Dynamic cosmic strings. I

PHYSICAL REVIEW D, VOLUME 63, 024011

Dynamic cosmic strings. I

K. R. P. Sjodin,* U. Sperhake,† and J. A. Vickers‡

Faculty of Mathematical Studies, University of Southampton, Southampton, S017 1BJ, United Kingdom~Received 30 June 2000; published 21 December 2000!

The field equations for a time dependent cylindrical cosmic string coupled to gravity are reformulated interms of geometrical variables defined on a (211)-dimensional spacetime by using the method of Gerochdecomposition. Unlike the 4-dimensional spacetime the reduced case is asymptotically flat. A numericalmethod for solving the field equations which involves conformally compactifying the space and including nullinfinity as part of the grid is described. It is shown that the code reproduces the results of a number of vacuumsolutions with one or two degrees of freedom. In the final section the interaction between the cosmic string anda pulse of gravitational radiation is briefly described. This is fully analyzed in the sequel.

DOI: 10.1103/PhysRevD.63.024011 PACS number~s!: 04.20.Ha, 04.20.Jb, 04.25.Dm, 04.30.Db

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I. INTRODUCTION

Cosmic strings are topological defects that formed durphase transitions in the early Universe. They are imporbecause they are predicted by grand unified theories andduce density perturbations in the early Universe that mibe important in the formation of galaxies and other larscale structures@1#. They are also important since they athought to be sources of gravitational radiation due to raoscillatory motion@2#. In the simplest case of a string moving in a fixed background one can take the thin string limand the dynamics are given by the Nambu-Goto action@3#which is known to admit oscillatory solutions. Howeverorder to fully understand the behavior of cosmic strings oshould study the field equations for a cosmic string coupto Einstein’s equations.

A cosmic string is described by a U~1! gauge vector fieldAm coupled to a complex scalar fieldF51/&Seif. The La-grangian for these coupled fields is given by

LM5 12 ¹mS¹mS1 1

2 S2~¹mf1eAm!~¹mf1eAm!

2l~S22h2!22 14 FmnFmn, ~1!

where Fmn5¹mAn2¹nAm ,e,l are positive coupling constants andh is the vacuum expectation value.

The Einstein-scalar-gauge field equations for an infinitlong static cosmic string have been investigated by Lagand Garfinkle@4#. Some geometrical techniques relatedthose used in this paper have also been used in the anaof cosmic strings by Peter and Carter@5# and by Carter@6#.In the present paper~and its sequel! we will investigate thebehavior of a time dependent cylindrical cosmic stricoupled to gravity. In particular we investigate the effect opulse of gravitational radiation on an initially static cosmstring and the corresponding gravitational radiation thaemitted as a result of oscillations in the string. Since weinterested in the gravitational radiation produced by

*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]

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string it is desirable to measure this at null infinity where tgravitational flux is unambiguously defined and one doesneed to impose artificial outgoing radiation conditions atedge of the numerical grid. However the infinite length of tstring in thez-direction prevents the spacetime from beiasymptotically flat. We therefore follow the approachClarke et al. @7# and use a Geroch decomposition with rspect to the Killing vector in thez-direction to reformulatethe problem in (211) dimensions. Note however that unlikClarke et al. @7# we apply the Geroch decomposition to thentire problem not just to the exterior characteristic regiThe gravitational degrees of freedom of the (311) problemare then encoded in two geometrically defined scalar fiedefined on the (211)-dimensional spacetime. These are tnorm of the Killing vectorn and the Geroch potentialt forthe rotation. We show in Sec. III that the energy-momenttensor of these fields describes the gravitational energy oforiginal cylindrical problem. An important feature of the reduced (211) spacetime is that it is asymptotically flat anthis allows us to conformally compactify the spacetime ainclude null infinity as part of the numerical grid.

In Sec. II we briefly describe the Geroch decompositiand the field equations that one obtains in the cylindricase. We also show how it is possible to rescalet, r, and thematter variables to simplify the equations. In order to deonstrate the numerical accuracy of the full code it is usefucompare it with either an exact solution or else with soother independently produced numerical results. Unfornately this is not possible for the full code where we musatisfy ourselves with the internal consistency of the codedemonstrated by convergence testing for example. Ifconsiders only the gravitational part of the code it is possito specialize to the case of a vacuum solution. The simpof these is the Weber-Wheeler solution@8#, but we also con-sider a rotating vacuum solution due to Xanthopoulos@9#which describes a thin cosmic string and a second rotavacuum solution due to Piranet al. @10#. In order to compareour results with these exact solutions we must first wrthem in terms of the Geroch potential, this is done in Sec.This description is useful not just for numerical purposesalso in interpreting these solutions since the two polarizatstates for the gravitational radiation have a simple interptation in terms of the Geroch variablesn andt. The numeri-

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K. R. P. SJO¨ DIN, U. SPERHAKE, AND J. A. VICKERS PHYSICAL REVIEW D63 024011

cal code used in this case, the convergence analysis ancomparison between the numerical results and the exaclution is given in Sec. IV. As far as the matter part of tcode is concerned there are no exact solutions and onecompare the code with other numerical results. The simpspecial case involves decoupling the matter variables fthe metric variables and considering the equations of moin Minkowski space. There do not seem to be other resavailable in the dynamic case, but the static solutions hbeen investigated by a number of authors@11#, @12#, and@13#. Another solution which has been investigated preously is a static string which is coupled to the gravitationfield. Finding such solutions is much harder than one misuppose due to the asymptotic behavior of the matter vables. As well as the physical solution to the equations, this an exponentially diverging nonphysical solution@13#which must be suppressed. By compactifying the radialordinate we can control the behavior of the solution at infiity in the static case by using a relaxation scheme. Thislows us to obtain solutions for all values of the radius ratthan the fairly restricted range ofr that had been previouslobtained, and also permits us to use the proper bounconditions at infinity. The static string is described in furthdetail in Secs. V and VI. Finally we briefly describe thnumerics for the dynamic string coupled to gravity. Heagain the asymptotics of the matter variables make it harwrite a stable code, but by using an implicit scheme weable to produce a code with long term stability and secoorder convergence which agrees with the previous resultthe special cases described earlier. From a physical poinview the most interesting feature of this code is that it is ato describe the interaction of a gravitational field with twdegrees of freedom with the full nonlinear cosmic striequations. So far we have only investigated the interactiothe cosmic string with an incoming Weber-Wheeler typulse of gravitational radiation with just one degree of fredom. We find that the pulse excites the cosmic string acauses the scalar and vector fields to vibrate with a frequewhich is roughly proportional to their respective massThis oscillation slowly decays and the string eventuallyturns to its previous static state. This is briefly describedSec. VII and in detail in the sequel@14#, which we hence-forth will refer to as paper II, where comparisons with othresults and a full convergence analysis is given.

