---

Revolving Circular Cosmic Strings

arXiv:gr-qc/9212007v1 10 Dec 1992gr-qc/9212007Weak-Field Gravity ofRevolving Circular Cosmic Strings∗Des J. Mc ManusSchool of Theoretical Physics, Dublin Institute for Advanced Studies,10 Burlington Road, Dublin 4, Ireland.†Michel A. VandyckPhysics Department, University College Cork, Ireland,andPhysics Department, Cork Regional Technical College,Bishopstown, Co. Cork, Ireland.AbstractA weak-field solution of Einstein’s equations is constructed. It is generated by a circularcosmic string revolving in its plane about the centre of the circle.

(The revolution is intro-duced to prevent the string from collapsing.) This solution exhibits a conical singularity,and the corresponding deficit angle is the same as for a straight string of the same linearenergy density, irrespective of the angular velocity of the string.PACS numbers: 04.20-q, 98.90+sTo appear in Physical Review D1

1.IntroductionOne of the most notable features of a straight cosmic string [1] is the presence in spacetimeof an angular deficit, the magnitude of which is related to the linear energy density µ of thestring by δψ = 8πGµ. Indeed, the deficit-angle structure of spacetime is central to manyproposals for the possible observation of the gravitational effects of cosmic strings [2].The deficit-angle model is generally accepted as a good approximation for describingthe exterior gravitational field of cosmic strings.Frolov, Israel, and Unruh (FIU) usedprecisely this approximation when they considered a thin circular string at a moment oftime symmetry [3].Recently, we investigated further [4] the problem of the deficit angle produced bycircular strings.We constructed a weak-field stationary solution of Einstein’s equationsgenerated by a thin circular string and established that external radial stresses had to beintroduced to support the ring against collapse (thus allowing a stationary solution to exist).The form of the radial stresses was completely determined by stress-energy conservation.The main result of our study was that a circular string produces conical singularities with thesame angular deficit as a straight string of identical linear energy density, fully supportingFIU’s assumption.

Thus, we demonstrated, in the weak-field limit, the validity of the FIUhypothesis directly from the field equations. Furthermore, the external radial stresses wereseen not to contribute to this angular deficit.In the present work, we ask the question whether it is possible to extend the previousresults to a self-supporting circular string, as opposed to an externally supported circularstring: can the stabilising role previously played by external radial stresses be played by aninternal mechanism?

Centrifugal force is the simplest candidate; it should be possible toprevent gravitational collapse by spinning the ring at an appropriate angular velocity.We construct a weak-field solution of Einstein’s equations corresponding to an in-finitely thin circular string revolving at a given angular velocity ω. We begin by examiningthe most general scenario where the string is partially supported by centrifugal force andpartially by external radial stresses.

The angular velocity is chosen arbitrarily and the radialstresses are then determined by stress-energy conservation. The angular deficit producedby this solution is found to be equal to the deficit produced by a straight string (of thesame linear energy density), irrespective of the value of the angular velocity ω (within thelimitation of the weak-field approximation).

Later, we also calculate and discuss the criti-cal angular velocity ωcrit at which the ring is totally self-supporting, namely the particularvelocity at which no radial stresses are necessary to support the ring. The latter is thensupported entirely by centrifugal force.2

In this paper, we use units in which ¯h = c = 1, take the metric to have signature(−, +, +, +) and adopt the geometrical conventions of Synge [5].2.The Stress-Energy TensorThe stress-energy tensor T µν generating the gravitational field of a revolving string partiallysupported against collapse by external radial stresses contains three contributions: MT µν,the contribution from the circular motion of the ring (excluding radial stresses requiredto maintain circular motion), AT µν, the contribution from the azimuthal flux through thestring (which corresponds to the T zz stresses for Vilenkin’s straight string), and ET µν, thecontribution from the external radial stresses. Given that the diameter of the core of astring arising from a spontaneously broken gauge theory [6] is microscopically small, it iswell justified to make the approximation that the stress-energy tensors MT µν and AT µν ofthe circular string be confined to the infinitely thin ring r = a, z = 0, where a denotesthe radius of the ring.

