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Flat wormholes from straight cosmic strings

We describe the analytical extension of certain static cylindrical multi--cosmic string metrics to wormhole spacetimes with only one region at spatial infinity, and investigate in detail the geometry of asymptotically Minkowskian wormhole spacetimes generated by one or two cosmic strings. We find that such wormholes tend to lengthen rather than shorten space travel. Possible signatures of these wormholes are briefly discussed.


Introduction
It has long been recognized that the equations of general relativity carry information not only about the local geometry of spacetime, but also about its possible global topologies. Early work on wormholes [1]- [4] was motivated by the hope that they might provide a way to evade the problems associated with point singularities in particle physics. A quite different motivation was provided by the analysis of Morris and Thorne, who first investigated in some detail the possibility of using traversable wormholes to travel in space [5], as well as in time [6].
Traversable wormholes may occur as solutions to the Einstein field equations with suitable sources violating the weak energy condition. When explicit solutions are discussed in the literature, these are usually static spherically symmetric Einstein-Rosen wormholes connecting two regions at spacelike infinity [7]. More relevant for the purpose of interstellar travel are Wheeler-Misner wormholes [2], with only one region at spatial infinity. Exact solutions of the time-symmetric initial-value problem of general relativity with such a topology have been constructed [3] [4], but these non-static wormholes are not traversable [5]. In a recent paper [8], Schein and Aichelburg have constructed a static Wheeler-Misner wormhole by matching, along two spherical shells S 1 and S 2 , an outer Majumdar-Papapetrou spacetime to an inner Reissner-Nordström spacetime; this is traversable only one way, from S 1 to S 2 .
Static Wheeler-Misner wormholes may be obtained by suitably extending a procedure, described by Visser, to construct models of flat Einstein-Rosen wormholes [9]. Remove from Euclidean space R 3 a volume Ω . Take a second, identical copy of R 3 − Ω, and identify these two excised spaces along the boundaries ∂Ω. The spacetime obtained by factoring the resulting space with the time axis R is a geodesically complete Einstein-Rosen wormhole (or multi-wormhole if Ω has several connected components), everywhere flat except on ∂Ω, where the stress-energy is concentrated. To similarly construct a Wheeler-Misner wormhole, remove from R 3 two non-overlapping volumes Ω and Ω ′ which are the image of each other under the involution (x, y, z) → (−x, y, z), and identify the boundaries ∂Ω, ∂Ω ′ (Fig. 1) (the diffeomorphism Ω → Ω ′ must reverse orientation if the resulting manifold is to be orientable [10]). In a further extension of this procedure, the boundaries ∂Ω, ∂Ω ′ are not identified, but connected by a cylindrical tube carrying equal energy per unit length and longitudinal tension (if the surface ∂Ω is compact and simply connected, it follows from the Gauss-Bonnet theorem that the energy per unit tube length is 1/2G). The geometry, as viewed by an "external" observer (in R 3 −Ω−Ω ′ ), does not depend on the "internal" distance (through the tube) between the two "mouths" ∂Ω, ∂Ω ′ , which may be arbitrarily large, so that the advantage for space travel is not so obvious.
In the case where ∂Ω is a cylinder, the internal tube is, as well as external space, flat. The curvature is then concentrated on the two wormhole mouths, each of which carries (again by the Gauss-Bonnet theorem) a mass per unit length and a longitudinal tension both equal to m = −1/4G. For instance, the static conical Einstein-Rosen wormholes generated by a circular cylindrical source [11] may thus, in the case of a vanishing deficit angle, be extended to Wheeler-Misner wormholes with zero tube length, which may easily be generalized to the case of an arbitrary tube length.
Let us now discuss the case where Ω is a polyhedron. Visser showed [4] that in this case the curvature of the boundary ∂Ω is concentrated on the edges, which each carry an equal energy per unit length and tension. A particular case of Visser's polyhedral wormholes is obtained in the limit of cylindrical polyhedra, i. e. configurations of p parallel cosmic strings of tension m i , with i m i = −1/2G. Following the procedure described above, these Einstein-Rosen wormholes can be straigthforwardly be extended to Wheeler-Misner wormholes generated by p straight cosmic strings in the case of zero tube length, or 2p cosmic strings for an arbitrary tube length.
