More about thin-shell wormholes associated to cosmic strings

Martın G. Richarte* and Claudio Simeone†

Departamento de Fısica, Facultad de Ciencias Exactas y Naturales, UBA Ciudad Universitaria,Pabellon I, 1428, Buenos Aires, Argentina

(Received 21 April 2009; published 23 June 2009)

Previous analysis about thin-shell wormholes associated to cosmic strings are extended. More evidence

is found supporting the conjecture that, under reasonable assumptions about the equations of state of

matter on the shell, the configurations are not stable under radial velocity perturbations.

DOI: 10.1103/PhysRevD.79.127502 PACS numbers: 04.20.Gz, 04.40.Nr, 11.27.+d

I. INTRODUCTION

Following the leading work by Morris and Thorne [1],traversable Lorentzian wormholes received considerableattention [2]. Wormholes would imply a nontrivial topol-ogy, connecting two regions of the Universe by a travers-able throat. If they actually exist, such configurations couldinclude some features of unusual interest, as, for example,the possibility of time travel [3]. However, the flareoutcondition to be fulfilled at the wormhole throat re-quires—in the framework of general relativity—the exis-tence of matter which violates the energy conditions(‘‘exotic matter’’) [1,2,4]. Conditions reducing matter sup-porting wormhole geometries, consequently, have de-served a detailed analysis. Besides (as most solutions ofthe equations of gravitation), traversable wormhole geome-tries result of physical interest mainly as long as they arestable, at least under a simple kind of perturbation. Thus,beyond the mere characterization of static wormhole solu-tions, their stability under perturbations should always beexplored.

Some years ago, the gravitational effects of topologicaldefects as domain walls and cosmic strings were the objectof a thorough study, because of their possible relevance instructure formation in the early Universe, and also for theirpossible manifestation as gravitational lenses (seeRef. [5]). On the other hand, present theoretical develop-ments suggest a scenario in which the fundamental build-ing blocks of nature are extended objects. In particular,one-dimensional objects—strings—constitute the candi-dates more often considered. Thus the interest in the gravi-tational effects of both cosmic and fundamental strings hasbeen renewed in the last years (see, for example, Ref. [6]).Recent studies regarding cylindrical wormholes includethe works by Clement [7,8], Aros and Zamorano [9],Kuhfittig [10], Bronnikov and Lemos [11], and byBejarano, Eiroa and one of us [12,13]. The thin-shellwormhole configurations associated to local and globalcosmic strings analyzed in Refs. [12,13] turned out to beunstable under velocity perturbations preserving the cylin-

drical symmetry. Moreover, it was noted [13] that thisfeature seemed to be independent of the particular geome-try considered, as long as the symmetry and the form of theequations of state of the static configurations were pre-served. Because for a far observer these wormholes wouldappear as cosmic strings, bounds on cosmic string abun-dance, which seem rather restrictive (see for instanceRef. [14]), could imply a first constraint on the existenceof such wormholes. Mechanical unstability would mean anadditional restriction to the possibility of finding theseobjects in the present day Universe.In the present work, we perform two extensions of these

previous analysis regarding cylindrical thin-shell worm-holes associated to cosmic strings. We first construct thin-shell wormholes associated with two different globalstrings; then, starting from static wormhole configurations,we consider small velocity perturbations and determine thecharacter of the subsequent dynamical evolution. Our re-sults are consistent with the conjecture introduced inRef. [13].

II. THIN-SHELLWORMHOLES

The wormhole construction follows the usual steps ofthe cut and paste procedure [15]. Starting from a manifoldM described by coordinates X� ¼ ðt; r; �; zÞwe remove theregion defined by r < a and take two copies Mþ and M�of the resulting manifold. Then we join them at the surface� defined by r ¼ a, so that a new geodesically completemanifold M ¼ Mþ [M� is obtained. The surface � is aminimal area hypersurface satisfying the flareout condi-tion: in both sides of the newmanifold, surfaces of constantr increase their areas as one moves away from �; thus onesays that M presents a throat at r ¼ a. On the surface �we define coordinates � ¼ ð�; �; zÞ, where � is the propertime. Then, to allow for a dynamical analysis, we let theradius depend on �, so that the surface � is given by thefunction H which fulfills the condition H ðr; �Þ ¼ r�að�Þ ¼ 0. As a result of pasting the two copies of theoriginal manifold, we have a matter shell placed at r ¼a. Its dynamical evolution is determined by the Einsteinequations projected on �, that is, by the Lanczos equations[16]

*martin@df.uba.ar†csimeone@df.uba.ar.