On the other hand, the radial stresses ET µν, being external, are notconfined to the core of the string. (See [4] for a more detailed discussion.

)For (pressure-free) dust of rest-energy density ρ0, moving with four-velocity uµ, thestress-energy tensor is MT µν = ρ0 uµ uν, where uµ uµ = −1. In cylindrical coordinatesxµ ≡(t, φ, r, z), the four-velocity uµ of an infinitely thin ring of radius a, centered at theorigin, lying in the x-y plane, and revolving with angular velocity ω is readily found asuµ=Γ (δµt + ω δµφ)(2.1)−Γ−2≡gtt(a, 0) + 2ω gtφ(a, 0) + ω2 gφφ(a, 0),(2.2)where gµν(a, 0) denotes the metric components evaluated at r = a, z = 0.

(The metricenters this equation because of the normalisation condition uν uν = −1. )Furthermore, the ring has its rest-energy density ρ0 purely localised:ρ0 = µ0 δ(r −a) δ(z),(2.3)where µ0 is the linear rest-energy density.Consequently, the stress-energy tensor MT µνproduced by the circular motion has the expressionMT µν = µ0 Γ2 (δµt + ω δµφ) [gνt(a, 0) + ω gνφ(a, 0)] δ(r −a) δ(z),(2.4)where g is the metric and no summation is implied over repeated indices t and φ.

(Thesummation convention will be suspended all throughout when t, φ or r appear as subscriptsor superscripts. )3

The stress-energy tensor giving the contribution from the azimuthal flux through thestring is elementary in the case of a non-revolving ring [4], the simple argument being thatAT φφ plays the role that T zz plays for a straight string. However, a more detailed reasoning isnecessary when the string is revolving since a revolving frame is not inertial.

The appropriatestress-energy tensor AT µν can be established by employing the general method applicable toanisotropic fluids [5]. (The “fluid” must be anisotropic since for physical reasons we expecta pressure along the φ axis but none along the r and z axes.

)The stress-energy tensor [5] of a fluid of rest-energy density ρ0, moving with a four-velocity uµ, isT µν = ρ0 uµ uν −Sµν,(2.5)in which the tensor Sµν, the stress tensor, satisfies Sµν uν = 0. The eigenvectors λµ(i), as-sumed space-like, and the corresponding eigenvalues −P(i), 1 ≤i ≤3 are called respectivelythe “principal axes” and the “principal stresses” of Sµν.

In terms of these quantities, thestress tensor can be expressed asSµν = −3Xi=1P(i) λµ(i) λ(i)ν,(2.6)where λµ(i) satisfiesuµ λµ(i)=0(2.7)λµ(i) λ(j)µ=δij. (2.8)The stress-energy tensor MT µν of (2.4) already takes into account the motion of thering; therefore, we only consider the stress part Sµν of (2.5).

The physical requirement thatthere be no pressure along the r and z axes implies that the principal pressures P(2) and P(3)vanish. Thus, the only non-zero principal pressure is the azimuthal pressure P(1) ≡P(φ).For an infinitely thin ring, the azimuthal pressure is confined to the core of the string, whichimpliesP(1) = k0 δ(r −a) δ(z),(2.9)where k0 is a constant with dimensions of force.

(We interpret −k0 as the tension in thering. )Moreover, the only relevant principal axis, λ(1), must point spatially along the φdirection.

The constraint (2.8), together with (2.7) applied to the velocity (2.1), determinesλ(1) asλµ(1) = K δµt + L δµφ,(2.10)4

where K and L are solutions of the systemK gtt(a, 0) + (L + ωK)gtφ(a, 0) + ωL gφφ(a, 0)=0(2.11)K2 gtt(a, 0) + 2KL gtφ(a, 0) + L2 gφφ(a, 0)=1. (2.12)Thus, we reach the conclusion that the stress-energy tensor AT µν generated by the azimuthalflux along the string readsAT µν = k0 (K δµt + L δµφ) [K gνt(a, 0) + L gνφ(a, 0)] δ(r −a) δ(z)(2.13)(with no summation on t or φ).