The purpose of this paper is to investigate in more detail the construction and geometry of flat cylindrical wormholes generated by straight cosmic strings, following an analytical method complementary to the geometrical method outlined above. In the second section we show how special multicosmic string metrics may be analytically extended [12][13] to Einstein-Rosen or Wheeler-Misner multi-wormhole, multi-cosmic string metrics. Because our spacetime is locally approximately Minkowskian, we shall be specially interested in asymptotically Minkowskian spacetimes. In the third section, we study in detail the topology and geometry of asymptotically Minkowskian, flat Wheeler-Misner wormholes generated by one or two straight cosmic strings. Geodesic paths through such wormholes are discussed in the fourth section, with applications to space travel and geometrical optics. Our results are summarized and discussed in the last section.

Wormholes from cosmic strings
We start from the well-known multi-cosmic string metric [14]- [16] where the 2-metric (ζ ≡ x + iy) may locally be transformed to the Cartesian form The spacetime of metric (2.1) is therefore flat outside the worldsheets of the cosmic strings ζ = a i (the conical singularities, with deficit angle 8πGm i , of the surface (2.2)), which carry an energy per unit length and a tension both equal to m i < 1/4G (for m i ≥ 1/4G the singularity ζ = a i is at spatial infinity). The metric (2.2) is also generically singular at the point at infinity in the complex ζ-plane, with deficit angle 8πG i m i . This last singularity is at infinite geodesic distance if In the generic case, the conical singularities of the metric (2.2) are logarithmic branch points which become branch points of order n for 4Gm i = 1/n (n integer). The conformal factor in (2.2) is analytical in the complex ζ-plane with cuts extending from the various branch points to infinity. Consider now the special case of the bicone with m 1 = m 2 = 1/8G, By choosing the cut to be the segment connecting the two branch points ζ = ±b, we can analytically extend this bicone to a geodesically complete surface: a cylinder. To show this [13], pinch the cylinder along a parallel. We thus obtain two identical bicones with deficit angles π at the two vertices and 2π at infinity, joined along the pinch. These two bicones are diffeomorphic to the two sheets of the Riemann surface of the metric (2.6) with the cut indicated (Fig. 2). The diffeomorphism is implemented by the transformation (by integration of (2.4)), which maps the cut into the circle u = 0 (v is an angular variable from (2.7)), and the two sheets into the two halves u > 0 and u < 0 of the cylinder. The cylinder with its two circles at infinity is the basic building block for Einstein-Rosen wormholes in two space dimensions. The general onewormhole flat metric is obtained by multiplying the right-hand side of (2.6) by an arbitrary conformal factor assumed to be regular at ζ = ±b. In the simplest case, take the cut to be the geodesic segment connecting the two branch points ζ = ±b; analytical continuation across this cut then leads to a surface with two symmetrical asymptotically conical sheets smoothly connected along a cylindrical throat, and two conical singularities -one in each sheet-at ζ = c. The corresponding spacetime (2.1) is therefore a two-cosmic string Einstein-Rosen wormhole. Note that the masses per unit length m of the sources are different from the "total" masses per unit length M determined from the asymptotic behaviour of the metric at either region at spatial infinity [17][13], the difference 1/4G being the topological contribution of the wormhole. For the spatial sections to be open (M ≤ 1/4G), m must be negative or zero.
In the special case m = −1/4G, the cosmic string and wormhole contributions to the total mass balance (for this reason we shall refer to the metric (2.10) as the "dipole" metric), so that the flat metric (2.1) is asymptotically Minkowskian. In Fig. 3 we show schematically the pattern of geodesics u = const. and v = const. for the dipole geometry (2.10) (in the case b and c real, |b| < |c|). The critical geodesic u = 0 which hits the singularity at ζ = c divides the ζ-plane in three regions. The geometry of the "left" region (u < 0) is that of a halfplane. In the "inner" region (u > 0), the geodesic cut u = L is surrounded by concentric closed geodesics u = const. with equal perimeter 2πc, until the geodesic segment u = 0 connecting the conical singularity with itself is reached; this region is the map of a truncated cylinder. Finally, the "right" region (u > 0) is again the map of a half-plane, however the geodesic distance between two distant geodesics v = D and v = −D is smaller in the "right" region than in the "left" region by a length 2l, with l = πc.
