Straight spinning string

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., [333 ] 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 [294 ] for observations of a recent candidate and [163 ] 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 $a$ and $k > 0$ . As usual, the azimuthal coordinate $φ$ is defined modulo $2π$ . For $a = 0$ and $k = 1$ , metric (133

) is the Minkowski metric in cylindrical coordinates. For any other values of $a$ and $k$ , the metric is still (locally) flat but not globally isometric to Minkowski spacetime; there is a singularity along the $z$ -axis. For $a = 0$ and $0 < k < 1$ , the plane $t = constant$ , $z = constant$ has the geometry of a cone with a deficit angle

(see Figure 23

); for $k > 1$ there is a surplus angle. Note that restricting the metric (133

) with $a = 0$ to the hyperplane $z = constant$ gives the same result as restricting the metric (115

) of the Barriola–Vilenkin monopole to the hyperplane $𝜗 = π∕2$ .

The metric (133 ) describes the spacetime around a straight spinning string. The constant $k$ is related to the string’s mass-per-length $μ$ , in Planck units, via

whereas the constant $a$ is a measure for the string’s spin. Equation (135

) shows that we have to restrict to the deficit-angle case $k < 1$ to have $μ$ non-negative. One may treat the string as a line singularity, i.e., consider the metric (133

) for all $ρ > 0$ . (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 $ρ∗$ one has to match the metric (133

) at $ρ = ρ ∗$ to an interior solution, thereby getting a metric that is regular on all of $ℝ4$ .

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Figure 23: On a cone with deficit angle $0 < δ < π$ , the point $p$ can be connected to every point $q$ in the double-imaging region (shaded) by two geodesics and to a point in the single-imaging region (non-shaded) by one geodesic.

Historical notes.
With $a = 0$ , 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 $a = 0$ 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, $a ⁄= 0$ , was worked out by Jensen and Soleng [169 ]. Already earlier, the restriction of the metric (133 ) with $a ⁄= 0$ to the hyperplane $z = 0$ 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, $f = 0$ . Hence, observers on $t$ -lines see each other without redshift. The Fermat metric $ˆg$ and Fermat one-form $ϕˆ$ read

As the Fermat one-form is closed, $dˆϕ = 0$ , the spatial paths of light rays are the geodesics of the Fermat metric $ˆg$ (cf. Equation (64

)), i.e., they are not affected by the spin of the string. $ˆ ϕ$ can be transformed to zero by changing from $t$ to the new time function $t − aφ$ . 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 $a ⁄= 0$ ), because $φ$ is either discontinuous or multi-valued on any region that contains a full circle around the $z$ -axis. As a consequence, $ϕˆ$ can be transformed to zero on every region that does not contain a full circle around the $z$ -axis, but not globally. This may be viewed as a gravitational analogue of the Aharonov–Bohm effect (cf. [308 ]). The Fermat metric (136

) describes the product of a cone with the $z$ -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 ].

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 $(t,φ ) ↦− → (t − aφ, kφ )$ 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 $s$ along all lightlike geodesics through some chosen event, with $φ$ taking values in $ℝ$ . In this way we get the following representation of the lightlike geodesics that issue from the observation event $(ρ = ρ ,φ = 0,z = 0,t = 0) 0$ into the past:

∘ ------------------------------- ρ(s) = s2sin2Θ + 2sρ0sinΘ cosΨ + ρ20, (138 ) tan(kφ (s)) = --s-sin-Θ-sinΨ----, (139 ) ρ0 + ssinΘ cos Ψ z(s) = scosΘ, (140 ) t(s) = − s + aφ (s). (141 )

The affine parameter $s$ coincides with $ˆg$ -arclength $ℓ$ , and $(Ψ, Θ )$ parametrize the observer’s celestial sphere,

| ( ρ (s )cosφ (s)) | ( cosΨ sin Θ) -d-( ) || ( ) ds ρ(s)sin φ (s) || = sinΨ sinΘ . (142 ) z (s ) s=0 cos Θ

Equations (138

, 139

, 140

, 141

) give us the light cone parametrized by $(s,Θ, Ψ )$ . 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 [122 ] (recall Section 2.1). For $k = 0.8$ and $a = 0$ , the light cone is depicted in Figure 24

; intersections of the light cone with hypersurfaces $t = constant$ (“instantaneous wave fronts”) are shown in Figure 27

. In both pictures we consider a non-transparent string of finite radius $ρ ∗$ , 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 $a$ is small. Large values of $a$ , however, change the picture drastically. For $a2 > k2ρ2 ∗$ , where $ρ ∗$ is the radius of the string, the $φ$ -lines become timelike on a neighborhood of the string. As the $φ$ -lines are closed, this indicates causality violation. In this causality-violating region the hypersurfaces $t = constant$ are not everywhere spacelike and, in particular, not transverse to all lightlike geodesics. Thus, our notion of instantaneous wave fronts becomes pathological.

