PHYSICAL REVIEW D VOLUME 45, NUMBER 2 15 JANUARY 1992

Gravitational effects of traveling waves along global cosmic strings

Athanasios EconomouFakultat fii r Physik der Uniuersitii t Konstanz, Postfach 5560, D 7750 Konstanz, Germany

Diego HarariInstituto de Astronomia y Fi'sica del Espacio, Casilla de Correo 67,

Sucursal 28, 1428 Buenos Aires, Argentina

Mars'a SakellariadouUni Uersite Libre de Bruxelles, R GGR, Faculte des Sciences, CP231, B 1050 Bruxelles, Belgium

(Received 17 June 1991)

Making use of the relationship between the corresponding field configurations, we derive the metricaround a straight global cosmic string with traveling waves in terms of the static metric (without travel-ing waves) in the weak-field limit. We discuss under which conditions the effect of the traveling waves

may overcome the repulsive gravitational potential of the static straight global string. We also extendthe calculation beyond the weak-field limit combining our result with the recent observation made byGarfinkle and Vachaspati that the exact solution must be of the generalized Kerr-Schild type.

PACS number(s): 98.80.Cq, 04.20.Jb

I. INTRODUCTION

Recent numerical simulations of gauge cosmic-stringnetworks suggest that intercommutation of string seg-ments leads to significant perturbations on scales muchsmaller than the correlation length of the network [1—3].On average over distances larger than the typical wave-length of the perturbations, a wiggly string appearssmooth, but with mass per unit length, p, larger and ten-sion T smaller than their unperturbed values, the productp, T remaining constant [4]. The exact nature and amountof small-scale structure on a string network is crucial tounderstanding its evolution and its output in gravitation-al radiation, which may place a crucial bound on thestring mass density through its potential observableeffects, such as in millisecond pulsar timing [5,6].

The simplest kind of structure, not necessarily of smallscale, are perturbations traveling in one direction at thespeed of light along an otherwise straight cosmic string.Although special, since they do not dissipate throughgravitational radiation [7] as more general small-scalewiggles do, they constitute a tractable and in some as-pects representative case of small-scale structure.

Vachaspati was the first to consider gravitationaleffects, in the weak-field limit, around a class of solutionsto the Nambu-Goto equations of motion representingtraveling waves along otherwise straight gauge cosmicstrings in the zero-thickness approximation [7]. At dis-tances p to the string core larger than the characteristicsize of the perturbations, the traveling wave exerts agravitational force proportional to 1/p. More recently,Garfinkle found an exact solution of vacuum Einstein'sequations which reduces to Vachaspati s cosmic-stringtraveling-wave metric in the weak-field limit, which heinterpreted as the metric of a strongly gravitating travel-ing wave along a gauge cosmic string in the zero-thickness limit [8].

Traveling-wave solutions of field theories are alsoknown for global cosmic strings [9]. In this paper we an-alyze their gravitational effects. Strings formed after thespontaneous breakdown of a global symmetry have gravi-tational effects quite different than their local counter-parts [10],their energy not being confined to a small tubebut rather extending into regions far beyond the centralcore [11]. A static, straight global string produces, in theweak-field limit, a repulsive logarithmic gravitational po-tential outside the core, in addition to an angular deficitsimilar to that of a gauge string but also logarithmicallydependent on the distance to the core [10]. It is thereforeof interest to analyze whether small-scale structure in theform of traveling waves may eventually overcome therepulsive form of the static global cosmic string.

Rather than directly solving Einstein's equations in theweak-field limit, we will exploit a relationship betweenthe field configurations for static infinite strings andstrings with traveling waves, respectively, noticed bothfor gauge as well as for global strings by Vachaspati andVachaspati [9]. We will see that there is a simpleprescription that allows us to obtain the linearized metricaround a gauge or global string with traveling wavesdirectly from the metric of the static string.