This tensor depends on two scalar parameters: ω, theangular velocity of the ring, and −k0, the tension in the ring. When ω vanishes, (2.13)becomes identical to the expression obtained in the non-revolving case [4], so that it isconsistent to identify the scalar k0 with the parameter k in [4].

Consequently, for stringmatter, it is reasonable to generalise the equation of state k = −µ of the non-revolving caseas k0 = −µ0 in the revolving case.In order to support the ring partially by external radial stresses, we assume theexistence of a radial component of ET µν confined to the x-y plane (as in the case [4] of anon-revolving ring):ET µν ≡∆(r, z) δµr δrν ≡f(r) δ(z) δµr δrν,(2.14)where f is a function to be determined later by stress-energy conservation, and there is nosummation over r.The complete stress-energy tensor T µν generating the gravitational field of the re-volving string is the sum of all the above contributions (2.4), (2.13), and (2.14):T µν =MT µν + AT µν + ET µν. (2.15)We now turn to the form of the spacetime metric g.3.The Metric and Field EquationsThe most general stationary metric produced by an axially-symmetric revolving source maybe written [7]ds2 = −e2ν dt2 + e2ζ−2ν r2 (dφ −Adt)2 + e2η−2ν (dr2 + dz2),(3.1)where ν, ζ, η, and A are functions of r and z only.5

We proceed as in the non-revolving case [4] and make the weak-field approximation.Thus, we calculate the Einstein tensor Gµν [7] for the metric (3.1) and retain only first-orderterms in ν, η, ζ, and A.Given that A is not dimensionless, it is not immediately apparent that terms oforder A2 may be neglected. However, for the stress-energy tensor (2.4), (2.13)–(2.15), thelinearised Einstein field equations show that this is indeed the case at first order in thedimensionless quantities Gµ0 and Gk0.

(A complete discussion is presented in AppendixA.) Therefore, in all our future considerations, the expression “weak-field approximation”will refer to the expansion of the field equations at first order in ν, η, ζ, A, Gµ0, and Gk0.With this approximation, the equations (2.2), (2.4), and (2.11)–(2.13) simplify greatly,and after some manipulations the field equations (see appendix A) reduce to∇2ν=4πG {(C1 + C2) δ(r −a) δ(z) + ∆(r, z)}(3.2)f∇2 η=8πG C2 δ(r −a) δ(z)(3.3)∇2 ζ + 1r ζr=8πG ∆(r, z)(3.4)∂∂r(rζ)=η −η0(3.5)∇2A + 2r Ar=−16πG C3 a−1 δ(r −a) δ(z),(3.6)in which ∆is the radial-stress function defined in (2.14), f∇≡∂2r + ∂2z, ∇≡f∇+ (1/r) ∂ris the Laplacian, η0 is an arbitrary constant of integration, and the constants Ci, 1 ≤i ≤3are related to the parameters µ0, k0, ω, and a of the problem byC1≡(µ0 + ω2a2 k0)Γ20(3.7)C2≡(ω2a2 µ0 + k0)Γ20(3.8)C3≡(µ0 + k0)Γ20 ωa(3.9)Γ−20≡1 −ω2 a2.

(3.10)(The quantities GCi, 1 ≤i ≤3, are dimensionless. )The compatibility condition for (3.3)–(3.5), or equivalently the stress-energy conser-vation law, determines the radial-stress function ∆of (2.14) as∆(r, z) = f(r) δ(z) = (C2/r) Θ(r −a) δ(z),(3.11)in which Θ denotes the Heaviside step function.