II. THE GEROCH DECOMPOSITION

As we explained above it is not possible for a cylindcally symmetric cosmic string to be asymptotically flat dto the infinite extent of the string in thez-direction. By fac-toring out this direction we can obtain a 3-dimensionspacetime that is asymptotically flat. If the Killing vectorthe z-direction is hypersurface orthogonal then one can sply project onto the surfacesS given byz5const. Howeverwe wish to consider cylindrical solutions which also haverotating mode and in this case the Killing vectorjm is nothypersurface orthogonal. Geroch@15# has shown how to factor out the Killing direction in this more general case. Tidea is to identify points which lie on the same integ

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curves of the Killing vector field and thus obtainS as aquotient space rather than as a subspace. There is thone-to-one correspondence between tensor fields onS andtensor fields on the 4-dimensional manifoldM which havevanishing contraction with the Killing vector and also vaishing Lie derivative alongjm. One may therefore use thfour dimensional metricgmn to define a metrichmn on Saccording to the equation

hmn5gmn1~jsjs!21jmjn . ~2!

The extra information in the 4-metric is encoded in two negeometric variables; the norm of the Killing vector

n52jmjm ~3!

~where we have introduced the minus sign to maken positivein the spacelike case! and the twist

tm52emntsjn¹tjs. ~4!

Geroch then showed how it is possible to rewrite Einsteiequations in terms of the 3-dimensional Ricci curvature(S,h) and equations involving the 3-dimensional covariaderivatives ofn and the twist. If we letDm define the covari-ant derivative with respect tohmn then one can show that

D [rts]5ersmnjmRtnjt, ~5!

whereRtn is the 4-dimensional Ricci tensor. It is clear th

this vanishes in vacuum so thatts is curl free and may bedefined in terms of a potential. It is a remarkable fact ththis remains true for spacetimes with a cosmic string enermomentum tensor so that even in this case we may writ

ta5Dat, ~6!

where we have introduced the convention of using Latindices to describe quantities defined onS.

We may now write Einstein’s equations for th4-dimensional spacetime (M ,g) in terms of the Ricci curva-ture of (S,h) and the two scalar fieldsn andt defined onS.We obtain

Rab5 12 n22@~Dat!~Dbt!2hab~Dmt!~Dmt!#

1 12 n21DaDbn2 1

4 n22~Dan!~Dbn!

18phamhb

n~Tmn2 12 gmnT!, ~7!

D2n5 12 n21~Dmn!~Dmn!2n21~Dmt!~Dmt!

116p~Tmn2 12 gmnT!jmjn, ~8!

D2t5 32 n21~Dmt!~Dmn!. ~9!

The transformation to the 3-dimensional description isonly mathematically convenient but is physically meaninful. If the Killing vector is hypersurface orthogonal thentvanishes and the gravitational radiation has only one poization which may be defined in a simple way in terms ofn.If there are both polarizations present then the1 mode isgiven in terms ofn while the3 mode is given in terms oft

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DYNAMIC COSMIC STRINGS. I PHYSICAL REVIEW D63 024011

@see Eq.~19! below#. One can simplify things by makingconformal transformation and using the metrichab5nhab inwhich case we can write the vacuum Einstein-Hilbert Lgrangian onM in terms of 3-dimensional variables onS,

I G5EM

RA2gd4x ~10!

5ES$R2 1

2 n22@ hab~Dat!~Dbt!

1hab~Dan!~Dbn!#%Ahd3x ~11!

and we see that the 4-dimensional gravitational field isscribed in three dimensions by the two scalar fieldsn andtconformally coupled to the 3-dimensional spacetime wmetric hab . Since the 3-dimensional spacetime has no Wcurvature it is essentially nondynamic and we see thatn andt encode the two gravitational degrees of freedom inoriginal spacetime. The corresponding ‘‘energy-momentutensor for these fields is

Tab5 12 n22@DatDbt2 1

2 habhcd~Dct!~Ddt!1DanDbn

2 12 habh

cd~Dcn!~Ddn!# ~12!

and if there is matter present in four dimensions therealso additional matter terms in three dimensions. As shoin Eq. ~23! this 3-dimensional ‘‘energy-momentum’’ tensofor n andt gives the correct expression for the 4-dimensiogravitational energy.

III. THE FIELD EQUATIONS

For the case of a cylindrically symmetric vacuum spatime one can write the metric in Jordan-Ehlers-KunKompaneets~JEKK! form @16,17#

ds25e2~g2c!~dt22dr2!2r2e22cdf22e2c~vdf1dz!2.

~13!

However this form of the metric is not compatible with thcosmic string energy momentum tensor so we follow Mar@18# by introducing an extra variablem into the metric andwriting it in the form

ds25e2~g2c!~dt22dr2!2r2e22cdf2

2e2~c1m!~vdf1dz!2. ~14!

This form of the line element enables us to make easy cparisons with the JEKK vacuum solutions previously consered numerically by Dubalet al. @19# and d’Invernoet al.@20#. The field equation form decouples from those for thother metric variables and it has a source term givenT002T11. The physical interpretation ofm is briefly dis-cussed by Marder@18#. In the static case one can show thC2 regularity on the axis implies thatm5 ln(n)2g. The met-ric given by Eq.~14! has zero shift and lapse determinedthe conditiongtt52grr . In this gauge the null geodesics a

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given by the simple conditionsu5t2r5const andv5t1r5const. The remaining coordinate freedom is giventhe freedom to relabel the radial null surfaces:u→ f (u) andv→g(v) wheref andg are arbitrary functions. We may fixthis by specifying the initial values ofm and its derivative.For example we can choosem to be equal to its static valueand m ,t to vanish, but due to time dependent matter souterms in the evolution equation form @see Eq.~28! below#this does not makem constant in time.

In terms of these variables we find the norm of the Killinvector in thez-directionjm5d3

m to be given by

n5e2~c1m! ~15!

and the twist potential is related tov by

Dst5r21e4c13m~v ,r ,v ,t,0,0!. ~16!

Finally the conformal 3-metrichab is given by

ds25e2~g1m!~dt22dr2!2r2e2mdf2. ~17!