The existence of a relatively simple relationship be-tween the static and traveling-wave cosmic string metricswas recently noticed by Garfinkle and Vachaspati, andexploited to obtain the exact metric around gauge cosmicstrings with traveling waves [12]. They concluded thatthe exact metric around a cosmic string with a travelingwave is of the generalized Kerr-Schild type, with the stat-ic string metric as the background. Their method actual-ly originates a family of solutions to the full Einstein'sequations, coupled to the Abelian-Higgs equation for thestring fields, of the form

45 433 1992 The American Physical Society

434 ECONOMOU, HARARI, AND SAKELLARIADOU 45

where the superscripts TW and st denote traveling waveand static metrics respectively, k„ is a null, hypersurfaceorthogonal, Killing field with respect to both metrics, andF is a scalar function that satisfies a well-defined equa-tion. The prescription of Garfinkle and Vachaspati tofind the exact metric around a gauge or global string withtraveling waves would be complete given a method tounivocally fix the function F in terms of the profiles of thetraveling waves. In the case of gauge cosmic strings theymanaged to fill this gap by imposing on F the asymptoticbehavior already known from the weak-field limit. Othervacuum solutions, also of the form (1.1) but with differentfunctions F, were shown to represent traveling wavesalong cosmic strings interacting with additional gravita-tional waves, or traveling waves along a set of parallelcosmic strings [13].

In the case of global strings imposing asymptotic con-ditions on Fwould be much more dificult, since the exactmetric presents a singularity at a finite proper distanceaway from the core [14]. Our prescription to obtain thetraveling-wave metric in the weak-field limit, both forgauge as well as global strings, provides this "missinglink, " since the weak-field limit can be used to determineuniquely the appropriate function F in (1.1) for givenprofiles of the traveling waves. The combination ofGarfinkle and Vachaspati's general result together withour weak-Geld calculation will thus allow us to write theexact metric around a global string with traveling waves.

II. COSMIC-STRINGTRAVELING-WAVE CONFIGURATIONS

In Minkowski spacetime with metric=diag(1, 1,1,—1) and Cartesian coordinates [x "]= [x,y, z, t] we consider gauge cosmic strings made of acomplex scalar field P and a gauge vector field A„via theusual Abelian Higgs Lagrangian

u =z —t =Z —T, v =z+t =Z+T . (2.3)

For definiteness, and without loss of generality, we willassume from now on that the functions f,g depend onlyon u. It would of course be the same to take them bothto be functions of v only. However they should not be afunction of both u and v; neither can one depend on u

and the other on v.Let [P"(x,y), A„"(x,y)] and [P"(x,y)J be the field

configurations that describe a straight static gauge andglobal cosmic string, respectively, located along the zaxis. Note also that because of the assumed symmetrieswe can choose a gauge such that A,"=A,"=0. Va-chaspati and Vachaspati have shown [9] that if the field

configuration above is an appropriate solution of the fieldequations representing a static straight string, then theconfiguration

(x,y, u}=P"(X,Y),XM

A (x,y, u)—: AM(X, Y}ax~

(2.4)

also solves the field equations. The fields

(x,y, u), A„(x,y, u)] represent traveling waves

moving with the speed of light along the cosmic string(i.e., along the z axis}, with profiles in the x and y direc-tions characterized by the functions f (u) and g(u), re-spectively. A note on our notation: In this paper, wedenote by q (X, Y) the function which results from a func-tion q (x,y) by changing its arguments x and y toX=x —f ( u ) and Y =y —g ( u ), respectively.