Figure 24: Past light cone of an event $pO$ in the spacetime of a non-transparent string of finite radius $ρ∗$ with $k = 0.8$ and $a = 0$ . The metric (133

) is considered on the region $ρ > ρ∗$ , and the light rays are cut if they meet the boundary of this region. The $z$ coordinate is not shown, the vertical coordinate is time $t$ . The “chimney” indicates the region $ρ < ρ ∗$ 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 $I− (pO)$ . Note that the light cone is not a closed subset of the spacetime.

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Figure 25: Past light cone of an event $p O$ in the spacetime of a transparent string of finite radius $ρ∗$ with $k = 0.8$ and $a = 0$ . The metric (133

) is matched at $ρ = ρ ∗$ 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 $z$ -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 $− I (pO)$ , the cusp ridge is on its boundary.

View Image

Figure 26: Close-up of the caustic of Figure 25

. The string is not shown. Taking the $z$ -dimension into account, the swallow-tail point is actually a spacelike curve (cusp ridge).

View Image

Figure 27: Instantaneous wave fronts in the spacetime of a non-transparent string of finite radius $ρ∗$ with $k = 0.8$ and $a = 0$ . The picture shows in 3-dimensional space the intersections of the light cone of Figure 24

with three hypersurfaces $t = constant$ , at values $t1 > t2 > t3$ . The vertical coordinate is the $z$ -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.

View Image

Figure 28: Instantaneous wave fronts in the spacetime of a transparent string of finite radius $ρ∗$ with $k = 0.8$ and $a = 0$ . 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.
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 $ρ = ρ0$ , $φ = 0$ , $z = 0$ , $t = 0$ , and we consider a light source whose worldline is a $t$ -line at $ρ = ρS$ , $φ = φS$ , $z = zS$ with $0 ≤ φS ≤ π$ . From Equations (138 , 139 , 140 ) we find that the images are in one-to-one correspondence with integers $n$ such that

They can be numbered by the winding number $n$ in the order $n = 0,− 1, 1,− 2,2,...$ The total number of images depends on $k$ . Let $N1 (k)$ be the largest integer and $N2 (k)$ be the smallest integer such that $N1(k) ≤ 1∕k < N2 (k )$ . Of the two integers $N1(k )$ and $N2 (k )$ , denote the odd one by $Nodd (k )$ and the even one by $Neven (k )$ . 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 $1 ≤ 1∕k < 2$ ). (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 $(Ψn, Θn )$ on the observer’s sky of an image with winding number $n$ and the affine parameter $sn$ at which the light source is met can be determined from Equations (138

, 139

, 140

). We just have to insert $ρ(s) = ρS$ , $φ (s) = φ + 2nπ S$ , $z(s) = z S$ and to solve for $tan Ψ = tan Ψ n$ , $tanΘ = tanΘ n$ , $s = s n$ :

tanΨ = ---ρSsin(k-(φS-+--2nπ))---, (146 ) n ρS cos(k(φS + 2n π)) − ρ0 ∘ -2----2-------------------------- tanΘ = ---ρS +-ρ-0 −-2ρS-ρ0cos(k(φS-+-2nπ)), (147 ) n zS ∘ -2----2----2-------------------------- sn = zS + ρS + ρ0 − 2ρSρ0 cos(k(φS + 2n π)). (148 )

The travel time follows from Equation (141

):

It is the only relevant quantity that depends on the string’s spin $a$ . With the observer on a $t$ -line, the affine parameter $s$ coincides with the area distance, $Darea(s) = s$ , because in the (locally) flat string spacetime the focusing equation (44

) reduces to $¨Darea = 0$ . For observer and light source on $t$ -lines, the redshift vanishes, so $s$ also coincides with the luminosity distance, $Dlum(s) = s$ , 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 $μ$ that is smaller than $10 −4$ . So, by Equation (135

), only the case $Nodd (k) = 1$ and $Neven(k) = 2$ is thought to be of astrophysical relevance. In that case we have a single-imaging region, $0 ≤ φS < 2π − π ∕k$ , and a double-imaging region, $2π − π∕k < φS ≤ π$ (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 $n = 0$ and a (“secondary”) image with $n = − 1$ . From Equations (147

, 148

) we read that the two images have different latitudes and different brightnesses. However, for $k$ close to 1 the difference is small. If we express $k$ by Equation (134

) and linearize Equations (146

, 147

, 148

, 149

) with respect to the deficit angle (134

), we find

ρ0π − ρS φS φSρS δ Ψ0 = ----------- − ------------ (150 ) ρS + ρ0 (ρS + ρ0)2π -ρS-δ-- Ψ −1 = Ψ0 + ρS + ρ0, (151 ) Θ− 1 − Θ0 = 0, (152 ) s−1 − s0 = 0, (153 ) T − T = 2aπ. (154 ) −1 0

Hence, in this approximation the two images have the same $Θ −$ coordinate; their angular distance $Δ$ on the sky is given by Vilenkin’s [332 ] formula

and is thus independent of $φS$ ; they have equal brightness and their time delay is given by the string’s spin $a$ via Equation (154

). All these results apply to the case that the worldlines of the observer and of the light source are $t$ -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 $a = 0$ and $1 < 1∕k < 2$ , 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.