It is also of interest to calculateTab in terms of thesevariables. We find

Tabtatb5

1

8e22~g1m!F S n ,u

n D 2

1S n ,v

n D 2

1S t ,u

n D 2

1S t ,v

n D 2G ,~18!

whereta is a unit timelike vector proportional to]/]t. Notethat the quantities

A5S n ,u

n D 2

, B5S n ,v

n D 2

, C5S t ,u

n D 2

, D5S t ,v

n D 2

,

~19!

are exactly the same as the quantitiesA,B,C, andD whichare given~by more complicated expressions! in terms ofcandv in Eqs.~4a!–~4d! of Piranet al. @21# and describe thetwo polarizations of the cylindrical gravitational fieldFurthermore if we consider the special case of vacuum stions and integrateTabt

atb over the regionV5$0<r<r0 ,t5t0% with respect to the volume formdV on t5t0 we find

E~ t0 ,r0!5E EVTabt

atbdV ~20!

5p

4 E0

r0e2gF S n ,u

n D 2

1S n ,v

n D 2

1S t ,u

n D 2

1S t ,v

n D 2Grdr ~21!

52pE0

r0g ,re2gdr ~22!

52p@12e2g~ t0 ,r0!#, ~23!

where we have used the vacuum field equations forg ,r @seeEq. ~31! below#. Note that this is the same as the ener

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obtained by Ashtekaret al. @22# but does not require theKilling vector to be hypersurface orthogonal. It differs frothe C-energy in general but agrees with it in the linearizcase.

As far as the matter variables are concerned we makeobvious generalization of the form used by Garfinkle@26#and write

F51

&S~ t,r!eif, ~24!

Am51

e@P~ t,r!21#¹mf. ~25!

We may now write the field equations for the complete stem. Since we are working in three dimensions we have thindependent evolution equations. After some algebra thmay be written as

hn2n21~t ,t22t ,r

2 2n ,t21n ,r

2 !2m ,tn ,t1m ,rn ,r

528pn@2ln21e2~g1m!~S22h2!2

1e22r22ne22m~P,t22P,r

2 !#, ~26!

ht12n21~t ,tn ,t2t ,rn ,r!2~m ,tt ,t2m ,rt ,r!50, ~27!

hm1r21m ,r2m ,t21m ,r

2

528p@2ln21e2~g1m!~S22h2!21r22e2gS2P2#,

~28!

whereh represents the flat spacetime d’Alembertian whin cylindrical coordinates is given by

h52]2

]t2 1]2

]r2 1r21]

]r. ~29!

There are also three constraint equations, one of whvanishes identically due to the rotational symmetry. Themaining equations give

g ,t5r

11rm ,r@m ,tr2m ,t~g ,r1m ,r!1 1

2 n22~t ,tt ,r1n ,tn ,r!

18p~S,tS,r1e22r22ne22mP,tP,r!#, ~30!

g ,r5r

r2m ,t22~11rm ,r!2 „~11rm ,r!

3$24p@2n21e2~g1m!l~S22h2!21~S,t21S,r

2 !

1e22r22ne22m~P,t21P,r

2 !1r22e2gS2P2#

2 14 n22~t ,t

21t ,r2 1n ,t

21n ,r2 !2r21m ,r2m ,rr%

1rm ,t@12 n22~t ,tt ,r1n ,tn ,r!1r21m ,t1m ,tr

18p~S,tS,r1e22r22ne22mP,tP,r!#…. ~31!

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Finally there are the equations for the matter variablesSandP which may be derived from the Euler-Lagrange equatiofor LM or alternatively from the contracted Bianchi identies.

hS2S,tm ,t1S,rm ,r5S@4ln21e2~g1m!~S22h2!

1r22e2gP2#, ~32!

hP2P,t~n21n ,t2m ,t!1P,r~n21n ,r2m ,r22r21!

5e2n21e2~g1m!PS2. ~33!

We also need to supplement these equation by bounconditions on the axis. For the 4-dimensional metric vaables the simplest condition is to require the metric to beC2

on the axis so that we have a well defined curvature tenThis gives the conditions

c~ t,r!5a1~ t !1O~r2!, ~34!

v~ t,r!5O~r2!, ~35!

m~ t,r!5a2~ t !1O~r2!, ~36!

g~ t,r!5O~r!. ~37!

In terms ofn andt this gives

n~ t,r!5a3~ t !1O~r2!, ~38!

t~ t,r!5O~r2!, ~39!

where we have chosen the additive constant in the definiof the potentialt so that it vanishes on the axis. In certasituationsC2 regularity is too strong and one must impothe weaker condition of elementary flatness@23#. Howevereven this is too strong for the Xanthopoulos solution whhas a conical singularity on the axis.

The boundary conditions forS andP on the axis are@26#

S~ t,r!5O~r!, ~40!

P~ t,r!511O~r2!. ~41!

In order to consider the behavior of the solution at ninfinity we first transform fromt to a null time coordinateu5t2r. We also wish to compactify the region and following Clarkeet al. @7# we make the transformation

y51

Ar. ~42!

In terms of these variables the equations become

hn1y3n21~t ,ut ,y2n ,un ,y!1 14 y6n21~t ,y

2 2n ,y2 !

1 12 y3~m ,yn ,u1m ,un ,y1 1

2 y3m ,yn ,y!

58pn@22ln21e2~g1m!~S22h2!2

1 12 e22y7ne22m~ 1

2 y3P,y2 12P,uP,y!#, ~43!

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ht2y3n21~t ,un ,y1t ,yn ,u1 12 y3t ,yn ,y!1 1

2 y3~m ,yt ,u

1m ,ut ,y1 12 y3m ,yt ,y!50, ~44!

hm2y2~m ,u1 12 y3m ,y!2 1

2 y3m ,y~12 y3m ,y22m ,u!

528p@2ln21e2~g1m!~S22h2!21y4e2gS2P2#, ~45!

g ,u5y2„~12 1

2 ym ,y!@2 12 y3m ,uy2m ,uu1m ,u

2

2 12 n22~t ,u

2 1 12 y3t ,ut ,y1n ,u

2 1 12 y3n ,un ,y!#

11

8y3m ,u@m ,y~12ym ,y!2ym ,yy#2 1

16 y4n22m ,u

3~t ,y2 1n ,y

2 !28p$~12 12 ym ,y!@S,u

2 1 12 y3S,uS,y

1e22y4ne22m~P,u2 1 1

2 y3P,uP,y!#

2 18 y4m ,u~S,y

2 1e22y4

3ne22mP,y2 !%…/$@y22~m ,u1 1

2 m ,y!#22m ,u2 %, ~46!

g ,y5@2 14 ~3y2m ,y1y3m ,yy!1 1

4 y3m ,y2 2 1

8 y3n22~t ,y2 1n ,y

2 !