The stress-energy-momentum tensor of the cosmic-string fields is

] = [X, YZ, T] = [x f—(t+z),y g—(t+z),z, t],(2.2}

where f,g are arbitrary functions of the null coordinatesu or v defined as

X= ——,'(D„$)(D"P)' ——,'F"'F„„—I A( ~P~

—rt

—) . (2.1) T„=(D(„(I})(D,)P)'+F„iF„+/at„„. (2.5)

Here D„=8„ieA„a—nd F„„=B„A„—B„A„. Hereafter,spacetime indices run from one to four unless otherwisestated, and we have adopted the summation conventionfor repeated indices. The Lagrangian (2.1) will be alsoused for the description of global cosmic strings by set-

ting the gauge field A„equal to zero.To discuss the cosmic-string traveling-wave solu-

tions it is useful to define the coordinate system[X~]= [X, Y,Z, T] related to [x"] via

Spv ~pv 2 Ipv~a (2.6)

and using Eqs. (2.4) and (2.5), we find this relation to be

The relationship (2.4) between the field configurationsrepresenting a static string and a string with travelingwaves, respectively, also implies a relation between theirenergy-momentum tensors T„„(x,y, u) and T&„(X,Y).Introducing the tensor S„defined in terms of the stress-energy-momentum tensor as

gXM ANS„(x,y, u) = —Q(X, Y)g„„+ [SM~(X, Y)+Q(X, Y)g~~],ax~ ax

where

Q(x,y)=S,", (x,y)= S"(x,y)= ,'[F F"—" A(~P~—g—)] . —

(2.7)

(2.8)

It is interesting to note that away from the string core, when ~P~~g and F„~O, then Q~O.

45 GRAVITATIONAL EFFECTS OF TRAVELING WAVES ALONG. . . 435

III. THE WEAK GRAVITATIONAL FIELD OFCOSMIC-STRING TRAVELING WAVES

Let us first review briefly the weak-field linearized Ein-stein equations [15]. In this limit, one can find coordi-nates {x"], such that the space-time metric is written as

g„„=2)„+h„({x ] ), (3.1)

where 2)z =i)" =diag(1, 1,1,—1) and ~h„~ & &1. Hereafter, within first order in ~h„„~, space-time indices will beraised and lowered using the Minkowski metric g„.

Preserving the weak-field form (3.1), one can alwayschoose nearly Cartesian coordinates {x&[ in such a waythat the harmonic gauge conditions

()"(h„„——2'rl„p 2 ) =0 (3.2)

are satisfied. In this gauge the linearized field equationsare simply

Oh„„=—16m GSp (3.3}

Here 6 is the gravitational constant.Let {S„",(x,y), h„„(x,y)J represent, according to the

field equations (3.2) and (3.3), the energy content and thegravitational field of a spacetime with a straight static(gauge or global) cosmic string along the z axis. With theassumed symmetries and in the weak-gravity limit itshould be possible to write the space-time metric in theform

Here 5," is the Kronecker delta while the indices i,j runfrom 1 to 2. The coordinate system {x"] has beenchosen to satisfy the gauge conditions (3.2) and the quan-tities h„,(x,y) satisfy the field equations (3.3}. In particu-lar, because of (2.8), the function w =w (x,y) =—h« = —h„satisfies

ds =[1—w(x, y)]( dt +—dz )+[5J+h,, (x,y)]dx'dxj .

(3.4)

In addition, froin the field equations (3.3) and Eq. (3.5),we find that the metric (3.6) has as a source precisely thetensor S„, (x,y, u) of Eq. (2.7). This is most easilyproved by expressing the 0 operator of Eq. (3.3) in the{X ] coordinate system of Eq. (2.2) and noting thatfor any function q(x,y, u), 0, M)q(X, Y, u)= {+(„~)q(x» ")J ~ =x,y = r

ds„= dt +—dz +dp +p (1—4Gp) d8 (3.7)

where p is the string mass per unit length. The metric(3.7} is actually flat, with a conical singularity. It de-scribes the spacetime away from the string core, when thefields that make up the string reach their asymptoticvalues, and is also valid beyond the linear approximation.Of course the exact solution to Einstein's equations cou-pled to the string fields is much more complicated, anddifficult to find analytically [18], except for very specialvalues of the parameters in the field-theory model for thestring [19]. But Eq. (3.7) is an excellent approximation tothe exterior metric, unless the string parameters are veryclose to the Planck scale [20]. Using harmonic coordi-nates, Eq. (3.7) can be written as

ds,', = dt +dz'+—[1+h(x,y)](dx +dy ), (3.8)