This expression, which is similar to the oneobtained in the non-revolving case [4], may now be substituted into (3.2), so that we havea complete set of equations of which the metric functions ν, ζ, η, and A are solutions.6

4.Angular DeficitOur main purpose in solving the field equations (3.2)–(3.6) is to investigate the metricfor conical singularities and to calculate the corresponding angular deficit. Because of thefact that conical singularities involve only the metric (3.1) at constant time t and constantazimuth φ, it is sufficient to restrict attention to obtaining explicitly the functions ν andη given by (3.2) and (3.3).

We solved these equations, for different values of the constantsCi, in our previous work on the non-revolving ring [4], and it is therefore not necessary torepeat the analysis here. We only recall that, in toroidal [8] coordinates (t, φ, σ, ψ), whichare related to the cylindrical coordinates (t, φ, r, z) byz/a ≡N−2 sin ψr/a ≡N−2sh σN2 ≡N2(σ, ψ) ≡ch σ −cos ψ,(4.1)the solutions near the string (namely for σ →∞) read:ν(σ, ψ)→−2G (C1 + C2) σ(4.2)η(σ, ψ)→−4G C2 σ.

(4.3)(As in the non-revolving case [4], the radial stresses (2.14) and (3.11) do not contribute tothese asymptotic forms for ν and η, and thus have no influence on the conical singularities. )It follows from (4.2) and (4.3) that the combination η −ν, which determines themetric (3.1) at constant t and φ, becomes, after substituting the definitions (3.7), (3.8),(3.10) and noting the non-trivial cancellation of the ω-dependent terms:η −ν →2G(µ0 −k0)σ.

(4.4)The fact that η −ν is proportional to the toroidal coordinate σ indicates the presence ofa conical singularity [3, 4]. The corresponding angular deficit δψ, which is related to theratio of the perimeter of a circle centered at the core of the string to the radius of this circle[3, 4], is given byδψ = 2π −limσ0→∞Z π−π dψ (N−2 eη−ν) |σ=σ0Z ∞σ0dσ N−2 eη−ν,(4.5)and is easily calculated asδψ = 4πG(µ0 −k0).

(4.6)As announced earlier, this expression is independent of the angular velocity ω atwhich the ring revolves. Moreover, the angular deficit is also identical with Vilenkin’s resultfor a straight string.

(For string matter, k0 = −µ0, as explained in [4].) We have thusdemonstrated, in the weak-field approximation, that a revolving circular string producesthe same angular deficit as a straight string of the same linear energy density.7

5.Self-supporting RingFinally, we address the problem of whether a revolving string can be totally self-supporting.Up to this point, we considered a ring partially supported by the Centrifugal Force producedby the revolution and partially supported by the external radial stresses ∆of (2.14) and(3.11).For the discussion that follows, the equation of state relating k0 and µ0 will conve-niently be written ask0 = (α −1) µ0,(5.1)where α is a parameter. ( This particular parametrisation excludes the possibility of thephysically uninteresting case µ0 = 0, k0 ̸= 0.) String matter is characterised by α = 0,whereas non-string matter has negative pressure for 0 < α < 1 and positive pressure ifα > 1.By definition the ring is self-supporting when no external radial stresses are necessaryto prevent collapse.

This happens, by virtue of (3.11), if and only if ω takes the criticalvalue ωcrit that forces C2 to vanish. Equations (3.8) and (3.10), upon inserting (5.1), implythat ωcrit is the solution of0 = µ0"−1 +α1 −(ωcrita)2#.

(5.2)The above constraint always has the trivial solution µ0 = 0, which in turn implies k0 = 0,so that spacetime is flat everywhere. For string matter α = 0, and this trivial solution isthe only solution that (5.2) admits.

Consequently, we have established that a ring of stringmatter cannot be made self-supporting exclusively by centrifugal force but that a certainamount of external stress is necessary to prevent collapse. In other words, trying to supporta ring of string matter purely by inducing a revolution requires the string to be massless,which is non physical.The solution of (5.2) in the physically interesting case, µ0 ̸= 0, is(ωcrita)2 = 1 −α .