22py3~S,y2 1e22y4ne22mP,y

2 !#/~y22 12 y3m ,y!, ~47!

hS1 12 y3S,um ,y1 1

2 y3S,y~m ,u1 12 y3m ,y!

5S@4ln21e2~g1m!~S22h2!1y4e2gP2#, ~48!

hP1P,u@ 12 y3~n21n ,y2m ,y!12y2#

1 12 y3P,y@n21~n ,u1 1

2 y3n ,y!2~m ,u1 12 y3m ,y!12y2#

5e2n21e2~g1m!PS2, ~49!

where in these coordinatesh is given by

h5y2

4 S 4y]2

]u]y1y4

]2

]y2 1y3]

]y24

]

]uD . ~50!

It is worth remarking that one can have solutions to thequations which are regular aty50 and which represen4-dimensional metrics in whichv diverges. Thus the notionof the 3-dimensional spacetime being asymptotically flaweaker than might first be supposed. The asymptotic behior of S andP at null infinity is given by

S~u,y!5h1O~y!. ~51!

P~u,y!5O~y!. ~52!

These are discussed in more detail in Sec. V.The field equations are solved numerically in two wa

Firstly an explicit second order Cauchy-characteristic mating ~CCM! scheme similar to that employed by Dubalet al.@19# and d’Invernoet al. @20# is used, but using a Gerocdecomposition in the whole spacetime~not just the charac-teristic portion! which allows one to use the geometricaldefined variables in both the interior and exterior regio

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This scheme works very well when compared to the exvacuum solutions but is less satisfactory when the materms are included~see below!. An alternative scheme withsimilar accuracy but with long term stability is a fully chaacteristic second order implicit scheme. This has the advtage that the scheme naturally controls the growth ofderivatives at infinity and hence automatically selectsphysical rather than the nonphysical solutions. The detailthis scheme are described in paper II.

IV. EXACT VACUUM SOLUTIONS

In this section we describe the exact vacuum solutiowhich are used to test the codes. The solutions we will csider are the Weber-Wheeler gravitational wave@8# whichjust has the1 polarization mode and two solutions dueXanthopoulos@9# and Piranet al. @10# which have both the1 and3 polarization mode.

The first exact solution we consider is the Weber-Wheegravitational wave originally investigated by Einstein aRosen@24#. It consists of a cylindrically symmetric vacuumwave with one radiational degree of freedom correspondto the1 polarization mode@21#. It describes a gravitationapulse originating from past null infinity and moving towarthez axis. After imploding on the axis, it emanates to futunull infinity.

This solution has no rotation so thatv and hencet vanishand the solution may be described in terms ofc which sat-isfies the wave equation. A solution to the wave equationcylindrical coordinates may be given in terms of Bessel futions and by superposing such solutions we may write

c~ t,r!52bE0

`

e2aVJ0~Vr!cos~Vt !dV, ~53!

wherea.0. For convenience we let

X5a21r22t2 ~54!

and one may show that Eq.~53! may be written in the alter-native form

n~ t,r!5expF2bA2~X1AX214a2t2!

X214a2t2 G . ~55!

The corresponding value ofg is obtained by integratingg ,r

5r(c ,t21c ,r

2 ) and usingg(t,0)50 and is found to be

g~ t,r!5b2

2a2 F122a2r2X224a2t2

~X214a2t2!22a21t22r2

AX214a2t2G .

~56!

The next solution we consider is one due to Xanthopou@9# which has a conical singularity on thez axis and thereforedescribes a rotating vacuum solution with a cosmic strtype singularity. Xanthopoulos derived the spacetimefinding a solution to the Ernst equation in prolate spheroicoordinates. To compare this with our numerical result

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FIG. 1. The exact Xanthopoulos solution for270<t<30, 0<w<3 anda50.5. Plots are from left to right:n(t,w), t(t,w), andg(t,w).One clearly sees the incoming pulses inn andt. As the pulse hits the axis the string loses energy through gravitational radiation asin g.

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must transform to cylindrical coordinates and also findGeroch potential. It is convenient to first define the followiquantities

Q5r22t211, ~57!

X5AQ214t2, ~58!

Y51

2@~2a211!X1Q#112aA2~X2Q!, ~59!

Z51

2@~2a211!X1Q#21, ~60!

where 0,uau,`. The solution derived by Xanthopoulothen becomes

c~ t,r!51

2ln

Z

Y, ~61!

v~ t,r!5Aa211~X1Q22!~Z2Y!

2aZ, ~62!

g~ t,r!51

2ln

Z

a2X, ~63!

where we have imposedg(t,0)50. A straightforward butrather tedious calculation shows that this satisfies Einstefield equations.~This and a number of other calculationsthe paper were checked using the algebraic computing page GRTensor II@25#!.

The norm of the Killing vector in thez direction is givenby

n~ t,r!5Z

Y. ~64!

The Geroch potential is easily obtained from the Ernsttential and is found to be

t~ t,r!52A2~a211!AX1Q

Y. ~65!

Note thatn andt satisfy Eqs.~26! and ~27! in vacuum, i.e.,

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hn2n21~t ,t22t ,r

2 2n ,t21n ,r

2 !50, ~66!

ht12n21~t ,tn ,t2t ,rn ,r!50. ~67!

Expressions for all these quantities may be obtained inexterior characteristic region by transforming to the (u,y)variables. Althoughc tends to zero as one approaches ninfinity, v diverges so that the 4-dimensional metric is nasymptotically flat even along null geodesics lying in tplanesz5const. However by contrast the Geroch potentiatvanishes as one approaches null infinity in the 3-dimensiospacetime. This is an example of the fact that the Gerpotential can be well behaved even ifv in the JEKK form ofthe 4-metric diverges as one goes outward in a null directWe also give an expression for the gravitational fluxinfinity which is given by E,u where E(u,y)52p@12e2g(u,y)#

limy→0

E,u524p

~11u2!@~112a2!A11u22u#,0. ~68!

Thus the string is losing energy through gravitational radtion.

To plot the solution for 0<r,` we introduce the radiavariable

w5H r for 0<r<1,

322/Ar for r.1,~69!

thus 0<w<3 where the infinite value ofr is mapped tow53. This choice is slightly different from that of Dubalet al.@19# and avoids discontinuities in the radial derivatives atinterface due to the square root in the definition ofy. Plots ofn, t, andg as given by Eqs.~63!–~65! are shown in Fig. 1.The error in the numerical results as computed usingCCM code are shown in Fig. 2.