We thus arrive at the following geometrical construction for the t = const., z = const. sections of the dipole wormhole: 1) Remove from the (u, v) plane the semi-infinite strip u > 0, −l < v < l, and glue together the edges (u > 0, v = l) and (u > 0, v = −l). This yields a flat surface which is the union of the "left" and "right" regions discussed above, and has a closed boundary geodesic Γ of length 2l (the segment (u = 0, −l ≤ v ≤ l) connecting the singularity (u = 0, v = ±l) to itself). 2) Take a second identical surface with boundary Γ ′ , and glue together the two boundaries Γ, Γ ′ to the two ends of a truncated cylinder of perimeter 2l and length 2L, with The resulting flat surface has the Einstein-Rosen wormhole topology, is asymptotically Euclidean with two conical singularities, and is mapped on the two-sheeted ζ-plane by the analytical extension of (2.10) described above.
In the limit L → 0 (c → b), the two singularities coalesce to a single conical singularity; this wormhole can be viewed as a very special case of Visser's polyhedral wormholes, a monohedron with one edge (the straight cosmic string) bounding one face (Γ × R) connecting the two sheets. The general n-wormhole metric is obtained by first making on the cylinder metric (2.6) the conformal transformation ζ → P n (ζ), where P n is a polynomial of order n, then multiplying the resulting n-cylinder metric by a conformal factor regular at the zeroes of P n (ζ) ∓ b. Multi-wormholes generated by a sigma-model field coupled repulsively to gravity are discussed in [13]. The metric for a flat n-wormhole spacetime generated by p cosmic strings is (2.1) with Various extensions of this metric across the n cuts are possible. In the symmetrical extension, the Riemann surface is made of two sheets joined along the n-component cut. The resulting spacetime is an n-wormhole Einstein-Rosen spacetime, with p cosmic strings in each sheet, and total mass per unit length in accordance with the Gauss-Bonnet theorem [13]. The simplest case after n = 1 is n = 2, p = 0: Then, M = 1/2G, so that the two-dimensional spatial sections of genus 1, orientable by construction, are compact and regular, i.e. are tori T 1 = S 1 × S 1 . To recover the symmetrical Riemann surface, pinch the torus along two opposite circles; this yields two tetracones with deficit angles π at each vertex, joined along the two pinches, which correspond to the two cuts of the Riemann surface. The flat metric (2.3) on the torus, with u and v periodical, is transformed to the metric (2.15) by , where sn is a biperiodical Jacobi function. However the metric (2.15) admits a more economical analytical extension to a topologically non trivial Riemann surface with only one sheet. Such a possibility derives from the observation [13] that the torus may be pinched only once into a single tetracone joined to itself by an identification of the two edges. This identification corresponds to an identification of the two cuts, leading to the identification of the two sheets, of the Riemann surface for the complex variable ζ(w), the point ζ in the first sheet of the symmetrical extension being identified with the point −ζ in the second sheet 1 . A large circle geodesic v = const. is thus mapped into a line connecting the two cuts either in the upper or in the lower half-plane of the ζ-plane, so that a particle going around the torus along this geodesic falls into e.g. the left-hand cut to come out again from the right-hand cut (Fig. 4).
Such a one-sheeted extension is possible whenever the distribution of both the n cuts and the p conical singularities of the flat metric (2.12) is invariant under the isometry ζ → −ζ, so that the two sheets of the symmetrical extension may be identified together. In the case n = 2 the resulting surface -a topological torus with a point at infinity (provided M < 1/4G) and p conical singularities-is a Wheeler-Misner wormhole. In fact our construction is the three-dimensional counterpart of Lindquist's [4] reinterpretation of a four-dimensional Einstein-Rosen manifold with two identical spherical bridges as a single Wheeler-Misner wormhole by identifying corresponding points on the two sheets of the Einstein-Rosen manifold . In the next section we investigate the various possible geometries for asymptotically Minkowskian (M = 0) Wheeler-Misner wormholes with p = 1 or 2.
3 Two-string and one-string asymptotically Minkowskian Wheeler-Misner wormholes The Wheeler-Misner wormhole generated by two cosmic strings is the onesheeted extension of a symmetrical n = p = 2 metric (2.12). In the asymptotically Minkowskian case, this metric depends on three complex parameters a, b, c. According to the relative values of these parameters, the non-extended geometrical configuration may belong to one of three possible generic types DD, AA, or Q.