(5.3)We observe that ωcrit is independent of G, and therefore of the gravitational field. Thisis a simple manifestation of the well-known [9] “Motion of the Source” problem in thelinearised Einstein equations: the linear approximation is sufficient to calculate the firstmetric correction produced by the source but neglects the back-reaction of gravity onto thesource, so that the source moves as is gravity were absent.

Taking this back-reaction into8

account requires using at least second-order terms as done, for instance, in the Einstein-Infeld-Hoffman procedure [10] for the motion of point-masses. It is important, however,to insist on the fact that, although the first-order framework is not appropriate to studygravitational influences on the motion of the source, it is perfectly valid to study gravitationalcorrections to the metric, and thus our result on the angular deficit holds.Further insight on the physical significance of (5.3) may be gained by studying, inClassical Mechanics, the equilibrium condition for a revolving ring of radius a, linear massdensity µ and tension T. Consider a small arc of angular width θ along the circle.

The twoextremities of this arc are subjected to a tension T which is tangential, but the resultantforce TR at the mid-point along the arc is purely radial and is given by TR = 2T sin (θ/2).On the other hand, the centrifugal force FC acting on the arc reads FC = µθω2 a2, andconsequently, equilibrium is attained when ω reaches the critical value ωcrit satisfying(ωcrit a)2 = Tµ limθ→0[(2/θ) sin (θ/2)] = Tµ. (5.4)This result is identical with (5.3) for T = −k0 and µ = µ0 since α = 1 + k0/µ0 by (5.1).It follows from (5.3) that matter must have a negative pressure to lead to a self-supporting ring [since α must be less than one for (5.3) to admit a solution for ωcrit].

This isphysically reasonable since a positive pressure would tend to create an expansion of the ring,whereas a negative pressure would tend to create a collapse. Only a collapse, and thereforea negative pressure, could be counteracted by Centrifugal Force.

(The gravitational forcedoes not contribute to the collapse at first order, as just mentioned. )Moreover, as the pressure becomes more negative and approaches −µ0, the criticalvelocity increases.

The extreme case of self-supporting string matter, namely α = 0, formallyimplies that ωcrita = ±1, which means that the string revolves at the speed of light. Thisis impossible for a massive body.

Thus, we reach the same conclusion as before that stringmatter cannot be made self-supporting by revolution only. (This may no longer hold ifgravity is explicitly taken into account in the motion of the source.) The present argument,albeit physically enlightening, is only formal since the weak-field approximation breaks downfor a massive body if ωa = ±1.We emphasise that the stress-energy tensor used in our treatment contains termsproportional to µ0 (1 −a2ω2)−1 and k0 (1 −a2ω2)−1.

Therefore, our results are valid whenthe dimensionless quantities Gµ0 (1−a2ω2)−1 and Gk0 (1−a2ω2)−1 are small. In particular,the value of ωa, for a ring partially supported by revolution and partially by external stresses,is not determined by any equation [in contrast with the totally self-supporting case (5.2),(5.3)], but is a free parameter.

Therefore, our main result, namely that the angular deficit9

produced by a revolving ring is the same as for a straight string of the same linear density,is valid, within the confines of the weak-field limit, for a large range of the values of theparameter aω.6.ConclusionIn this work, we extended our previous results [4] on the deficit-angle structure of spacetimeof a non-revolving circular string to the case of a revolving circular string. The circular stringwas prevented from collapse partially by the centrifugal force produced by the revolutionand partially by external radial stresses that were determined by stress-energy conservation.We established, in a weak-field treatment, that a conical singularity exists along the stringand that the magnitude of the corresponding deficit angle is the same as that produced bya straight string of the same linear energy density.We also investigated whether a revolving circular ring could be totally self-supporting,that is whether there existed a critical angular velocity ωcrit at which the ring is in equilibriumwithout the presence of external radial stresses.

