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Thin-shell wormholes associated with global cosmic strings

Cecilia Bejarano,1,* Ernesto F. Eiroa,1, and Claudio Simeone2,1Instituto de Astronoma y Fsica del Espacio, C.C. 67, Suc. 28, 1428, Buenos Aires, Argentina

2Departamento de Fsica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria Pab. I, 1428, Buenos Aires, Argentina

(Received 25 October 2006; published 12 January 2007)

In this article we construct cylindrical thin-shell wormholes in the context of global cosmic strings. Westudy the stability of static configurations under perturbations preserving the symmetry and we find thatthe throat tends to collapse or expand, depending only on the direction of the velocity perturbation.

DOI: 10.1103/PhysRevD.75.027501 PACS numbers: 04.20.Gz, 11.27.+d, 04.40.Nr

I. INTRODUCTION

The study of traversable Lorentzian wormholes hasreceived great attention since the leading paper by Morrisand Thorne [1]. These objects are solutions of the equa-tions of gravitation that consist in two regions connectedby a throat, which for static wormholes is a minimal areasurface satisfying a flare-out condition [2]. Wormholes canjoin two parts of the same universe or two separate uni-verses [1,3], and they must be threaded by exotic matterthat violates the null energy condition [14]. It was shownby Visser et al. [5] that the amount of exotic matter neededaround the throat can be made as small as desired byappropriately choosing the geometry of the wormhole. Aphysically interesting class of wormholes is that of thin-shell ones, which are constructed by cutting and pastingtwo manifolds [3,6] to form a geodesically complete newone with a throat placed in the joining shell. In this case,the exotic matter needed for the existence of the configu-ration is located at the shell. Poisson and Visser [7] per-formed a linearized stability analysis under sphericallysymmetric perturbations of a thin-shell wormhole madeby joining two Schwarzschild geometries. Later, Barceloand Visser [8] applied this method to wormholes con-structed using branes with negative tensions and Ishakand Lake [9] analyzed the stability of transparent spheri-cally symmetric thin-shells and wormholes. Eiroa andRomero [10] extended the linearized stability analysis toReisner-Nordstrom thin-shell geometries, and Lobo andCrawford [11] to wormholes with a cosmological constant.Also, Lobo and Crawford considered the stability of dy-namical thin-shell wormholes [12]. Five-dimensional thin-shell wormholes in Einstein-Maxwell theory with a Gauss-Bonnet term were studied by Thibeault et al. [13].

Recently, there has been a renewed interest in the studyof cosmic strings as a consequence of new developments insuperstring theory which suggest that fundamental stringscan play the role of cosmic strings, and also because of newobservational evidence which may support their existence

[14]. In the presence of a complex scalar field, spontaneoussymmetry breaking results in the formation of cosmicstrings; these can be global or local, depending on thecharacter of the symmetry which is broken in the phasetransition [15]. In the case of local strings, a gauge fieldexists besides the complex scalar field, and the stress-energy tensor turns to be concentrated in a very thincylinder. The gravitational effects of such objects havebeen extensively studied [16], its main features are that aconstant deficit angle is induced in the space around, andthe trajectories of non relativistic particles are not affected(in the absence of wiggles or kinks along the string [17]and if a second phase transition leading to the appearanceof charge carriers has not taken place [18]). The energy ofglobal strings, instead, extends beyond the core, and staticsolutions with boost invariance along the symmetry axishave a physical singularity at a finite distance [19,20]. Ifthe condition of boost invariance is relaxed, the outersingularity can be avoided [21]. Global cosmic stringshave near the core a deficit angle which grows with theradial coordinate, then decreases and finally turns into anexcess angle that diverges at the singularity [15]. Also, thegeodesics around a global string show the peculiarity of arepulsive effect [22] pushing matter towards the singularity[19]. These important differences make clearly non trivialan extension to global strings of any analysis regarding thephysics of local strings; in particular, wormholes associ-ated with them deserve a separate and detailed study.

Wormholes associated with cosmic strings have beenpreviously investigated by several authors. Clement [23]studied traversable multiwormhole solutions where themetric asymptotically tends to the conical cosmic stringmetric, and later, he extended cylindrical multiple cosmicstrings metrics to wormhole spacetimes with only oneregion at spatial infinite, and he thoroughly analyzed thegeometry of asymptotically flat wormhole spacetimes pro-duced by one or two cosmic strings [24]. Aros andZamorano [25] obtained a solution which can be under-stood as a traversable cylindrical wormhole inside the coreof a global cosmic string. Eiroa and Simeone [26] studiedcylindrical thin-shell wormholes associated with local cos-mic strings. In this article we analyze thin-shell wormholes

*Electronic address: cbejarano@iafe.uba.arElectronic address: eiroa@iafe.uba.arElectronic address: csimeone@df.uba.ar

PHYSICAL REVIEW D 75, 027501 (2007)

1550-7998=2007=75(2)=027501(4) 027501-1 2007 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.75.027501

constructed from the spacetime metrics of global cosmicstrings. We do not intend to explain the mechanisms thatmight provide the exotic matter to them, but, instead, wefocus on the geometrical aspects of these objects. In Sec. IIwe construct the wormholes by applying the cut and pasteformalism, in Sec. III we study the stability of the configu-rations under radial perturbations, and finally, in Sec. IV,the results are discussed. Units where c @ G 1 areused.

