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., [334] 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 [295]
for observations of a recent candidate and [164] for a discussion of
the general perspective). Basic lensing features for various string
configurations are briefly summarized in [9]. 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 functions.
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 [135].) 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 [213, 214]. 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 [308] as an example for a
conic singularity. Later Vilenkin [332, 333
] showed that within
the linearized Einstein theory the metric (133) with
describes the spacetime outside a straight
non-spinning string. Hiscock [159],
Gott [144
], and
Linet [207] 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 [333
] and Gott [144]. The matching to
an interior solution for a spinning string,
, was worked out by Jensen and Soleng [170].
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 [192
] for the (3 +
1)-dimensional case). For geodesics in string metrics one may also
consult Galtsov and Masar [131].
The metric (133) can be generalized to
the case of several parallel strings (see Letelier [205] for the
non-spinning case, and Krori, Goswami, and Das [192] 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
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. [309]). 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 [115, 29, 155].
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:
The affine parameter
coincides with
-arclength
, and
parametrize the observer’s celestial sphere,
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 [123] (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.
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
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 [333] 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.
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.