We took string matter to have the equationof state k0 = −µ0, where −k0 and µ0 denote the tension and the linear rest-energy ofthe string, respectively, and found that it was impossible to have a self-supporting string.However, for non-string matter characterised by the equation of state k0 ̸= −µ0, the criticalangular velocity was established to be(ωcrita)2 = −k0/µ0,(6.1)where a is the radius of the ring. [The weak-field treatment was shown to be valid as longas the dimensionless quantities Gµ0(1 −a2ω2)−1 and Gk0(1 −a2ω2)−1 are small comparedto unity.]Acknowledgements:M.V.

gratefully acknowledges the Royal Irish Academy for a grant from the Research ProjectDevelopment Fund and the Cork Regional Technical College (in particular Mr. R. Langfordand Mr. P. Kelleher) for Leave of Absence. It is a pleasure to thank S.J.

Hughes and L.O’Raifeartaigh for enlightening discussions about this problem.10

Appendix A.Frame components of the Field equa-tionsThe most general stationary metric produced by an axially-symmetric source, namely (3.1),may be written in terms of an orthonormal basis, {θ(ˆµ)} asds2 = ηµν θ(ˆµ) ⊗θ(ˆν),(A.1)where ηµν = diag(−1, 1, 1, 1) andθ(ˆ0) ≡eν dtθ(ˆ1) ≡reζ−ν (dφ −Adt)θ(ˆ2) ≡eη−ν drθ(ˆ3) ≡eη−ν dz. (A.2)We define the transformation matrix eˆµν (and its inverse eµˆν) byθ(ˆµ) ≡eˆµν dxν.

(A.3)The components of the Einstein tensor and the stress-energy tensor in the frame θ(ˆµ) arethen simplyGˆµˆν = Gαβ eαˆµ eβˆν;Tˆµˆν = Tαβ eαˆµ eβˆν,(A.4)and Einstein’s equations read Gˆµˆν = −8πG Tˆµˆν.We now turn to the weak-field approximation. The metric (A.1) is flat if ν = η =ζ = 0 and A is constant.

Furthermore, the quantities ν, η, ζ are dimensionless, whereasA has dimensions of 1/length. Therefore, it is not immediately clear that a meaningfulapproximation to Einstein’s equations can be obtained by neglecting terms of order A2.To clarify this point, we begin by calculating the Einstein tensor in the orthonormalframe (A.2) up to terms of first order in ν, η, ζ and all orders in A.

The results areGˆ0ˆ0=−2(∇2ν) + (∇2 + 1r ∂r)ζ + f∇2η + r24 (1 + 2ǫ) (A2r + A2z)(A.5)Gˆ1ˆ1=−f∇2η + 3r24 (1 + 2ǫ) (A2r + A2z)(A.6)Gˆ2ˆ2=−ζzz −ηrr + r24 (1 + 2ǫ) (A2z −A2r)(A.7)Gˆ3ˆ3=−ζrr + 1r(ηr −2ζr) + r24 (1 + 2ǫ) (A2r −A2z)(A.8)Gˆ2ˆ3=ζrz + 1r (ζz −ηz) −r22 (1 + 2ǫ) Ar Az(A.9)Gˆ0ˆ1=−r2 LA,(A.10)11

withL ≡e−2(η+ζ−2ν)∂rhe(3ζ−4ν) ∂ri+ ∂zhe(3ζ−4ν) ∂zi+ 3rhe(3ζ−4ν) ∂ri,(A.11)where ǫ ≡ζ −ν −η, f∇2 ≡∂2r + ∂2z and ∇2 ≡f∇2 + (1/r) ∂r is the Laplacian operator. [Inequation (A.11), the exponentials are only to be taken to first order in their arguments.

]The stress-energy tensors (2.4) and (2.13) are proportional to µ0 and k0, respectively,and thus, their contributions to the Einstein equations are proportional to the dimensionlessquantities Gµ0 and Gk0. Therefore, the metric appearing in the expressions for the stress-energy tensors (2.4) and (2.13) may be replaced by the flat-space metric, namely ν = η =ζ = 0, A =const, if attention is restricted to a first-order treatment in Gµ0 and Gk0.