The final exact solution we consider is one due to Piet al. @10# which also has two degrees of freedom represeing the two polarization states. As in the case of Xanthoplos’ solution, it represents two incoming pulses that imploon the axis and then move away from it. Piranet al.obtainedtheir solution by starting with the Kerr metric in BoyeLinquist form, transforming to cylindrical polar coordinate

1-6

t


FIG. 2. Pointwise error for theXanthopoulos solution. From lefto right: n(t,w)3105 and t(t,w)3105.

Kli

ho-e

o-y.

nss

ns.nto

eas-

and then swapping thet andz coordinates~and introducingsome factors ofi to maintain a real Lorentzian metric!. See@10# for details. The resulting metric may be written in JEKform. The solution is rather complicated but may be simpfied by introducing the following additional quantities

R5b21@Ab21~ t2r!22t1r#, ~70!

S5b21@Ab21~ t1r!21t1r#, ~71!

T511RS12a21A~a221!RS ~72!

and

X5~11R2!~11S2!, ~73!

Y5a2T21~R2S!2, ~74!

Z5a2~12RS!21~R1S!2, ~75!

where 1<a,` and 0<b,`. The metric coefficients arethen given by

c~ t,r!51

2ln

Z

Y, ~76!

v~ t,r!5bAa221F2S 11Aa221

a D 2~R1S!2T

ARSZG ,

~77!

02401

-

g~ t,r!51

2ln

Z

X. ~78!

Notice that Minkowski space is obtained in the limit thata→1, and that we can also consider the caseb→0 in whichcase the rotation vanishes. This is not true for the Xantpoulos solution which is only real for a sufficiently largrotation.

The solution is regular on the axis, but like the Xanthpoulos solutionv diverges as one approaches null infinitAgain the answer is to transform to then, t variables whichare regular both on the axis and at null infinity. Findingn isstraightforward; however solving the differential equatiofor the Geroch potentialt for such a complicated metric iextremely difficult, butt may be found by first finding theGeroch potential for thetimelike Killing vector of the Kerrsolution and then making the appropriate transformatioNote that the same process transforms the Killing vector ione along thez axis. One then finds

n~ t,r!5Z

Y, ~79!

t~ t,r!524A~a221!RS~R2S!

@2A~a221!RS1a~11RS!#21~R2S!2.

~80!

The corresponding results in the characteristic region areily found by transforming to (u,y) coordinates. Plots ofn, t,andg as given by Eqs.~78!–~80! are shown in Fig. 3. The

FIG. 3. The exact Piranet al. solution for 270<t<30, 0<w<3, a54 andb52. Plots are from left to right:n(t,w), t(t,w), andg(t,w). Notice thatn does not have a double ridge as found in the Xanthopoulos solution.

1-7


FIG. 4. Pointwise error for thePiran et al. solution. From left toright: n(t,w)3105 and t(t,w)3105.

CM

ehin

inpte45ye

fin

rm

gralt

has

os-ofn

-

fm

in-to

onn

ber

error in the numerical results as computed using the Ccode are shown in Fig. 4.

We now briefly describe the accuracy and the convgence analysis for the explicit CCM version of our code. Tresults for the implicit version are similar and are givenpaper II. We define the pointwise error at thei th time sliceand j th grid point for some functionf by

j ji ~ f !5 f ~ t i ,wj !exact2 f ~ t i ,wj !computed. ~81!

The pointwise error for the vacuum solutions is shownFigs. 2, 4, and 5 for 600 grid points and 10 000 time stecorresponding to 0<t<15. The code is stable and accurafor at least 20 000 time steps with a Courant factor of 0.Beyond this point the metric functions have almost decato zero and the dynamical behavior is very slow.

In order to analyze the convergence of the code we dethe spacetimel 2-norm for some functionf as

l 2@ f N2

N1#5A( i , j@j ji ~ f !#2

N1•N2, ~82!

whereN1 is the number of time slices andN2 is the numberof grid points on each slice. We also define the relative no

f r5A( i , j@j ji ~ f !#2

( i , j@ f j ,exacti #2. ~83!

Convergence testing of the code is done by doubling thesize keeping the Courant factor constant. We thereforeneed to double the number of time steps. We measureconvergence through

02401

r-e

s

.d

e

idsohe

f c5l 2@ f N2

N1#

l 2@ f 2N2

2N1#. ~84!

For a convergent second order code one should havef c54,and we can clearly see from Tables I–III that the codeachieved second order convergence in time and space.

V. THE COSMIC STRING IN MINKOWSKI SPACETIME

In this section we examine the field equations for the cmic string. The simplest case is to look at the equationsmotion on a fixed Minkowski background. If we do this thethe Euler-Lagrange equations for~1! give

hS5S@4l~S22h2!2r22P2#, ~85!

hP22r21P,r5e2S2P, ~86!

where h is the d’Alembertian in cylindrical polar coordinates given by Eq.~29!.

It will turn out that the solutions of this simpler set oequations are qualitatively similar to those of the full systeof equations for a dynamic cosmic string coupled to Estein’s equations. However the full system enables oneperturb a static string with a pulse of gravitational radiatiand in turn look at the effect of the string’s oscillations othe gravitational waves.

A special case of Eqs.~85! and~86! is when one looks fora static solution. This has been looked at before by a numof authors, for example, Garfinkle@26#, Lagunaet al. @11#,Laguna-Castilloet al. @12#, and Dyeret al. @13#. For a staticstring Eqs.~85! and ~86! reduce to

FIG. 5. Pointwise error for theWeber-Wheeler solution. Fromleft to right: c(t,w)3106 andg(t,w)3107.

1-8

oun

ufodi

ioeri-

oo

lifhin

is

be-

al, is

be-

by


rd

dr S rdS

dr D5S@4lr2~S22h2!1P2#, ~87!

rd

dr S r21dP

dr D5e2S2P. ~88!

This pair of coupled second order equations requires fboundary conditions. For the physically relevant finite eergy solution these are

S~0!50, limr→`

S~r!5h, ~89!

P~0!51, limr→`

P~r!50. ~90!

It is not possible to obtain an exact solution to these eqtions but one can investigate the asymptotic behaviorlarge r. The solution satisfying the above boundary contions has asymptotic behavior given by@27#

S~r!;h2K0~A8lhr!;h2A p

A32lhr21/2e2A8lhr,

~91!

P~r!;rK1~ehr!;A p

2ehr1/2e2ehr. ~92!