II. THIN-SHELL WORMHOLES

We consider global strings with no currents presentalong the core and boost invariance along the string axis[19]. The metric of these cosmic strings, in coordinatesX t; x; ; z, takes the form

ds2 fxdt2 dz2 gxdx2 hxd2; (1)where

fx 1 lnxlnxs

; (2)

gx 2eln2x= lnxs1 lnx

lnxs

1=2; (3)

and

hx gxx2: (4)This metric is valid outside the core, where the stress-energy tensor due to a Goldstone boson field has theform T @@ 1=2gj@j2 H:c:, with Fei. The parameter can be determined by match-ing the metric given above with the metric inside the core.The radius of the core is xcore 1 and an outer physicalsingularity [19] occurs at lnxs 8F21, with F deter-mined by the scale of the symmetry breaking leading to theappearance of the topological defect. For F much less thanthe Planck mass MP, xs is very large; for example, whenF 2 1017 GeV, then lnxs 137 and the singularity issituated at a distance of the order of the cosmologicalhorizon [19]. The area per unit of z coordinate for a fixedvalue of x is given by 4

fxhxp . This area is an in-

creasing function of x from xcore to xwh xs exp

lnxs

p=2 and decreases with x from xwh to xs,

where it is zero and the outer singularity is reached.Starting from the geometry given by Eq. (1), we choose aradius a between xcore and xwh, and we take two copies ofthe region with x a:

M fX=x ag; (5)and paste them at the hypersurface defined by

fX=x a 0g; (6)to make a geodesically complete manifold M M [M. The area per unit of z coordinate defined above is anincreasing function for x 2 a; xwh xcore; xwh; there-fore the flare-out condition is satisfied and this construction

creates a cylindrically symmetric thin-shell wormhole1

with two regions connected by a throat at . On themanifold M we can define a new radial coordinate l

Rxa gxp dx, where the positive and negative signs cor-respond, respectively, to M and M, with jlj represent-ing the proper radial distance to the throat, which is placedin l 0. To study this traversable wormhole we use thestandard Darmois-Israel formalism [27,28]. The throat ofthe wormhole is placed at the shell , which is a synchro-nous timelike hypersurface. We can adopt coordinatesi ; ; z in , with the proper time on the shell.For the analysis of stability under perturbations preservingthe symmetry, we let the radius of the throat be a functionof the proper time, that is a a. The position of thethroat is then given by the equation

: H x; x a 0: (7)The extrinsic curvature associated with the two sides of theshell is:

Kij n@2X

@i@j

@X

@i@X

@j

; (8)where n are the unit normals (nn 1) to in M:

n

g @H@X @H@X

1=2@H@X : (9)In the orthonormal basis fe; e; ezg, defined by e 1=fxp et, e 1=hxp e, ez 1=fxp ez, the metric

has the form g{ | { | diag1; 1; 1; thus the nonvanishing components of the extrinsic curvature read

K gap

2ga _a2 1

p 2 a _a2f0afa

g0aga

f0a

faga; (10)

K

h0a

ga _a2 1

p2ha gap ; (11)

and

Kz z

f0a

ga _a2 1

p2fa gap ; (12)

where the dot stands for the derivative with respect to .The Einstein equations on the shell (Lanczos equations)take the form:

K{ | Kg{ | 8S{ |; (13)where K{ | K{ | K{ |, K g{ |K{ | is the trace ofK{ | and S{ | diag;#;#z is the surface stress-energy tensor, with the surface energy density and #;zthe surface tensions. Then replacing Eqs. (10)(12) in

1If, besides, one demands that geodesics within a plane or-thogonal to the string open up at the throat, then the conditionthat hx is an increasing function of x is also required.

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Eq. (13) we have

ga _a2 1

p8

gap

f0afa

h0aha

; (14)

#z gap

8ga _a2 1

p 2 a _a2f0afa

g0aga

h0aha

1ga

f0afa

h0aha

; (15)

# gap

8ga _a2 1

p 2 a _a22f0afa

g0aga

2f0a

faga: (16)

The flare-out condition implies that the product faha isan increasing function of a so from Eq. (14) is immediateto see that the surface energy density is negative, whichreflects the presence of exotic matter at the throat. Thetensions #, #z and the energy density satisfy theequation

# #z f0aha fah0af0aha fah0a: (17)

Equations (14)(16) plus one of the equations of state# or #z determine the dynamics of the shell. Weemphasize that these dynamic equations are valid under theassumption that the velocity of the throat is small so thatthe geometry outside the shell remains static and theemission of gravitational waves can be neglected.