A. Gauge cosmic strings

Equation (3.6) expresses, in the weak-field limit, themetrics of cosmic strings with traveling waves, be themgauge or global, in terms of the corresponding staticmetrics, which are known. Gauge cosmic strings withtraveling waves have already been discussed by Va-chaspati [7] in the weak-field limit and Garfinkle [8,12] inthe general case. Let us anyhow consider gauge cosmicstrings in this subsection, just for completeness and forlater comparison with the global strings.

The metric around a static, straight gauge cosmicstring along the z axis reads [16,17]

Qw (x,y) = —16m.GQ (x,y } . (3.5) with

Consider now in the {x"J coordinate system the metric

g„„(x,y, u) = [1—w (X, Y)]7)„„

aXM ax"+ [hM)v(X, Y)+w(X, Y)rlMtt] .ax~ ax"

(3.6)

This metric represents the gravitational field of the cosm-ic string modulated by a traveling wave moving with thespeed of light in the z direction with profiles in the x and

y directions characterized by the functions f (u) and

g (u), respectively. Indeed, it is not very difficult to checkthat, since the harmonic conditions (3.2} are satisfied inthe metric (3.4), the same is also true in the metric (3.6).

' —4Gp,x +y

h (x,y)= —1=—4G)uln[(x +y )/y ] .r'

x 2+y 2 —y2[( 1 4Gp ) /y ]2/(1 —4GP) (3.10)

Using Eq. (3.6) (notice that now w =0 and h; =h5,") weget, for the traveling-wave metric,

(3.9)

Here r is an arbitrary length scale, that can be changedrescaling x and y. It should get fixed by matching the ex-terior solution to a realistic core, and is expected to be ofthe order of the core width. The relation of the harmoniccoordinates x and y to the radial proper distance p isgiven by

dsrw= dt +dz +dx +—dy +h(X, Y)[(f +g )(dz dt) 2fdx(dz dt) 2gd—y—(dz dt)+—dx —+dy ]—. (3.11)

436 ECONOMOU, HARARE, AND SAKELLARIADOU 45

Here an overdot denotes a derivative with respect to theargument u =z t.—Our derivation of Eq. (3.6) is onlyvalid in the weak-Geld limit, so we should takeh = —4Gpln[[(x f—) +(y —g) ]/y ] in Eq. (3.11), andthe result agrees with Vachaspati's direct calculation [7].It is nevertheless interesting to remark that the metric(3.11), with the exact function h (X, Y) given by (3.9), isfor (X, Y)W(0,0) an exact solution of vacuum Einstein'sequations, as shown by Garfinkle [8], who also noticedthat it is of the type known as pp waves. It can be inter-preted as the exact metric of a gauge cosmic string withtraveling waves in the limit of zero thickness of the stringcore.

The geodesic equations for a nonrelativistic test parti-cle in the traveling-wave metric (3.11) were already dis-cussed in Ref. [7], while motion of light rays was con-sidered in Ref. [21]. Consider a localized traveling wave(a traveling pulse), acting upon a test particle that is ini-tially at rest with respect to the string, while the pulse is

making its approach from far away. When the pulse hasalready passed by and is far away in the other direction,the particle will be moving towards the string at a speedproportional to the integral of f +g over the pulseprofile [22]. In order to visualize this effect in a simplecase, consider the acceleration felt by a slowly movingtest particle located at y =0 at a distance x to the stringmuch larger than the amplitude of the traveling waves,x » f,g. Notice however, that now radial trajectoriesare not necessarily geodesics, since the traveling wavesbreak the rotational symmetry around the string axis,and may also force test particles to move in the directionparallel to the string. To lowest order there is nodifference between t and proper time, and then

ence quite large forces. In fact, as shown by Garfinkle[23], for the gauge cosmic-string traveling-wave metric(3.11), the tidal forces diverge at the string's core(X, Y) =(0,0).