1) Dipole-dipole (DD). The sequences of closed "u-geodesics" surrounding each of the two symmetrical geodesic cuts terminate in two disjoint geodesic segments, each connecting one of the singularities with itself. A single v-geodesic segment, bissecting the angles formed by the continuations of these critical u-geodesics, connects the two singularities together. An instance of this case is a, b, c real, a 2 > b 2 + c 2 (Fig. 5(a)). The analytical extension of this geometry to the Riemann surface obtained by identification of the two cuts, as described in the previous section, leads to the DD wormhole geometry. The geometrical construction of this wormhole (Fig. 5(b)) follows closely that of the dipole Einstein-Rosen wormhole, except that the two copies of the (u, v) plane deprived of a semi-infinite strip are replaced by a single Euclidean plane deprived of a rectangular strip. Two opposite edges, of length 2d, of this rectangle are glued together, while the other two edges, of length 2l, are glued to the two ends of a truncated cylinder of circumference 2l and length 2L.
2) Antidipole-antidipole (AA). In this type again, two disjoint critical ugeodesic segments connecting each of the two singularities with itself enclose concentric closed u-geodesics surrounding a cut. However the two singularities are now connected by two symmetrical v-geodesic segments, bissecting the two angles formed by a critical closed u-geodesic segment and its continuation to infinity ( Fig. 6(a), drawn for a, b, c real, b 2 < a 2 < c 2 − b 2 ). The corresponding Wheeler-Misner wormhole geometry turns out to be equivalent to that of the Q wormhole, as we shall presently explain.
3) Quadrupole (Q). In this case the sequences of closed u-geodesics surrounding the two cuts terminate in two contiguous geodesic contours made from three u-segments connecting the two singularities together ( Fig. 6(b), drawn for a and b real with b 2 < a 2 , and c imaginary). The critical vgeodesics bissecting the two angles formed by this self-intersecting u-geodesic are v = d and v = −d, where 2d is the geodesic distance between the two singularities along the "external" segments of the u-geodesic u = 0. The distance between the two singularities along the "central" segment u = 0 is 2(l − d), where l is again the perimeter of the closed u-geodesics. The geometrical construction of the Q Wheeler-Misner wormhole resulting from identification of the two cuts is shown in Fig. 7. The Euclidean plane is incised along a segment AB of length 2d. A torus of "small" perimeter 2l (l > d) and "large" perimeter 2L is also incised along a matching segment A ′ B ′ (length 2d) of a small circle. Finally, the torus and the plane are glued together along the two edges of the cuts AB, A ′ B ′ . These two edges correspond to the two external geodesic segments u = 0 in Fig. 6(b); the complementary small-circle segment connecting A ′ and B ′ is mapped into the central segment u = 0 in Fig. 6(b), while the antipodal small circle on the torus is mapped into the two identified cuts of the complex plane. Now we show the equivalence of the Wheeler-Misner analytical extensions of the AA and Q metrics of Fig. 6, by showing that they correspond to two dual maps for the same basic geometry in Fig. 7. Instead of cutting the torus along the antipodal small circle (which leads to the Q map of Fig. 6(b)), cut it along the large circle through O in Fig. 7. In the Q map, this large circle is, as all large circles, a v-geodesic connecting together the two identified cuts of Fig. 6(b). In the AA map, the same large circle now corresponds to the two identified cuts of Fig. 6(a), which are connected together by a sequence of small circles -now v-geodesics -terminating in the two sides v = ±l of the segment A ′ B ′ (AB); the two critical geodesics v = ±d of Fig. 6(b) -large circles going through A ′ and B ′ -correspond to the two critical geodesics u = ±d of Fig. 6(a). In other words, the AA and Q maps are transformed into each other under the duality u ↔ v exchanging the two circles of S 1 × S 1 .
Asymptotically Minkowskian wormholes generated by a single cosmic string may be obtained from the two-cosmic string case by taking limits such that the two cosmic strings (the two singularities in the two-dimensional spatial sections) coincide. Two inequivalent geometries may result, DD 0 or 8. Consider first the DD wormhole geometry, Fig. 5(b), and take the limit L → 0. The resulting "DD 0 " geometry may be directly obtained from the Euclidean plane deprived of a rectangular strip by gluing together, first two opposite edges of the rectangle, then the other two edges. Clearly, by construction the singularity is connected to itself by only two geodesics of lengths 2d and 2l. A metric which leads to this geometry after a one-sheeted analytical extension is (3.1) with a, b, c real, a 2 = b 2 + c 2 .