PHYSICAL REVIEW D 79, 127502 (2009)

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� ½Kji � þ ½K��j

i ¼ 8�Sji ; (1)

where Kji is the extrinsic curvature tensor defined by

K�ij ¼ �n��

�@2X�

@�i@�j þ ����

@X�

@�i

@X�

@�j

����������; (2)

with n�� the unit normals (n�n� ¼ 1) to � in M:

n�� ¼ ���������g�� @H@X�

@H@X�

���������1=2@H

@X� : (3)

The bracket ½Kji � denotes the jump Kjþ

i � Kj�i across the

surface �; ½K� ¼ gij½Kij� is the trace of ½Kij�; and Sji ¼diagð�; p�; pzÞ is the surface stress-energy tensor, withthe surface energy density and p�, pz the surface pressures.

III. WORMHOLES ASSOCIATED TO COSMICSTRINGS

A. Nonsingular global string

We first construct a thin-shell wormhole associated to aglobal cosmic string whose metric, differing from the casetreated in [13], presents no curvature singularity; this is aconsequence of relaxing the demand of boost invariancealong the axis of symmetry [17]. The geometry has theform

ds2 ¼ FðrÞð�dt2 þ dr2Þ þHðrÞdz2 þGðrÞd�2; (4)

where

FðrÞ ¼�Pþ 2

B2lnr

r0

�1� ffiffi

2p; (5)

HðrÞ ¼�Pþ 2

B2lnr

r0

� ffiffi2

p; (6)

GðrÞ ¼ 4B2�2

�r

r0

�2�Pþ 2

B2lnr

r0

�2� ffiffi

2p: (7)

The constant r0 is the core radius, and P, B and � areintegration constants fulfilling B2 ¼ 4�v2=�2, with v de-fining the value of the field associated to the string [17]. Itis clear from the metric (4) that for P> 0, which corre-sponds to the case of no outer singularity, the area per unitlength is an increasing function of the radial coordinate forr > r0. Then the flareout condition required for the worm-hole construction is fulfilled. Also, note that the geodesicswithin a plane orthogonal to the string open up at the throatof the wormhole construction, because the metric compo-nent g��ðrÞ is itself an increasing function of r.

In terms of these functions the components of the ex-trinsic curvature read

K��� ¼ �

ffiffiffiffiffiffiffiffiffiffiFðaÞp

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ FðaÞ _a2p

�2 €aþ F0ðaÞ

F2ðaÞ þ 2 _a2F0ðaÞFðaÞ

�; (8)

K��� ¼ �G0ðaÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ FðaÞ _a2p2GðaÞ ffiffiffiffiffiffiffiffiffiffi

FðaÞp ; (9)

and

Kz�z ¼ �H0ðaÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ FðaÞ _a2p2HðaÞ ffiffiffiffiffiffiffiffiffiffi

FðaÞp ; (10)

where the dot means d=d� and the prime indicates aderivative with respect to r. Replacing these expressionsin the Lanczos equations we obtain the surface energydensity ¼ �S�� and the pressures p� ¼ S�� and pz ¼ Szz:

¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ FðaÞ _a2p8�

ffiffiffiffiffiffiffiffiffiffiFðaÞp

�G0ðaÞGðaÞ þ

H0ðaÞHðaÞ

�; (11)

p� ¼ 1

8�ffiffiffiffiffiffiffiffiffiffiFðaÞp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ FðaÞ _a2p�

�2FðaÞ €aþ FðaÞ _a2

�H0ðaÞHðaÞ þ

2F0ðaÞFðaÞ

�þH0ðaÞ

HðaÞþ F0ðaÞ

FðaÞ�; (12)

pz ¼ 1

8�ffiffiffiffiffiffiffiffiffiffiFðaÞp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ FðaÞ _a2p�

�2FðaÞ €aþ FðaÞ _a2

�G0ðaÞGðaÞ þ

2F0ðaÞFðaÞ

�þG0ðaÞ

GðaÞþ F0ðaÞ

FðaÞ�: (13)

From these equations we find that for the static situation_a ¼ €a ¼ 0 the pressures and the surface energy densitysatisfy the equations of state

p� ¼ �GðaÞ½FðaÞH0ðaÞ þ F0ðaÞHðaÞ�FðaÞ½GðaÞH0ðaÞ þG0ðaÞHðaÞ� ; (14)

pz ¼ �HðaÞ½FðaÞG0ðaÞ þ F0ðaÞGðaÞ�FðaÞ½GðaÞH0ðaÞ þG0ðaÞHðaÞ� : (15)

If we are interested in small velocity perturbations, it islicit to assume that the evolution of the matter on the shellcan be described as a succession of static states. Thus weshall accept, as done before [12,13], that the form of theequations of state corresponding to the static case is keptvalid in the dynamical evolution.1 With this assumption,the equations above lead to the equation of motion

2FðaÞ €aþ F0ðaÞ _a2 ¼ 0: (16)

By writing €a ¼ _ad _a=da we can recast this equation (for

1Besides, the perturbative treatment avoids possible subtletiesregarding the validity of the static geometry for r > a in thepresence of a cylindrical moving shell.

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_a � 0) as

2FðaÞ d _a

daþ F0ðaÞ _a ¼ 0: (17)

This equation is solved by

_að�Þ ¼ _a0

ffiffiffiffiffiffiffiffiffiffiffiffiFða0ÞFðaÞ

s; (18)

where a0 and _a0 are, respectively, the initial wormholeradius and its initial velocity. This shows that the sign ofthe velocity is determined by its initial sign, that is, after asmall velocity perturbation the throat undergoes a monoto-nous evolution; no oscillatory behavior exists, at leastunder the approximations assumed. Note that, because Fis a decreasing function of a, then the absolute value of thethroat velocity decreases when _a0 is negative, while thevelocity grows when _a0 points outwards (so, after a finitetime the assumption of a low velocity is no more valid).

B. Global string with curvature singularity

After studying the nonsingular metric case, we nowaddress a generalization of the analysis carried out inRef. [13]. The starting point is the metric [18]

ds2 ¼ �UðrÞdt2 þWðrÞdz2 þ VðrÞðdr2 þ r2d�2Þ (19)

with the functions U, W, and V defined as follows:

UðrÞ ¼�1� lnr

lnrs

�1þ!

; (20)

WðrÞ ¼�1� lnr

lnrs

�1�!

; (21)

VðrÞ ¼ �2

�1� lnr

lnrs

�ð1=2Þð!2�1Þexp

�� ln2r

lnrs

�: (22)

This metric presents a curvature singularity placed at finiteproper distance rs ¼ ð8�F2Þ�1, where F is the vacuumexpectation value of the scalar field whose symmetrybreaking leads to the global string. The parameter ! isrelated to the mass excess (per unit length and as a fractionof M2

P, being MP the Planck mass) resulting from theexistence of a timelike current along the string. The con-stant � can be determined by matching the metric (19) withthe metric inside the core; the radius of the core is rcore � 1(see Ref. [18]). We shall assume 0<!< 1; in the limit! ! 0 the case studied in Ref. [13] is recovered.

The extension of our analysis to this metric is slightlymore complicated than the preceding one. It requires somecare with the preliminary study of the original manifold,

because of the existence of a curvature singularity at finiter and because the flareout condition is not fulfilled every-where for arbitrary values of the parameters. However, thesteps leading to an equation allowing for a qualitativeunderstanding of the dynamical evolution turn out to bestraightforward. The area per unit of z coordinate, for agiven value of the radius, is determined by the product ofW, V and r2. Taking the corresponding derivative withrespect to r, it is easy to show that the flareout conditionrequired for the possible existence of the associated thin-shell wormhole can be satisfied. Indeed, we find that the

flareout condition is fulfilled as long as a < rs expð�j!�1j ffiffiffiffiffiffiffiffiffiffiffiffiðlnrsÞp