B. Global cosmic strings

Consider now global cosmic strings. In the weak-fieldlimit the metric of a straight static string reads [10]

ds„= 1 —46p ln++c +—5 ( dt'—+dz')

+dp +p 1 —8Gp ln +c dg5

(3.13)

l"w =h« = —h„=4Gp 1n—,

Here p=n. rt for the model Lagrangian of Eq. (2.1) (withno gauge fields), 5=(g A, )

' is the core width, and c is aconstant of order unity that takes into account the effectof the string core. Remember that for the global string,the energy per unit length in the Nambu-Goldstone-boson mode up to a distance p outside the core of thestring is pin(p/5), and that for most relevant cases thislogarithmic factor is large (as large as 130 if p is thepresent Hubble distance and rt =10' GeV). The constantfactor proportional to (c +5/4) in front of ( dt +dz —)

corresponds to a choice of scaling on t and z appropriatefor later comparison with the exact metric considered inthe next section.

Harmonic coordinates can be chosen such that theabove metric takes the form of Eq. (3.4) with

4G„f'+g'

x Bt y(3.12)

h „=—2Gp ln —+21n ——2sin 0+a2 ~ 2

r y(3.14)

The first term on the right-hand side is a 1/x attractiveforce, analogous to what is expected from an additionalmass per unit length of string proportional to p(f +g ).The second term can have different signs at differenttimes. However, being a time derivative it has no neteffect upon integration between times before and after atraveling pulse has passed by the particle. The force onthe test particle increases as we approach the string. Al-though Eq. (3.12) holds for sufficiently large distancesfrom the string, the 1/x dependence of the accelerationclearly indicates that. near the string test particles experi-

h = —2Gp ln —+21n ——2 cos 0++T 2 r 2VV y r

h„~ = —4Gp sin8 cos8,

where now r =x+y, 8—=arcta—n(y/x), and y=5exp( —c ——', ). In h„„and h»», a is an arbitrary con-

stant which can be transformed away with an appropriaterescaling of r. In what follows, we will set a=0. It isnow straightforward to obtain the metric of a globalstring with traveling waves using Eq. (3.6). The resultcan be written as

dsrw =[rt„+h,(X, Y)]dx "dx +du [f(w+h„)(fdu 2dx )+g(w—+h )(gdu 2dy )+2h„(fgdu——gdx fdy ), —

(3.15)

where, as already explained, the h„here are the samefunctions as in (3.14) but with X=x f and Y =y —g as-arguments, rather than x and y. As usual, u =z —t.

Let us now evaluate, as we did for gauge strings, theacceleration in the x direction felt by a slowly movingtest particle located at y =0, assuming the distance to thestring core much larger than the amplitude of the travel-

I

ing waves, x »f,g. We will also neglect terms of orderunity against lnx/y, which we assume to be large. Then

dx 1 f +g x 5 2x f-=2Gp ——2 ln —+2—f lndt' x x y ~t y

(3.16)

45 GRAVITATIONAL EFFECTS OF TRAVELING WAVES ALONG. . . 437

IV. STRONGLY GRAVITATING TRAVELING WAVES

The metric around a static and cylindrically symmetriccosmic string that lies along the z axis, clearly admits twonull hypersurface orthogonal Killing vectors pointingalong the z+t directions. As the weak field Eq. (3.6) im-plies, traveling waves propagating along the (z t) di—rec-tion break the symmetry corresponding to the Killingvector 8, +, but not the one corresponding to 8, „sincethe latter remains a Killing vector in the traveling-wavemetric. In fact this property holds also beyond theweak-field limit. Garfinkle and Vachaspati have shown[12] that the exact metric of cosmic strings with travelingwaves must be of a generalized Kerr-Schild type withbackground the static cosmic-string metric; i.e., that themetric must have the form of Eq. (1.1). As already men-tioned in Sec. I, k„of Eq. (1.1) is a null andhypersurface-orthogonal Killing vector with respect toboth static and traveling-wave metrics. F is a scalar func-tion which satisfies, with respect to the static metric, thecondition

kPV„F=0,

and the wave equation

Q(e "F)=0 .