Taking the limit d → 0 in the DD geometry of Fig. 5(b) is obviously equivalent to taking the limit d → l in the Q geometry of Fig. 7. The resulting "8" geometry ( Fig. 8) is obtained by incising the Euclidean plane along a segment of length 2l, bringing together the two vertices so that the two edges make a figure 8, then gluing these two edges to the two ends of a truncated cylinder of circumference 2l and length 2L. The singularity is connected to itself by (2 + N) geodesics, two geodesics of length 2l (the two edges just mentioned), and a denumerable family of geodesics of length 2 √ L 2 + n 2 l 2 going from one end of the cylinder to the other while winding n times around it. Two examples of metrics (3.1) leading to this geometry are given by real parameters a, b, c with b 2 < a 2 , and either c 2 = 0 or (because of the equivalence AA ↔ Q) c 2 = a 2 + b 2 .
The two preceding one-cosmic string geometries depend on two parameters. By taking the further limit d = 0 in the DD 0 geometry, or L = 0 in the 8 geometry, we obtain the "I" geometry, which corresponds simply to a plane with two points identified. A single geodesic, of length 2l, connects the resulting conical singularity to itself. This limiting geometry may be obtained from the one-sheeted extension of the metric

Space travel and geometrical optics
Because our wormhole spacetimes are (almost everywhere) flat, as well as asymptotically Minkowskian, they do not classically scatter test particles or light rays. A test particle going through a wormhole will emerge with a worldline parallel to its ingoing worldline. However the outgoing worldline will generically be shifted, in space as well as in time. Shifts in space lead to non-trivial geometrical optics effects, while shifts in time might be relevant for, e.g., intergalactic travel. We first consider shifts in time, with a view to address the question, raised in the Introduction, whether traversable wormholes are really advantageous for long distance space travel. Because of the simple form of our spacetime metric (2.1), time shifts only depend on the three-velocity of the test particle and on the geodesic distance travelled in two-dimensional sections t = const., z = const. So we consider some given two-dimensional geodesic as "start" line, and another, parallel geodesic as "finish" line, and compare the geodesic distance between these two lines along two paths, one which "goes through the wormhole", and another which does not go through the wormhole. First we have to give a workable definition of "going through the wormhole". These questions are usually addressed in the context of Einstein-Rosen wormholes, where a path which goes from one point at spatial infinity to the other obviously "goes through the wormhole". More generally, consider a space E with N points at spatial infinity. We compactify this space to a closed topological space E, and define a "path going through a wormhole" as a path, going from spatial infinity to spatial infinity, which is not homotopic to zero. This definition covers in particular both the case of the Einstein-Rosen wormhole, a topological sphere with two points at infinity (paths connecting these two points are trivially non contractible), and that of the Wheeler-Misner wormholes of the previous section, topological tori with one point at infinity (closed paths are non-contractible if they wind around one or both circles).
For simplicity, we first deal with the case of one cosmic string. Fig. 9 shows two parallel geodesics 1 and 2 going through a DD 0 wormhole of parameters (l, d); an effect of the shifts in space mentioned above is that geodesic 1 comes in to the right of geodesic 2 but comes out to its left. These geodesics cannot be deformed to the spectator geodesics 3 or 4 without crossing the singularity. Both are shorter than the spectator geodesics, the path being shorter by 2d cos θ for geodesic 1, and by 2l sin θ for geodesic 2 (θ is the incidence angle of geodesic 1). So we have here a model of a one-cosmic string wormhole which does indeed shorten space travel. There is however a hazard: a moving object, such as a spaceship, assumed to have a size of the order of the dimensions l, d of the wormhole, and to be light enough so as not to affect the geometry, would be cut in four pieces by the cosmic string.