=2Þ is selected as the wormhole throat radius.Applying the cut and paste procedure, the components ofthe extrinsic curvature now turn to be

K��� ¼ � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVðaÞ

1þ VðaÞ _a2s �

2 €aþ _a2�U0ðaÞUðaÞ þ

V0ðaÞVðaÞ

�

þ U0ðaÞUðaÞVðaÞ

�; (23)

K��� ¼ � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ VðaÞ _a2

VðaÞ

s �2

aþ V 0ðaÞ

VðaÞ�; (24)

and

Kz�z ¼ � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ VðaÞ _a2

VðaÞ

sW 0ðaÞWðaÞ : (25)

The resulting expressions for the energy density and pres-sures are

¼ � 1

8�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ VðaÞ _a2

VðaÞ

s �2

aþW 0ðaÞ

WðaÞ þV 0ðaÞVðaÞ

�; (26)

p� ¼ 1

8�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVðaÞ

1þ VðaÞ _a2s �

2 €aþ _a2�U0ðaÞUðaÞ þ

V0ðaÞVðaÞ þ

W 0ðaÞWðaÞ

�

þ 1

VðaÞ�U0ðaÞUðaÞ þ

W 0ðaÞWðaÞ

��; (27)

pz ¼ 1

8�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVðaÞ

1þ VðaÞ _a2s �

2 €aþ _a2�U0ðaÞUðaÞ þ 2

V0ðaÞVðaÞ þ

2

a

�

þ 1

VðaÞ�U0ðaÞUðaÞ þ

V 0ðaÞVðaÞ þ

2

a

��: (28)

In the static case _a ¼ €a ¼ 0, the pressures are related withthe energy density by the equations of state

pz ¼ �U0ðaÞVðaÞWðaÞaþUðaÞV 0ðaÞWðaÞaþ 2UðaÞWðaÞVðaÞUðaÞVðaÞW 0ðaÞaþUðaÞV 0ðaÞWðaÞaþ 2UðaÞWðaÞVðaÞ ; (29)

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p� ¼ �U0ðaÞVðaÞWðaÞaþUðaÞVðaÞW 0ðaÞa

UðaÞVðaÞW 0ðaÞaþUðaÞV0ðaÞWðaÞaþ 2UðaÞWðaÞVðaÞ : (30)

If we propose, as before, that the static equations of stateremain valid for the dynamic case (as long as the samehypothesis above are satisfied) then, after some algebraicmanipulations, we obtain the equation of motion

2VðaÞ €aþ V 0ðaÞ _a2 ¼ 0: (31)

The form of the equation of motion is the same of thatobtained before, and the solution for the throat velocity isthen

_að�Þ ¼ _a0

ffiffiffiffiffiffiffiffiffiffiffiffiVða0ÞVðaÞ

s: (32)

Therefore, after a small velocity perturbation the wormholethroat undergoes a monotonous evolution. In other words,no oscillatory behavior exists after perturbing an initiallystatic configuration. Within the range r� < r < rþ withr� ¼ ffiffiffiffi

rsp

expð� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnrsðlnrs þ!2 � 1Þp

=2Þ the function Vdecreases with r. For 0<!< 1, the radius r� turns tobe smaller than the core radius, while rþ results greaterthan the singularity radius. Thus for such values of ! we

have V 0 < 0 for any physically meaningful radius beyondthe core and satisfying the flareout condition. Con-sequently, the throat contracts decelerating if _a < 0, whilein the case of an initial velocity pointing outwards theresult is an accelerated expansion (as before, the lowvelocity assumption is eventually no more fulfilled).

IV. CONCLUSION

Summarizing, for both extensions of previous analysiswithin the framework of thin-shell wormholes associatedto cosmic strings, we find that no oscillatory solutions existfor the class of equations of state adopted. A small initialvelocity pointing to the axis of symmetry leads to a decel-erated contraction of the wormhole throat, while an initialslow expansion is accelerated. Our results, thus, providemore evidence supporting the recent conjecture that thisclass of wormholes would not be stable under small veloc-ity perturbations, as long as the cylindrical symmetry andthe static equations of state for matter on the shell arepreserved.

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