(4.1)

(4.2)

Here A is a scalar determined from V„k„=kt,V„~A.The existence of A follows from the fact that k„ isa Killing and hypersurface-orthogonal vector field.Equivalently, if we write the static metric in coordinatesadapted to the two Killing fields, say [u, v,y', y ], thene " is the g„, component of the metric written in the form

The first term, independent on the traveling waves, is therepulsive 1/x force that a static global string exerts [10].The second term, proportional to (f +g ), is an attrac-tive 1/x force analogous to the gauge-string case. Butnow this term has an additional factor lnx/y. This extralogarithmic dependence can be understood taking intoaccount that while for a gauge string the traveling wavesaffect the stress-energy tensor just along the string core,in the global-string case it is affected everywhere, and glo-bal strings have mass per unit length and tensions thatdepend logarithmically on the distance to the string core.The last term has different signs at different times, but asin the gauge-string case is a time derivative that can bedropped off when evaluating the integrated effect of atraveling pulse.

In the global-string case the situation might appear, atfirst sight, even more problematic [12],because the staticmetric becomes singular at a finite proper distance fromthe core. Hence, it may not be possible to give asymptot-ic conditions on the function F. In this section we showhow to select the function F among all possible solutionsby imposing the appropriate weak-field limit. Our stra-tegy will be to explicitly write the weak-field traveling-wave metric of Eq. (3.6) as a generalized Kerr-Schildmetric with background the corresponding static one. Inso doing, we will find the weak-field value of the functionF. Nothing new will be learned for the gauge-string case,already discussed in Ref. [12]. For the global string,though, this will allow us to choose the function F ade-quate to the exact metric with traveling waves.

Let us first express the metric (3.6) in the coordinates[X,Y, u, v] defined in Eqs. (2.2) and (2.3). We easily findthat

we find

dsrw =2(1—w)du dv+(5,"+h;.)dX'dXJ

—2(1 w)( fX+g—Y)du (4.6)

In this form, the cosmic-string traveling-wave metric canbe compared to the static one (3.4). We observe that itcontains the static part (3.4), with the coordinatesX, Y, v+u/2 and v —u/2 playing, respectively, the roleof x, y, z, and t. The modification with respect to thestatic metric is of the (generalized) Kerr-Schild type sincethe metric (4.6) can be written as

gp~ gp~ +Fkpk~ (4.7)

with k"=5", a null, hypersurface orthogonal Killing field

with respect to both static and traveling-wave metrics.Since k„=(1—w)5„", we conclude by comparison of (4.6)and (4.7) that the function F in (4.7) is in the weak-gravitylimit equal to

fX+g Y1 —w

dsrw =(1—w)du dv +(5,"+h; )dX'dXJ

+(1—w)[(f +g )du +2fdX+2gdY]du, (4.4)

with i,j running from one to two. By changing in a laststep the v coordinate to v,

2v=v+ f (f '+g')du+2fX+2gY, (4.5)

g( 1 2)ds„=2e"' ' 'du dv+g, "(y',y )dy'dy~, (4.3) A.. Gauge cosmic strings

with i,j running from one to two. Here one of the obvi-ous null Killing vectors is k"=5"„,k„=e "5„"(the other isobtained from k„by interchanging u and v). Thus in thiscoordinate system, Eq. (4.1) simply states that F is in-dependent of v.

The function F is not uniquely determined from Eq.(4.2) alone. One needs further information, such asasymptotic conditions, to appropriately choose the solu-tion that describes the spacetime around a string withtraveling waves characterized by the functions f and g.