Consider now the case of the 8 wormhole (Fig. 8). Obviously a geodesic hitting, under the incidence θ, one edge of the incision in the Euclidean plane then goes the full length L of the cylinder, while winding a number of times around it before emerging from the other, contiguous edge. So travel through the wormhole will always be longer in this case. We find that the path excess is, for a geodesic path, where n = [(L/l) tan θ] is the winding number of the path, i.e. the integer part of the number of turns inside the cylinder. This number increases without limit as the incidence angle θ nears π/2, so that ∆ ≃ 2L cos θ for θ → π/2 (4.2) becomes arbitrarily large. Finally, the I wormhole can be obtained as a limit of both the DD 0 and the 8 wormholes so that, while technically a wormhole according to our general definition, it is without effect on space travel. Now for the two-cosmic string wormholes. In the DD geometry of Fig.  5(b) there are, as in the DD 0 case, two kinds of paths through the wormhole, corresponding to the two circles of S 1 × S 1 . Geodesic paths crossing once the two identified segments of length 2d in Fig. 5(b) are "shorter" (than they would be in Euclidean space) by 2l sin θ. Geodesic paths crossing once the two circular junctions of length 2l may wind n times around the cylinder; they can be shorter if L < d, but are always longer if L > d, the path excess being given in terms of the winding number by ∆ = 2(L − d) cos θ + 2nl sin θ. So the path excess becomes very large in the limit of a very small gatewidth (d ≪ l) or of a grazing incidence (θ ≃ π/2); when both limits are taken, we obtain from Eqs. (4.4)-(4.6) Light rays from one geometrical point (e. g. a galactic source S) to another (e. g. an observer O) may similarly follow a variety of homotopically inequivalent optical paths (geodesics), leading to an array of geometrical images S ′ . From Fig. 9, the DD 0 wormhole behaves as a rectangular prism of infinite refractive index, and thus gives two images of a point source. The 8 wormhole behaves rather as a parallel plate with partially reflecting faces, multiple reflections being replaced by multiple turns around the cylinder; the result is that a point source gives rise to a one-dimensional periodical array of images. In the case of the DD geometry, the observer would see a single "near" image (due to light rays crossing the two identified segments of length 2d in Fig. 5(b), equivalent to a parallel plate of infinite index) together with a one-dimensional array of increasingly distant images (due to light rays winding around the cylinder). Finally, in the case of the Q wormhole, the possibility of light rays winding around both the large circle and the small circle of the torus leads to a two-dimensional array of images S ′ , as shown in Fig. 10 (where the observer and source are assumed to be in the same plane z = const.).

Conclusion
We have shown that certain analytical maps may be extended to describe Wheeler-Misner wormholes (with only one region at spatial infinity) which are everywhere flat except for parallel cosmic string singularities. We have used these analytically extended maps to investigate the asymptotically Minkowskian one-wormhole geometries generated by two cosmic strings (these are either of the DD or Q type), or by a single cosmic string (of the DD 0 or 8 type). As anticipated in the Introduction, it appears that such wormholes would have on the mean the effect of lengthening rather than shortening space travel.
Because of this lengthening, which could be arbitrarily large, and of the unpleasant consequences of accidentally hitting one of the cosmic strings, a space traveller might wish to avoid these wormholes altogether. The presence of such hypothetical wormholes as well as their type could in principle be inferred from the pattern of images of a source viewed through the wormhole. The DD 0 wormhole (the only one to always shorten space travel) would give only two images, while other wormholes (which generically lengthen space travel) would give one-or two-dimensional arrays of images. Such a characterization is incomplete: most of the images could be too faint to be detected, or hidden behind other objects. A wave-optics treatment should make possible a better characterization. One anticipates non-trivial effects arising both from diffraction by the topological defects (cosmic strings) and resonance due to periodicity conditions in the cylinder (or torus).
Our construction of static wormholes generated by staight cosmic strings could be extended in two directions. A first extension should be to construct wormholes generated by non-parallel moving straight cosmic strings, along the lines of the analytical construction [18] of spacetimes generated by multiple moving crossed cosmic strings, and to investigate the causal structure of these spacetimes.
Another possible extension would be to investigate wormholes generated by closed cosmic strings or rings. As pointed out by Visser [9], the polyhedon Ω mentioned in the Introduction can be collapsed to a dihedron, or an irregular two-sided disk connecting two copies of R 3 . The resulting Einstein-Rosen wormhole is generated by the cosmic ring circumscribing the disk. Some time ago, Zipoy [19] constructed analytical static solutions to the vacuum Einstein equations with a circular ring singularity and a double-sheeted topology. More recently, Bronnikov and co-workers [20] have similarly constructed static Einstein-Rosen wormhole solutions to D-dimensional gravity with a circular ring singularity. However, the analytical construction of Wheeler-Misner ring wormholes has not yet been attempted. Hopefully, this problem could be addressed along the lines followed here for straight cosmic strings.