III. STABILITY ANALYSIS

In this section, we analyze the stability of the staticsolutions under perturbations preserving the symmetry.The equations for a static shell are obtained by replacing_a 0 and a 0 in Eqs. (14)(16):

18

gap

f0afa

h0aha

; (18)

# 1

4gap

f0afa ; (19)

#z 1

8gap

f0afa

h0aha

: (20)

Equations (19) and (20) can be recast in the form

# 2f0aha

fah0a f0aha; (21)

#z ; (22)thus, for a given value of the throat radius, the functions f,g and h determine the equations of state # and #zof the exotic matter on the shell. Because the analysis is

restricted to small velocity perturbations around the staticsolution, the evolution of the exotic fluid in the shell can beconsidered as a succession of static states. Therefore weassume that the form of the equations of state for the staticcase is preserved in the dynamic case, thus # and#z are given by Eqs. (21) and (22). Then, replacingEqs. (14) and (15) in Eq. (21) [or Eqs. (14) and (16) inEq. (22)], the following second order differential equationfor a is obtained:

2ga a g0a _a2 0: (23)It is easy to see that

_aga

q _a0

ga0

q(24)

satisfies Eq. (23), with 0 an arbitrary (but fixed) time.Integrating both sides of Eq. (24) from 0 to and makingthe substitution da _ad, we have

Z aa0

ga

qda _a0

ga0

q 0: (25)

Replacing Eq. (3) in Eq. (25), and performing the integra-tion, this gives

0 j j xsp lnxs5=825=8 _a0

ga0

p 38;1 lna

lnxs

2 lnxs

2

3

8;1 lna0

lnxs

2 lnxs

2

; (26)

where ; z R1z u1eudu is the incomplete gammafunction. Inverting this relation the evolution of the radiusof the throat a turns to be

a xs exp

21

3

8; A B 0

lnxs

s ; (27)

with ! R10 u!1eudu the Euler gamma function,1; z the inverse (for fixed ) of the incompletegamma function,

A 3

8;1 lna0

lnxs

2 lnxs

2

; (28)

and

B 25=8 _a0

ga0

pj j xsp lnxs5=8 : (29)

Equation (27) remains valid as long as the velocity givenby Eq. (24) is small. The qualitative behavior of a is noteasily seen from Eq. (27), but it can be obtained fromEqs. (23) and (24). Following Eq. (23), the sign of theacceleration of the throat is determined by the sign of thederivative of the radial function ga. It is clear that g0a isnegative if a 2 x1; x2, with

x1;2 xs

pexp

12

lnxslnxs 1

q ; (30)

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where the minus and the plus signs correspond, respec-tively, to x1 and x2. As lnxs 1, we have that x1 1 andx2 xs, so that the acceleration of the throat is alwayspositive. From Eq. (24), we can see that the sign of theinitial velocity determines the sign of the velocity at anytime. Therefore, when the initial velocity _a0 is positivethe wormhole throat presents an accelerated expansion andif _a0 is negative it has a decelerated collapse. We want toemphasize that the temporal evolution allows only forthese two possibilities because the sign of the velocity,which is given by its initial value, remains unchanged.

IV. DISCUSSION

In this paper we have studied thin-shell traversableLorentzian wormholes constructed from the metric of astatic global cosmic string with boost invariance. An ob-server outside the throat could not distinguish by localmeasurements between the wormhole geometry consid-ered here and the global string spacetime. We have pre-sented a general stability analysis under perturbationspreserving the symmetry of these wormholes. We haveassumed that the equations of state of the exotic matter atthe throat are linear relations between the surface tensionsand the surface energy density, and their form is the samein the dynamic case as in the static one. We have found thatthe temporal evolution of the wormhole throat is basically

determined by the sign of its initial velocity. If it is positive,the wormhole throat expands monotonically; when it isnegative, the wormhole throat collapses to the core radiusin finite time; and, if it is null, the wormhole throat remainsat rest. Then we conclude that there are no oscillatingsolutions. We have explicitly shown static solutions forany value of the radial coordinate but these solutions arenot stable under radial perturbations in the velocity. Weremark that this analysis is valid as long as the velocity ofthe throat is much smaller than the velocity of light toguarantee a static spacetime outside the throat, so that theemission of gravitational waves can be neglected. Globalstring geometries without boost invariance [21] have ametric that can be written in a similar form of those studiedin Ref. [26]. It is straightforward to see that for thin-shellwormholes constructed from these strings the stabilityproperties are the same as those obtained here and inRef. [26]. All these results seem to be a general conse-quence of cylindrical configurations and the equations ofstate assumed.

ACKNOWLEDGMENTS

This work has been supported by Universidad de BuenosAires and CONICET. Some calculations were done withthe help of the package GRTensorII (freely available athttp://grtensor.org).

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