However as well as these physical solutions, the equatadmit nonphysical solutions which have exponentially divgent behavior asr→`. It is the existence of these nonphyscal solutions which make the problem rather delicate fromnumerical point of view and makes a method such as shing hard to apply.

Before proceeding further we follow Garfinkle@26# byintroducing rescaled variables and constants which simpthe algebra~and are also important when considering the t

TABLE II. Convergence test: the Xanthopoulos solution.

Grid pts. n r t r g r nc tc gc

300 1.7031026 5.7131026 1.6631026 - - -600 2.9931027 1.0131029 2.9331027 4.01 4.00 4.01

1,200 5.2831028 1.7831027 5.1831028 4.01 4.00 4.012,400 9.3231029 3.1531028 9.1631029 4.00 4.00 4.004,800 1.6531029 5.5731029 1.6231029 4.00 4.00 4.00

TABLE I. Convergence test: the Weber-Wheeler solution.

Grid pts. n r g r Time steps nc gc

300 6.2931027 9.3831026 5,000 - -600 1.1131027 1.6631026 10,000 4.01 4.01

1,200 1.9631028 2.9431027 20,000 4.00 4.002,400 3.4731029 5.1931028 40,000 4.00 4.004,800 6.14310210 9.1831029 80,000 4.00 4.00

02401

r-

a-r

-

ns-

at-

y

string limit!. Provided we rescale the time coordinate thalso simplifies the fully coupled system. Let us introduce

X5S

h, ~93!

r 5Alhr, ~94!

t 5Alht, ~95!

a5e2

l. ~96!

Thus a represents the relative strength of the couplingtween the scalar and vector field given bye compared to theself-coupling of the scalar field given byl. Critical coupling,when the masses of the scalar and vector fields are equgiven by a58. In the theory of superconductivity,a58corresponds to the interface between type I and type IIhavior @27#.

With the above rescaling, Eqs.~87! and ~88! become

rd

dr S rdX

dr D5X@4r 2~X221!1P2#, ~97!

rd

dr S r 21dP

dr D5aX2P. ~98!

Note the rescaled version of Eqs.~26!–~33! may be foundsimply by making the replacements

S→X, ~99!

r→r , ~100!

t→ t , ~101!

e2→a, ~102!

l→1. ~103!

The boundary conditions of the cosmic string are given

X~0!50, limr→`

X~r !51, ~104!

P~0!51, limr→`

P~r !50. ~105!

TABLE III. Convergence test: the Piranet al. solution.

Grid pts. n r t r g r nc tc gc

300 1.0731025 3.9431026 2.7631026 - - -600 1.8931026 6.9631027 4.8731027 4.00 4.00 4.01

1,200 3.3531027 1.2331027 8.6131028 4.00 4.00 4.002,400 5.9231028 2.1731028 1.5231028 4.00 4.00 4.004,800 1.0531028 3.8431029 2.6931029 4.00 4.00 4.00

1-9


TABLE IV. Convergence test: static cosmic string coupled to gravity.

n m g X P

l 2( f 1200) 1.2831027 2.5131026 2.3931026 4.1631027 5.9531027

l 2( f 150)/ l 2( f 300) 3.56 3.59 3.58 3.37 4.04l 2( f 300)/ l 2( f 600) 3.76 3.79 3.78 3.60 4.19l 2( f 600)/ l 2( f 1200) 4.58 4.61 4.60 4.44 4.98

irtouc

e

e

ax

edorfo

r-aencofio

s

d

n ass

the

atico

-

Because of the boundary conditions at infinity it is desable to introduce a new coordinate which brings in infinitya finite coordinate value. For this purpose we again introdan inner region (r<1) and an outer region (r>1) where weuse the radial coordinatey given by

y51

Ar. ~106!

In order to solve Eqs.~97! and~98! numerically we intro-duce a spatial grid consisting ofn1 points in the inner regionandn2 points in the outer region. The pointsr 1 ,...,r n1

cover

the range 0<r<1, and yn111,...,yn11n2cover the range

1>y>0. Thus,r 515y is represented by two points. Thestwo points form the interface of the code wherer derivativesare transformed intoy derivatives. The static equations in thouter region are

yd

dy S ydX

dyD54XF4~X221!

y4 1P2G , ~107!

yd

dy S y5dP

dy D54aX2P. ~108!

In order to apply boundary conditions at bothr 50 andy50 the equations were solved numerically using a relation scheme~as described in@28#, for example!. For thispurpose we wrote the equations as a first order systemboth regions~see paper II! and used second order centerfinite differencing. Solutions were computed in this way fdifferent resolutionsN5n11n2 to check the convergence othe code. Since there is no exact solution available the cvergence was checked by calculating thel 2-norm with re-spect to a high resolution result forN52400~1200 points ineach region!

l 2@D f N#5A( i 51N ~ f i

N2 f i2400!2

N, ~109!

wheref stands for eitherX or P. For a second order convegent code one would expect thel 2-norm to decrease byfactor of four if the number of grid points is doubled. Wfind that the code shows clear second order convergeSince the convergence in this case is very similar to thatstatic string coupled to gravity considered in the next sectwe only give the results for the latter case in Table IV.

In Fig. 6 we plotX andP for a50.125,1,8,64. The resultshow that for fixed values of the self-couplingl and vacuumexpectation valueh of the scalar field, both the vector an

02401

-

e

-

in

n-

e.a

n,

scalar fields become more concentrated towards the origithe coupling between the fieldse increases. One also findthat for a fixed ratio of the coupling constantsa the fieldsbecome more concentrated towards the origin as eitherself-couplingl or the vacuum expectation valueh of thescalar field increases.

VI. THE STATIC COSMIC STRING COUPLEDTO GRAVITY

The next class of solutions we wish to consider are stsolutions of the fully coupled equations. In the case of ntdependence, Eqs.~26!–~33! reduce to

1

r~rn ,r ! ,r52n ,rm ,r1

n ,r2 2t ,r

2

n

18ph2Fn2e22mP,r

2

ar 222e2~g1m!~X221!2G ,~110!

1

r~r t ,r ! ,r5t ,r S 2

n ,r

n2m ,r D , ~111!

1

r 2 ~r 2m ,r ! ,r52m ,r2 28ph2

3Fe2gX2P2

r 2 12e2~g1m!

n~X221!2G ,

~112!

FIG. 6. Cosmic string in Minkowski spacetime.X andP for a:0.125, 1, 8, 64. Asa increases, bothX andP become more concentrated towards the origin.