This case has already been discussed in the literature[8]. Here we will just briefly reproduce the result for theexact metric around a gauge cosmic string with travelingwaves in the zero-thickness approximation, for complete-ness and to exemplify how our method works beforemoving into global strings.

The exact metric around the static string is given byEq. (3.8) with the exact function h of Eq. (3.9). Compar-ing it with Eq. (4.3) we find that A =0. It should then beobvious that the function F which satisfies Eq. (4.2) and

438 ECONOMOU, HARARI, AND SAKELLARIADOU 45

has as weak-field limit the Eq. (4.8) with w =0, is simplyF= 2—(fX+gY). This result agrees with that of Ref.[8], obtained there in a different way.

The weak-field limit is obtained for

(~go»1 . (4.11)

B. Global cosmic stringsTo see this, let us first perform a coordinate transforma-tion of the radial coordinate g to R:

The exact metric around a static global string reads[14] g

—go

= — 1 — ln—1 R

2' (4.12)

ds„= ( dt +—dz )0

' 1/2

+ y2 exp (dg +d8 ) .ko

—0'

0(4.9)

Here, with regard to the expression that appears in Ref.[14], we have made the notational changes j u, u o J~ [g,go]. The parameter y is the same as in Eq. (3.14).Finally, the parameter go is

1—=4GR «10

(4.10)

with Gp the same as in Eqs. (3.13) and (3.14). We pointout here that the metric in Eq. (4.9), as well as its weak-field liinit in Eq. (3.13), are not solutions of Einstein equa-tions coupled to the equations for the scalar field thatmakes up the string. Rather, they are solutions to Ein-stein equations with the scalar field configuration fixed toits asymptotic value everywhere outside the core, whichis in most cases a sensible approximation.

The metric (4.9) has two singularities, one at g= ao andanother at /=0. The former is located at the center ofthe string's core, where the metric (4.9) is not expected todescribe correctly the spacetime. Its validity starts fromthe string's core at (=go and ranges down to the outersingularity at (=0.

~,[g~g(gF)]+~e[gae(gF)] =0 . (4.13)

According to Eq. (4.8), in the weak-field limit (4.11) Fmust behave, to first order in go ', as

R(f cos8+g sin8)1 —( I/go) ln(R /y)

(4.14)

Guided by this large-g behavior, Eq. (4.13) can be easilysolved by separation of variables and an intermediate an-satz F=P/g, leading to a zero-order modified Besseldifferential equation for the g-dependent part of P. Theresult is

It is then not very difficult to check that with the usualcoordinate transformation X =R cose, Y=R sine weobtain in the limit (4.11), to first order in go, the weak-field metric for the static global string as in Eq. (3.14).Consequently, the coordinates t, z, (R,8), X, and Y ofthis subsection, agree in the weak-field limit (4.11) withthe corresponding t, z, (r, 8),x,y harmonic coordinates ofSec. III B.

Comparing Eqs. (4.9) and (4.3) we see that e "~g.Therefore, using Eq. (4.2) and the metric (4.9), we findthat the function F for the global string satisfies

Ko(g)F= —2Q (f cos8+g sin6), with Q =1/2

2ko ye' 1+ 1

8 o(4. 15)

Here Ko(g) denotes the zero-order modified Bessel function. That this expression reduces appropriately to (4.14), canbe checked using Eq. (4.12) and the large-argument asymptotic behavior of Ko(g):

1/2

Ko(g)~8g

e & 1 — +O(g )

Thus, according to Eq. (4.7), the exact metric for a global string with traveling waves can be written as

dsT =2+du du+y — exp (dg +d8 )— (Ko(g)(f cos8+g sin8)du (4. 16)

To go back to the x,y, (r, 8),z, t coordinate system, where

f and g attain their natural interpretation as profiles ofthe traveling waves in the x and y direction, respectively,one has to perform in reversed order the transforrnationsof Eqs. (4.5) and (2.2), using (4.12) to go from $,6 to X, Y.

