Exact gravitational lensing by cosmic strings with junctions

Teruaki Suyama

Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan(Received 5 June 2008; published 18 August 2008)

We point out that the results by Brandenberger et al. [Phys. Rev. D 77, 083502 (2008)] that the

geometry around the straight cosmic strings with stationary junctions is flat to linear order in the string

tension can be immediately extended to any order.

DOI: 10.1103/PhysRevD.78.043532 PACS numbers: 98.80.Cq

I. INTRODUCTION

Cosmic strings may be produced in the early universe. Ifthese strings are fundamental strings, F-strings, or D1-branes, D-strings, which are generic end products of thebrane inflation, these strings can combine to form boundstates of p F-strings and q D-strings, (p,q)-strings [1–5].When two (p,q)-strings intersect, a junction is formed [6–9].

In [10], the authors gave a simple geometrical prescrip-tion to obtain the location of images of the source lensed bythe strings with a junction. In [11], the explicit form of themetric around the strings with stationary junctions wasprovided within the linear perturbation. It was also shownthat, like in the case of an infinite straight string [12,13],the space-time is locally flat and has deficit angles aroundthe strings determined by the string tensions.

In this short note, we point out that the flatness of thegeometry can be shown without invoking a perturbativeexpansion.

II. FLATNESS OF THE GEOMETRY

We start from the action of the strings with junctions

S ¼ � XNA¼1

�A

Zd2�A

ffiffiffiffiffiffiffiffiffiffi��A

p: (1)

Here N is the total number of the strings, �aAða ¼ 0; 1Þ is

the two-dimensional coordinate on the A-th string worldsheet and

�Aab � g��ðXAÞ @X�A

@�aA

@X�A

@�bA

; (2)

is the induced metric on the A-th string.�A is the tension ofthe A-th string. The strings are assumed to be infinitelythin. Taking the variation of Swith respect to g�� yields the

energy-momentum tensor of the strings, which is given by

T��ðxÞ ¼ � XNA¼1

�Affiffiffiffiffiffiffiffiffiffiffiffiffi�gðxÞp Zd2�A

ffiffiffiffiffiffiffiffiffiffi��A

p�abA

@X�A

@�aA

� @X�A

@�bA

�ð4Þðx� XAÞ: (3)

We are interested in the metric around the static stringswith junctions. For the junction to be static, the forcesexerted on the junction from attached strings must balance,which we will assume hereafter. Derivation of the forcebalance condition will be given later. The static conditionimplies that there exists a coordinate system in whichg0i ¼ 0 and the other metric components do not dependon t. As the metric around the strings, we use the followingansatz:

ds2 ¼ �dt2 þ hijð ~xÞdxidxj: (4)

g00 is not perturbed by the strings, which was explicitlyshown at the linearized level [11]. Here we assume that thisholds in fully nonlinear treatment, which will turn out to beconsistent with basic equations.Let us choose the time coordinate on each world sheet as

�0A ¼ t, i.e. X0

Að�AÞ ¼ �0A. Then the static condition means

that Xið�AÞ is independent of �0A. As for the spatial coor-

dinate, we take �1A so that it represents the length of the

corresponding string, which gives ðd�1AÞ2 ¼ hijdX

iAdX

jA.

In these world sheet coordinates, the induced metrics aregiven by

�A00 ¼ �1; �A01 ¼ 0; �A11 ¼ 1: (5)

Let us first consider the Einstein equations,

R�� ¼ 8�GS�� � 8�GðT�� � 12g��TÞ: (6)

Substituting Eqs. (4) and (5) into Eq. (3) yields S0� ¼ 0.

From the metric (4), we find R0� ¼ 0 for arbitrary hijð ~xÞ.Hence 0�� components of the Einstein equations areautomatically satisfied. Meanwhile, i� j components ofthe Einstein equations give equations for hijð ~xÞ,

Rð3Þij ¼ 8�GSij; (7)

where Rð3Þij is the Ricci tensor calculated from the three-

dimensional metric hijð ~xÞ. At the linear order, expanding

the spatial metric as hij ¼ �ij þ �hij, the solution of

Eq. (7) can be written as �hijð ~xÞ ¼ �4GRd3ySijð ~yÞ=j ~x�

~yj, which agrees with the results in [11]. Solving Eq. (7) forhijð ~xÞ beyond the linear approximation is cumbersome.