1-10


FIG. 7. Cosmic string coupled to gravity.a51 andh51023. On the left the metric variables:g, n21 andm are plotted multiplied by105. On the rightX andP are plotted.

i-

iod

uusiagau-

erafe

sult

inl-o

llyith

of

tial.

he

os-ly

in

i-oth-icalM

aleer-withthelity.ode

g ,r5r

11rm ,rH 1

4n2 ~t ,r2 1n ,r

2 !14ph2

3FX,r2 1ne22m

P,r2

ar 22e2gX2P2

r 2

22e2~g1m!

n~X221!2G J 2m ,r , ~113!

1

r~rX ,r ! ,r52X,rm ,r1XF4

e2~g1m!

n~X221!1e2g

P2

r 2 G ,~114!

r S 1

rP,r D

,r

5P,r S m ,r2n ,r

n D1ae2~g1m!

nPX2.

~115!

Notice that ~111! with the corresponding boundary condtions is satisfied by the trivial solutiont50. One also hasfrom the field equations

~rg ,r ! ,r52rg ,rm ,r1m ,r18ph2S e2gX2P2

r1e22mn

P,r2

ar D .

~116!

This equation is a direct consequence of the other equatand will not be used in the calculations but is instead usea check for the code. The equations in the outer regionwell as the resulting first order system, the interface eqtions and boundary conditions are given in paper II. Wethe same grid and numerical method as in the Minkowskcase. In order to check the code for convergence we acompute thel 2-norm with respect to a high resolution calclation. The results are shown in Table IV fora51 and thelarge valueh50.1 and clearly indicate second order convgence. Small deviations from a convergence factor of 4expected since we compare against a high resolution re

02401

nsasasa-enin

-rer-

ence solution rather than an exact solution. The same rehas been obtained for other choices ofa andh.

The metric and matter variables fora51 andh51023

are shown in Fig. 7. We find that the behavior ofX andP isvery close to that obtained for a static cosmic stringMinkowski spacetime shown in Fig. 6. For physically reaistic values ofh the metric variables at infinity are close ttheir Minkowskian values although the nonzero value ofg05 limr→` g(r ) indicates that the spacetime is asymptoticaconical, that is Minkowski spacetime minus a wedge wdeficit angleDf52p(12e2g0). Thus a string witha51andh51023 has an angular deficit of about 231025 whichcorresponds to a grand unified symmetry breaking scaleabout 1016GeV @27#. For larger values ofh, however, thedeviation from the Minkowskian case becomes substanClose to critical coupling the deficit angle exceeds 2p forvalues ofh greater than about 0.28 which explains why tcode converges well forh&0.28.

VII. THE DYNAMIC COSMIC STRING COUPLEDTO GRAVITY

In this section we discuss the interaction between the cmic string and the gravitational field. Here we will simpoutline the numerical scheme, the full details are givenpaper II. The field equations in the (t,r) coordinates aregiven by Eqs.~26!–~33! while those in (u,y) coordinates aregiven by Eqs.~43!–~49!. In both cases there are two addtional Einstein equations which are consequences of theers and are used to check the code. In fact two numerschemes were employed. The first was an explicit CCscheme similar to that employed by Dubalet al. @19# andd’Inverno et al. @20#. However the use of the geometricvariablesn andt in both the interior Cauchy region and thexterior characteristic region significantly improves the intface and results in a genuinely second order schemegood accuracy even with both polarizations present. Forvacuum equations the code also exhibits long term stabiHowever when the matter variables are included this c

1-11


FIG. 8. On the left:P(u,w) for a51, h51023 andu58.5. On the right: The corresponding contour plot ofP(u,r ) for 0<r<200 and28<u<16. Note the use ofr for values greater than one in this plot.

ns-he

p

yth

vmoreti

on

firnnkplir

teutth

ed

dilitedrethcu

mm

omitymaty-

eeow

de.

nt.

eingns

.

-ren-ver,

performs less satisfactorily. This is because of the existeof exponentially growing nonphysical solutions. It is posible to control these diverging solutions by multiplying tu-derivatives of the matter variables by a smooth ‘‘bumfunction’’ which vanishes aty50 but is equal to 1 fory.c ~where c is a parameter!. This produces satisfactorresults but the bump function introduces some noise intoscheme which eventually gives rise to instabilities.

A much better solution is to control the asymptotic behaior at infinity by using an implicit scheme. The main problewith the system of differential equations is the irregularitythe equations at both the origin and null infinity. Therefojust as in the static case considered in the previous secthe scheme employed divides the spacetime into two regian inner regionr<1 where coordinates (u,r ) are used andan exterior regionr>1 where the (u,y) coordinates areused. The equations in both regions are written as aorder system connected by an interface and the evolutioaccomplished using a code based on the implicit CraNicholson scheme. This implicit scheme provided a simway of implementing the boundary conditions and thus ccumventing all problems with the irregularities. The ouboundary conditions as well as the equations in the oregion, the first order system used for the numerics andinterface are given in paper II. The implicit code showvery good agreement with both the exact~vacuum! and pre-viously obtained~static! numerical solutions. It also showeclear second order convergence and very long term stabFor convenience, characteristic coordinates were also usthe inner region but we do not think that their use wassponsible for the good features of this code and believean implicit CCM scheme would have produced similar acracy, convergence, and long term stability.

The code is able to consider the interaction of the cosstring with a gravitational field with two degrees of freedohowever here we simply describe the interaction withWeber-Wheeler wave which has just one degree of freedWe consider a pulse which comes in from past null infinand interacts with a cosmic string in its static equilibriuconfiguration. This interaction causes the string to oscillwhich in turn affects the gravitational field as measured bnand t. The oscillations in bothX and P decay as one ap

02401

ce

e

-

f

ons,

stis-

e-rere

y.in-at-

ic,a

.

e

proaches null infinity~i.e., asr→` for fixed u! and also forfixed r as u→`. After the oscillation has died away thstring variablesX and P return to their static values. Nothowever that this decay is rather slow and being able to shthis effect depends upon the long term stability of the coAlthough oscillations are observed in bothX andP the char-acter and frequency of these oscillations is rather differe

In Fig. 8 we plotP for a51 andh51023 at a timeu58.5 ~left panel!. The oscillations out to large radii can bclearly seen. The contour plot on the right shows the ringbehavior of the string and the slow decay of the oscillatioin P. In contrast, the oscillations inX displayed in Fig. 9. arerestricted to small radii and decay on a shorter timescale

An investigation of the frequenciesf X and f P of the os-cillations of the scalar fieldX and the vector fieldP, ~in therescaled unphysical variables! indicates that they are relatively insensitive to the value ofh and the Weber-Wheelepulse which excites the string. They are also largely indepdent of the radius at which they are measured. Howealthough f X is also independent ofa, we find that f P isproportional toAa. When we convert to the physical fieldsSandP and use the physical coordinates~t,r! we find that the

FIG. 9. X(u,w) for a51, h51023.