An interesting point to note here is that gKo(g') van-ishes at /=0 as ging. As mentioned above, at /=0 there

is a curvature singularity of the static global-string metric[14]. This singularity appears to be unavoidable in stan-dard field-theoretical models of a global string [24]. Non-singular solutions around a static global string (whichhave an event horizon rather than a singularity) can befound only if invariance under boosts in the string direc-tion is given up [25,26]. It is thus worthwhile to investi-

45 GRAVITATIONAL EFFECTS OF TRAVELING ~AVES ALONG. . . 439

gate what is the effect of traveling waves on the singularstructure of the static global-string metric. In the Ap-pendix we evaluate the nonvanishing components of theRiemman tensor for the spacetime around a global stringwith traveling waves. We conclude that more com-ponents of the Riemann tensor diverge at the singularity(/=0). Thus, we may say that the effect of the travelingwaves is to increase the strength of this singularity, atleast in the sense of increasing the tidal forces in itsneighborhood. However one should note that curvaturescalars, like the one given in the Appendix, appear to beentirely insensitive to the presence of the traveling waves.

exact metric around a string with traveling waves mustbe of the generalized Kerr-Schild type, i.e., as in Eq. {4.7),with our result for the linearized metric, we found the ex-act metric around a global string with traveling waves tobe specifically that of Eq. {4.16). The metric around astatic global string is singular at a finite proper distanceoff the core. Traveling waves can not remedy this situa-tion. On the contrary, more components of the Riemanntensor diverge at the singularity. Curvature invariants,though, blow up at the singularity exactly as in the staticcase.

ACKNOWLEDGMENTS

V. SUMMARY AND CONCLUSIONS

A traveling wave along an otherwise straight cosmicstring exerts, in the linear approximation to general rela-tivity, a net attractive effect upon a nonrelativistic testparticle. We have seen that in the case of global cosmicstrings these effects are larger than for gauge strings by afactor with logarithmic dependence on the distance to thestring core. This is not so surprising, since ihe mass perunit length and tensions of the static global string alsogrow logarithmically. The static global string exerts arepulsive force, inversely proportional to the distance offthe core. The traveling waves may have even larger grav-itational effects if (f +g )1n(ply) is larger than unity.

Combining Garfinkle and Vachaspati's proof that theI

We are grateful to T. Vachaspati and A. Vilenkin forpointing out a mistake in our manuscript, and for veryuseful comments about the gravitational effects of a trav-eling pulse. This work was supported by the DirectorateGeneral for Science, Research and Development of theCommission of the European Communities under Con-tract No. C 11-0540-M(TT). The work of D.H. was alsosupported by Consejo Nacional de InvestigacionesCientificas y Tecnicas (Argentina) (CONICET), Universi-dad de Buenos Aires and Fundacion Antorchas.

APPENDIX

Here we give for the metric (4.16) the nonvanishingcomponents of the Riemann tensor:

&/2 (2 g2R = (2goy ) + exp

0 0

R„~„~=—

4koWo{k)+ {4k'+ 3')Ico{()Q(f sin8 —g cos8),

0

4k'+CoR eee 8/2

{4k'+ko)W'o{4)+ l44'{Co+1)—kol&o{4)R„(„(= Q(g sln8+ j'cos8),

4kok

{4k'+Co)W'o{k)+(4k'{to+1)+koÃo{k)Q (g sin8+ f cos8),

4kok(A 1)

, 4k' —ko

Renege=l'

4& &~

' 1l20

expko

—0'

ko

=2R ee=~

(A2)

From Eq. (Al) we can obtain after a lengthy butstraightforward calculation the curvature scalar

Here prime denotes derivative with respect to g.The only nonvanishing component of the Ricci tensor

1s(A3)

Note that it has no dependence on the traveling waves. Itblows up at (=0 and g= 0D demonstrating thus the realnature of these singularities.

ECONOMOU, HARARI, AND SAKELLARIADOU 45

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