However, the knowledge of the explicit expression for

PHYSICAL REVIEW D 78, 043532 (2008)

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hijð ~xÞ is not necessary for our argument and we skip

solving Eq. (7).The Riemann tensor R���� for the metric (4) is given by

R���0 ¼ 0, Rijk‘ ¼ Rð3Þijk‘, where R

ð3Þijk‘ is the Riemann ten-

sor for hij. In three-dimensional space, the Riemann tensor

is completely written by the Ricci tensor,

Rð3Þijk‘ ¼ �2ðhi½‘Rð3Þ

k�j þ hj½kRð3Þ‘�i � Rð3Þhi½kh‘�jÞ; (8)

where ½� � �� means the antisymmetrization with respect to

the indices in the bracket. Outside the strings, Rð3Þij vanishes

from the Einstein Eq. (7), which gives Rð3Þijk‘ ¼ 0. Therefore

the space-time is flat around the strings.We have implicitly assumed that the strings are straight

and force balanced. Here we show that these assumptionsare validated from an equation of motion for the strings anda boundary condition at the junction. Variation of Eq. (1)with respect to X�

A yields,

@

@�aA

� ffiffiffiffiffiffiffiffiffiffi��A

p�abA

@X�A

@�bA

�þ ffiffiffiffiffiffiffiffiffiffi��A

p�abA ��

��

@X�A

@�aA

@X�A

@�bA

¼ 0;

(9)

with the boundary condition

XA0�A0

ffiffiffiffiffiffiffiffiffiffiffi��A0p

naA0@X

�A0

@�aA0

��������junction¼ 0; (10)

for each junction.1 Here A0 means that we take the sum

only for strings attached to the junction we are considering.Imposing the static condition, Eq. (9) becomes

d2XiAð�1

AÞd�1

Ad�1A

þ �ijk

dXjAð�1

AÞd�1

A

dXkAð�1

AÞd�1

A

¼ 0; (11)

which is the geodesic equation in the three-dimensionalspace. Because the space is locally flat, the straight con-figuration of the strings is a solution of the equations. Theboundary condition becomes

XA0�A0

dXiA0

d�1A0

��������junction¼ 0; (12)

which is exactly the force balance condition. To maintainthe static condition, the force balance condition must besatisfied. Note that the force balance condition can also bederived from the conservation law r�T

�� ¼ 0.

The space-time is not globally flat and has conicalstructure. The deficit angle �A around the A-th string can

be written as �A ¼ 12

Rplane d

2xffiffiffiffiffiffiffiffigð2Þ

qRð2Þ, where the metric

and the Ricci scalar in the integral are defined on the two-dimensional plane perpendicular to the string. After the use

of the Einstein equations to replace Rð2Þ with the energy-momentum tensor of the string, the well-known result�A ¼ 8�G�A is obtained [11–13].We have considered the static configurations. The ex-

tension to the stationary case where all the strings movewith a constant velocity can be obtained by the boosttransformation from the static coordinate. Because all thecomponents of the Riemann tensor vanish in the staticcoordinate, they vanish in the stationary coordinate.Therefore, the geometry is also flat for the stationaryconfigurations.

III. CONCLUSION

We showed, without invoking the perturbative expan-sion, that the geometry around the stationary strings withjunctions is flat, which is a nonlinear generalization of [11].

ACKNOWLEDGMENTS

The author would like to thank Masahiro Kawasaki foruseful comments.

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1The boundary condition can be derived as follows. Forsimplicity, we assume that there is only one junction in thestring network (Extension to the case of multijunctions is trivial).We first attach a point mass of its mass m to the junction.Denoting the coordinate of the point mass as Z�, we have toadd the terms

�mZjunc

d�XA

Zjunc

d�A�ðX�A ð�AðÞÞ � Z�ðÞÞ;

to Eq. (1), where is the proper time and �AðÞ denotes theworld sheet coordinate of the junction. X

�A ð�AðÞÞ ¼ Z�ðÞ are

the constraints that the strings are attached to the point mass. �A�are the Lagrange multipliers. Because no mass point is assumedto be attached to the junction in the main text, we have to takethe limit m ! 0. By taking the limit m ! 0 in the equation ofmotion for Z�, we obtain a relation

PA�A� ¼ 0. Substitution of

this relation into the boundary conditions which are derived fromthe variation of X

�A yields Eq. (10).

TERUAKI SUYAMA PHYSICAL REVIEW D 78, 043532 (2008)

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EXACT GRAVITATIONAL LENSING BY COSMIC STRINGS . . . PHYSICAL REVIEW D 78, 043532 (2008)

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