Patricio S. Letelier
Departamento de Matematica Aplicada-IMIiCC Universidade Estadual de Campinas
13081 Campinas, S.P.. Brazil
We find that the strings solutions is not plane fronted waves and the interaction between vortex.
existence of either vortices or cosmic, affected by the presence
óf
gravitationalthat curvature singularities appear due to the wave and either the string or the
The metric associated to a finite number of parallel cosmic strings and its generalization for a continuum of parallel cosmic strings was found by the author without making reference to Its field theory origin! Since cosmic strings are produced by symmetry breaking in early stages of the evolution of the Universe a c01nsisLent way to define cosmic strings is to consider the Einstein equations coupled to the Yang-Mills-Higgs field equations! A solution to the previous equations that can be interpreted as a finite number of parallel vortex lines, or a finite number of parallel cosmic strings was considered by Linet3 A similár solution was studied by Comtet and Gibbons , together with solutions to the Einstein equations coupled with a-model type of field theories. The existence of the above mentioned solutions, as well as the multiple vortex solutions, relays on the fact that
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in a particular curve spacetime, albeit sufficiently general to contain the cosmic strings, the 13ogomol'nyi equations 5 obtained in the Bognmol'nyi limit are essentially the same equations that in Aiiiikowski spacetlrne 3 ' 4 .
The purpose of this communication is to study the Einstein equations coupled with either an Abelian gauge field interacting with a charged scalar field In presence of the usual symmetry breaking potential or a nonlinear Q.-model type of field equations for the metric
ds 2'. Ildu 2+Zdudv + 2Adudx +213dudy - e-4V
(dx 2+ dy 2 ), (11
where II, A and B, are functions of u, x, and y; V is a function of x, and y only. In particular, we shall be interested in the solutions that can be interpreted as cosmic strings. In Refs. 3 and 4 the existence of cosmic strings solutions were studied for the special case of a spacetime (1) with II=A=B=O. We shall consider the case in which the functions A, B, and 11 arc restricted by
A - 13 = 0, A + B = 0, 11 + H O. (2)
.Y .x ,x .Y ,xx .YY
When V =0 the metric (l) with the restrictions (2) represents a plane fronted wave (' with a constant wave vector O. The metric (I) is a particular case of the general metric that admits a null vector with zero covariant derivative?
The Einstein tensor for the metric (1) with the restrictions (21 can be cast as
Glrv'-2e4Y(V.xx+ V.
YY)(ipko+ I
Vkp), (3)
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where k=(II/2)a L +SÚ+A SÚ +136', 1 u =S co=e: Sú, lt nd a n cu Z f II á'
is and orthonormal vicrbcin.
The Lagrangean for the U(1) gauge field that we shall consider Is the the covariant generalization of the Giuzgurg-Landau model,
L=-(l/4)I41tf jcv+(1/2)(81c " -leA ^^)(8le +ieAN..) -f1(I.I 2- (d 712)2, (4) where Fib,= O A- a^Ali; e, A, and y are three coupling constants.
Assuming that A li= (0,0,A 1 ), and that A i s(Ax ,Ay ), and coagp i +iwi are functions of x 1= (x,y) only, we get
L =-(1/4)) 7Jm
FIJFinn-
(1/2)7tJD^iPnDJI;^ a(1^12- 712)21 (5) where 71J= eov 6 1J, and Dt'IbE 819c CC
libA1vb, with c ty -c 21=I. and C = C =0.
11 22
When the coupling constant are related by e 2=8a and the fields by F1 = e 71
1J(I l I 2- 712)12, DJ^pe=
lIJkCsb?
kl Dl
Pb, (6)
where Jk = a-wc, the Lagrangean (5) is a total divergence and In consequence the solutions. of the first order equations (6) (I3ugoenul'nyi equations) are solutions of the second order Euler-Lagrange field equations derived from (5). Furthermore, by direct substitution one can verify that T 11= 0. In this case we can cast the EMT associated to (5) as e
'l ipar=-L( Ipkv+ I^k
l1), (7)
with -L . =1I/4)e ina1Ja la
.1(-iei2+ *I2lni.i2). Defining the orthonormal vectors 7/20' = kN+ 111 V2z1'= 01- i . .we can put (7) In
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the form of the EMT that represents a cloud of strings,
= p(lNty-zpzv), (8)
with p=-L. When -U0. p represents the density of the cloud. For multiple vortex solutions p is a distribution with support on straight lhms. From (3). and (7) we have that the Einstein equations reduce to the Laplace equation and can be explicitly integrated. Moreover, one can show for the field equations (6) the existence of solutions for the boundary conditions that define one or several vortICe sx,4.9
Nuw we shall consider a a-model with target metric on a Kehler manifold. Let 04(x') a map from S into a 2n-dimensional Kabier manifold M with metric GAD
( )
and complex structure Ja
i.),A=1,? n,. The Lagrangean for this 2-dimensional model is
L=-(1/2)p 2Gaa 8010" elOB 7t1. (9)
The quantity +• is another coupling constant. When the fields are related b 0#A
a JA i, 0 Oc, the Lagrangean is a topological invariant and in consequence the Euler-Lagrange equations associated to (9) are identically satisfied. Again, one can show by direct substit u Lions that Ttl-Q. Thus, when the field are holomorphic the
uir
for the u-model (9) can be cast as (7), i.e., as a cloudor
cosmic strings.Since in the interaction of cosmic strings with plane fronted gravitational waves the spacetlme can develop nontrivial curvature singularities'° we have that in the cosmic string limit the vortex
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and the liolomorfic o'-mode( solutions will present the same singular behavior. In other words we have proved that the singularities studied In Rcf.10 have a physical origin.
REFERENCES
1. P.S. Letelier, Class. Quantum Gray. 4, L75 (1987).
2. For a review see A. Vilenkin, Phys. Rep. 121, 263 (1985).
3. D. Linct, Gen. Re1.Gray. 20, 451 (1988).
4. A. Comtet and G.W. Gibbons, Nucl. Phys. D 299. 719 (19891.
5. E.D. Dogonnoi'nyl, Soy. Journ. Nucl. Plays. 24, 449 (1977).
6. W, Kundt, Z. Phys, 163. 77 (19611.
7. See for instance, P.S. Lowlier Gen. Rel. Gray. 11, 367 (1979) and references therein.
8. N.S. Letelier, Class. Quantum Gray. 8, L137-L140 (1991).
9. C.11. Taubes, Comm. Math. Phys. 72, 277 (1980).
10. P.S. Letelier. Phys. Rev. Lett. 66. 268 (1991).
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