cone
with the -line. Its geodesics (spatial paths of light rays) are straight lines if we cut the cone
open and flatten it out into a plane (see Figure 23). The metric of a cone is (locally) flat but
not (globally) Euclidean. This gives rise to another analogue of the Aharonov–Bohm effect, to
be distinguished from the one mentioned above, which was discussed, e.g., in [114, 29, 154].
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
The affine parameter coincides with -arclength , and parametrize the observer’s celestial
Equations (138, 139, 140, 141) give us the light cone parametrized by . The same equations
determine the intersection of the light cone with any timelike hypersurface (source surface) and thereby the
exact lens map in the sense of Frittelli and Newman  (recall Section 2.1). For and
, the light cone is depicted in Figure 24; intersections of the light cone with hypersurfaces
(“instantaneous wave fronts”) are shown in Figure 27. In both pictures we consider
a non-transparent string of finite radius , i.e., the light rays terminate if they meet the
boundary of the string. Figures 25 and 28 show how the light cone is modified if the string is
transparent. This requires matching the metric (133) to an interior solution which is everywhere
regular and letting light rays pass through the interior. For the non-transparent string, the
light cone cannot form a caustic, because the metric is flat. For the transparent string, light
rays that pass through the interior of the string do form a caustic. The special form of the
interior metric is not relevant. The caustic has the same features for all interior metrics that
monotonously interpolate between a regular axis and the boundary of the string. Also, there is
no qualitative change of the light cone for a spinning string as long as the spin is small.
Large values of , however, change the picture drastically. For , where is the
radius of the string, the -lines become timelike on a neighborhood of the string. As the
-lines are closed, this indicates causality violation. In this causality-violating region the
hypersurfaces are not everywhere spacelike and, in particular, not transverse to
all lightlike geodesics. Thus, our notion of instantaneous wave fronts becomes pathological.
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:
Figure 24: Past light cone of an event in the spacetime of a non-transparent string of finite
radius with and . The metric (133) is considered on the region , and
the light rays are cut if they meet the boundary of this region. The coordinate is not shown,
the vertical coordinate is time . The “chimney” indicates the region which is occupied
by the string. The light cone has no caustic but a transverse self-intersection (cut locus). The cut
locus, in the (2 + 1)-dimensional picture represented as a curve, is actually a 2-dimensional spacelike
submanifold. When passing through the cut locus, the lightlike geodesics leave the boundary of the
chronological past . Note that the light cone is not a closed subset of the spacetime.
Figure 25: Past light cone of an event in the spacetime of a transparent string of finite radius
with and . The metric (133) is matched at to an interior metric, and
light rays are allowed to pass through the interior region. The perspective is analogous to Figure 24.
The light rays which were blocked by the string in the non-transparent case now form a caustic. In
the (2 + 1)-dimensional picture the caustic consists of two lightlike curves that meet in a swallow-tail
point (see Figure 26 for a close-up). Taking the -dimension into account, the caustic actually
consists of two lightlike 2-manifolds (fold surfaces) that meet in a spacelike curve (cusp ridge). The
third picture in Figure 2 shows the situation projected to 3-space. Each of the past-oriented lightlike
geodesics that form the caustic first passes through the cut locus (transverse self-intersection), then
smoothly slips over one of the fold surfaces. The fold surfaces are inside the chronological past
, the cusp ridge is on its boundary.
Figure 26: Close-up of the caustic of Figure 25. The string is not shown. Taking the -dimension
into account, the swallow-tail point is actually a spacelike curve (cusp ridge).
Figure 27: Instantaneous wave fronts in the spacetime of a non-transparent string of finite radius
with and . The picture shows in 3-dimensional space the intersections of
the light cone of Figure 24 with three hypersurfaces , at values . The
vertical coordinate is the -coordinate which was suppressed in Figure 24. Only one half of each
instantaneous wave front is shown so that one can look into its interior. There is a transverse
self-intersection (cut locus) but no caustic.
Figure 28: Instantaneous wave fronts in the spacetime of a transparent string of finite radius
with and . The picture is related to Figure 25 as Figure 27 is related to Figure 24.
Instantaneous wave fronts that have passed through the string have a caustic, consisting of two cusp
ridges that meet in a swallow-tail point. This caustic is stable (see Section 2.2). The caustic of the
light cone in Figure 25 is the union of the caustics of its instantaneous wave fronts. It consists of
two fold surfaces that meet in a cusp ridge, like in the third picture of Figure 2.
Lensing by a non-transparent string.
They can be numbered by the winding number in the order The total number
of images depends on . Let be the largest integer and be the smallest
integer such that . Of the two integers and , denote the
odd one by and the even one by . Then we find from Equation (143)
Thus, the number of images is even in a wedge-shaped region behind the string and odd everywhere else. If
the light source approaches the boundary between the two regions, one image vanishes behind the string
(see Figure 23 for the case ). (If the non-transparent string has finite thickness, there is also a
region with no image at all, in the “shadow” of the string.) The coordinates on the observer’s
sky of an image with winding number and the affine parameter at which the light source is
met can be determined from Equations (138, 139, 140). We just have to insert ,
, and to solve for , , :
The travel time follows from Equation (141):
It is the only relevant quantity that depends on the string’s spin . With the observer on a -line, the
affine parameter coincides with the area distance, , because in the (locally) flat string
spacetime the focusing equation (44) reduces to . For observer and light source on -lines, the
redshift vanishes, so also coincides with the luminosity distance, , owing to the general
law (48). Hence, Equation (148) gives us the brightness of images (see Section 2.6 for the
relevant formulas). The string metric produces no image distortion because the curvature tensor
(and thus, the Weyl tensor) vanishes (recall Section 2.5). Realistic string models yield a mass
density that is smaller than . So, by Equation (135), only the case and
is thought to be of astrophysical relevance. In that case we have a single-imaging region,
, and a double-imaging region, (see Figure 23). The occurrence
of double-imaging and of single imaging can also be read from Figure 24. In the double-imaging
region we have a (“primary”) image with and a (“secondary”) image with .
From Equations (147, 148) we read that the two images have different latitudes and different
brightnesses. However, for close to 1 the difference is small. If we express by Equation (134)
and linearize Equations (146, 147, 148, 149) with respect to the deficit angle (134), we find
Hence, in this approximation the two images have the same coordinate; their angular distance on
the sky is given by Vilenkin’s  formula
and is thus independent of ; they have equal brightness and their time delay is given by the
string’s spin via Equation (154). All these results apply to the case that the worldlines of
the observer and of the light source are -lines. Otherwise redshift factors must be added.
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