Hence,without making any approximation on the order of A, we calculate that the Tˆ0ˆ1 componentof the total stress-energy tensor (2.15) isG Tˆ0ˆ1 = G(µ0 + k0)a(A −ω)1 −a2(A −ω)2 δ(r −a) δ(z). (A.12)Consequently, the Einstein equation Gˆ0ˆ1 = −8πGTˆ0ˆ1 becomesLA = 16πG(µ0 + k0)A −ω1 −a2(A −ω)2 δ(r −a) δ(z).

(A.13)We now introduce the dimensionless function B byA ≡−16πG(µ0 + k0) ωB. (A.14)Note that A tends to zero with ω, so that the above parametrisation guarantees compatibilitybetween the revolving case and the non-revolving case [4], where A vanishes identically.Thus, the following equation for B is equivalent to (A.13):LB =1 + 16πG(µ0 + k0) ωB1 −a2ω2[1 + 16πG(µ0 + k0)B]2 δ(r −a) δ(z).

(A.15)Expanding B in powers of the dimensionless quantity G(µ0 + k0), we clearly seefrom (A.15) that the leading term 0B is of zeroth order in G(µ0 + k0) and satisfiesL 0B =11 −a2ω2 δ(r −a) δ(z). (A.16)As a result, by virtue of (A.14), the leading term in A is proportional to G(µ0 + k0).Therefore, we may neglect second powers of A and products of A with ν, η, or ζ in ourapproximation scheme, since we only retain first powers of Gµ0 and Gk0.12

∗Address after September 1, 1992: Dept. of Mathematics, Statistics, and ComputerScience, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4H6.†Research Associate of the Dublin Institute for Advanced Studies.References[1] Ya.

B. Zel’dovich, Mon. Not.

Roy. Astron.

Soc. 192, 663 (1980); A. Vilenkin, Phys.

Rep.121, 263 (1985); R.H. Brandenberger, Physics Scripta T36, 141 (1991); A. Vilenkin,Phys. Rev.

Lett. 46, 1169, 1496E (1981); A. Vilenkin, Phys.

Rev. D 23, 852 (1981).

[2] A. Vilenkin, Phys. Rev.

D 24, 2082 (1981); A. Vilenkin, Phys. Lett.

107B, 47 (1981); A.Vilenkin, Ap. J.

282, L51 (1984); J.R. Gott, Ap. J.

288, 422 (1985); N. Kaiser and A.Stebbins, Nature 310, 391 (1984); C.J. Hogan and M.J. Rees, Nature 311, 109 (1984);T. Vachaspati and A. Vilenkin, Phys.

Rev D 31, 3052 (1985); R. Brandenberger, A.Albrecht and N. Turok, Nuc. Phys.

B277, 605 (1986); F. Accetta and L. Krauss, Nuc.Phys. B319, 747 (1989); N. Sanchez and M. Signore, Phys.

Lett. 219B, 413 (1989).

[3] V.P. Frolov, W. Israel, and W.G.

Unruh, Phys. Rev.

D 39, 1084 (1989). [4] S.J.

Hughes, D.J. Mc Manus, and M.A.

Vandyck, Weak-Field Gravity of Circular Cos-mic Strings, submitted to Phys. Rev.

D (1992). [5] J.L.

Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1960), Chap.8, p. 310 and Chap. 4, p.

186. [6] J. Preskill Architecture of the Fundamental Interactions at Short Distances, edited byP.

Ramond and R. Stora (North Holland, Amsterdam, 1987). [7] S. Chandrasekhar and J.L.

Friedman, Astrophys. J.

175, 379 (1972). [8] H. Bateman, Partial Differential Equations of Mathematical Physics (Cambridge Univ.Press, Cambribge, 1952), p.

461. [9] H.C. Ohanian, Gravitation and Spacetime (Norton and co., New York, 1976), Chap.

3,p. 103.

[10] A. Einstein, L. Infeld, and B. Hoffmann, Ann. Math.

39, 65 (1938); A. Einstein and L.Infeld, Canad. J.

Math. 1, 209 (1949).13

---

출처: arXiv:9212.007 • 원문 보기