1-12

an

tki

yui

th

cfonm

c-a

retri-

mici-

in-

heyof

ee

od

s ofer-de-

nsal-tringfor

hevi-the

ailsbe-in

-


frequencies in natural units are given by

f S;Alh, ~117!

f P;eh. ~118!

If we now use the fact that the masses of the scalarvector fields are given byMS

2;lh2 andM P2;e2h2 @27# this

gives

f S;mS, ~119!

f P;mP . ~120!

In fact these frequencies are also obtained by consideringsimpler model of a dynamic cosmic string in Minkowsspace with equations of motion~76! and ~77! provided onegives S and P similar initial conditions to that produced bthe interaction with the pulse of gravitational radiation. Ththe main role of the gravitational field as far as the stringconcerned is to provide a mechanism for perturbingstring. This is discussed more fully in paper II.

VIII. CONCLUSION

In this paper we have shown how the method of Gerodecomposition may be used to recast the field equationstime dependent cylindrical cosmic string in four dimensioin terms of fields on a reduced 3-dimensional spacetiThis has the advantage that it has a well defined notionnull infinity. It is therefore possible to conformally compatify the 3-dimensional spacetime and avoid the need fortificial outgoing radiation conditions. An additional featuof this approach is that it naturally introduces two geomecally defined variablesn andt which encode the two gravitational degrees of freedom in the original spacetime.

-

. D

02401

d

he

sse

hr ase.of

r-

-

We have described how the field equations for a cosstring coupled to gravity may be solved numerically by dviding the 3-dimensional spacetime into two regions; anterior regionr<1, and an exterior regionr>1 in which thecoordinatey is used. The use of the geometric variablesnandt greatly simplifies the transmission of information at tinterfacer 51. Although an explicit CCM code worked vereffectively in the vacuum case, the asymptotic behaviorthe matter fieldsS and P made it less effective when thstring is coupled to the gravitational field. An alternativimplicit fully characteristic scheme however exhibited goaccuracy and long term stability.

In this paper we have demonstrated the effectivenesthe CCM codes in reproducing the results of the WebWheeler solution and two vacuum spacetimes with twogrees of freedom due to Xanthopoulos and Piranet al. Thisinvolved calculating the Geroch potential for these solutioand using it to compare with the numerically computed vues. We have also described a code for a static cosmic swhich uses a relaxation scheme and provides initial datathe dynamic code.

In the final section of the paper we briefly described tinteraction of the cosmic string with a Weber-Wheeler gratational wave. The pulse of gravitational radiation excitesstring and causes the fieldsSandP to oscillate with frequen-cies proportional to their respective masses. The full detof the code, the convergence testing, and the interactiontween the string and the gravitational field are describedpaper II.

ACKNOWLEDGMENTS

We would like to thank Ray d’Inverno for helpful discussions and Denis Pollney for help with GRTensor II.

l.,

D

n-

L:

@1# Y. B. Zel’dovich, Mon. Not. R. Astron. Soc.192, 663 ~1980!.@2# A. Vilenkin and A. E. Everett, Phys. Rev. Lett.48, 1867

~1982!.@3# T. Goto, Prog. Theor. Phys.46, 1560~1971!.@4# D. Garfinkle and P. Laguna, Phys. Rev. D39, 1552~1989!.@5# B. Carter and P. Peter, Phys. Lett. B466, 41 ~1999!.@6# B. Carter, inFormation and Interactions of Topological De

fects, edited by R. Brandenberger and A-C. Davis~Plenum,New York, 1995!, pp. 303–348.

@7# C. J. S. Clarke, R. d’Inverno, and J. A. Vickers, Phys. Rev52, 6863~1995!.

@8# J. Weber and J. A. Wheeler, Rev. Mod. Phys.29, 509 ~1957!.@9# B. C. Xanthopoulos, Phys. Rev. D34, 3608~1986!.

@10# T. Piran, P. N. Safier, and J. Katz, Phys. Rev. D34, 331~1986!.

@11# P. Laguna and D. Garfinkle, Phys. Rev. D40, 1011~1989!.@12# P. Laguna-Castillo and R. A. Matzner, Phys. Rev. D36, 3663

~1987!.@13# C. C. Dyer and F. R. Marleau, Phys. Rev. D52, 5588~1995!.@14# U. Sperhake, K. R. P. Sjo¨din, and J. A. Vickers, following

paper, Phys. Rev. D63, 024012~2001!.@15# R. Geroch, J. Math. Phys.12, 918 ~1971!.@16# P. Jordan, J. Ehlers, and W. Kundt, Abh. Math.-Naturwiss. K

Akad. Wiss. Lit., MainzK1, 2 ~1960!.@17# A. S. Kompaneets, Zh. Eksp. Teor. Fiz.34, 953 ~1958! @Sov.

Phys. JETP7, 659 ~1958!#.@18# L. Marder, Proc. R. Soc. London244, 524 ~1958!.@19# M. R. Dubal, R. A. d’Inverno, and C. J. S. Clark, Phys. Rev.

52, 6868~1995!.@20# R. A. d’Inverno, M. R. Dubal, and E. A. Sarkies, Class. Qua

tum Grav.17, 3157~2000!.@21# T. Piran, P. N. Safier, and R. F. Stark, Phys. Rev. D32, 3101

~1985!.@22# A. Ashtekar, J. Bicak, and B. G. Schmidt, Phys. Rev. D55,

669 ~1997!.@23# J. P. Wilson and C. J. S. Clarke, Class. Quantum Grav.13,

2007 ~1996!.@24# N. Rosen and A. Einstein, J. Franklin Inst.223, 43 ~1937!.@25# P. Musgrave, D. Pollney, and K. Lake, GRTensor II, UR

http://grtensor.phy.queensu.ca/

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re,

P.-

d,


@26# D. Garfinkle, Phys. Rev. D32, 1323~1985!.@27# E. P. S. Shellard and A. Vilenkin,Cosmic Strings and Othe

Topological Defects~Cambridge University Press, CambridgEngland, 1994!.

02401

@28# W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B.Flannery,Numerical Recepies in C: The Art of Scientific Computing ~Cambridge University Press, Cambridge, Englan1992!.

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Dynamic cosmic strings. I