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A vortex-line model for infinite straight cosmic strings
Volume 124, number 4,5
PHYSICS LETTERS A
28 September 1987
A VORTEX-LINE MODEL FOR INFINITE STRAIGHT COSMIC STRINGS B. LINET UnitéAssociée au CNRS No. 769, Université Pierre et Marie Curie, Inst itut Henri Poincaré, 11, rue Pierreet Marie Curie, 75231 Paris Cedex 05, France Received 12 June 1987; accepted for publication 30 July 1987 Communicated by J.P. Vigier
Within the framework of a simple scalar-gauge theory in general relativity, we consider the vortex-type solutions representing infinite straight cosmic strings. The lower bound of energy can be attained when a particular relation between the coupling constants is satisfied. In this case, we determine exactly the asymptotic metric in terms of the energy of these solutions.
Cosmic strings could be produced at a phase transition in the early universe [1,21. In order to study the gravitational field of an infinite straight cosmic string some simple models have been used [3—51. In particular the spacetime described by the metric (1) ds2 =dt2 —dz2 —dp2 —B2p2 d~ in a coordinate system (t, z, p, q) with p>~Oand 0 ~ ~ <2it, induces on the axis p = 0 a singular line source [6,7] of the Einstein equations having the following energy—momentum tensor,
uates approximately the asymptotic metric (see also ref. [9]). In this letter we start from Garfinkle’s formulas adding a possible winding number n different from 1 and we shall study the solutions when a particular relation between the coupling constants is satisfied. The static metrics with cylindrical symmetry can be written dS2 ~ B d 2 2 C d 2 —e Z —dp —e ~ where A, B and C are functions of p only satisfying
1 —B o~ T’,=T~=——, T~ =T9 =0 4 I’ ~
the following conditions on the axis p_—O, A(0)=B(0)=0, limp2eC=1.
(2)
-
—
—
where (5(2) i,/j is the Dirac distribution at the origin on the two-surface t = const and z = const, being the
The complex scalar field CP and the potential A
determinant of the induced metric. The linear mass density ~ of the cosmic string is obtained thereby [5]
the form c1 = Re~’, with neZ,
(6)
,~
9 take
p=~(l—B), with 0
(3)
We have chosen units in which G=c=h= 1. Nevertheless the only consistent way to find the metric describing an infinite straight cosmic string is to solve the coupled Einstein-scalar-gauge field equations of a simple theory, for example by considering an abelian gauge field A9 and a charged scalar field cb with a coupling constant e and a potential (4) This has been done by Garfinkle [8] and he eval240
A~=e~(P—n), A~=A~.=A~=0,
(7)
where R and P are functions of p only. We look for vortex-type solutions [10] such that the scalar field and the magnetic field are finite everywhere. To get solutions with a finite energy, the appropriate boundary conditions are R(p)’-.’~, P(p)-~0 as p—~cx. (8) The magnetic flux is quantized and it is easy to see that
0375-960l/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 124, number 4,5
PHYSICS LETFERS A
P(0)=n.
(9)
The field equations for R and Pare d2R ld + ~—~(A+B+C)~ —R[4A(R2 d2P
+
ld
-~—~-
+e~’P2] 0,
~2)
(A+B—C)
—e2R2P=0.
The non-vanishing components energy—momentum tensor are
(10) of
the
dR
28 September 1987
~~C12RP~
(14)
where ~=l for n>0 and ~=—l for n<0. Putting A = B = 0, the field equations (10) are automatically satisfied by the solutions to eqs. (14). As in the flat spacetime [121, for such solutions we can verify that — T~° =0. Therefore it is consistent to consider ~ only the gravitational potential C. Introducing the notation $2 = eC (15) the Einstein equations reduce to
T’,=T~=
[(~)2+e_CR2P2+2i~(R2
~
~2)2
—
T +
j
e_C /dP\21 (.....)
T9
+
22(R2
j
e_C/dP\21
i
r
p~0. We can establish for eqs. (14) and (16) the asymptotic behavior for large p,
(dR\2 +e~’R2P2—2t(R2
R(p)_7+Rop_i~2exp(_2~Ji.~,1p),
~j2)2
= ~ L—
P(p)’—’P
2 exp(—2\/~,7p), 0p”
e~~’ fdp\21
~
(16)
conditions ing number(6) which andsatisfies (8) andthe which prescribed is regular boundary everywhere, in particular R(0)=0 and fl~dP/dpfinite as
2) 2
[(dR) (s,)
8irfl dp2
where Tt~is given by (11). There exists surely a unique solution to eqs. (14) and (16) for each wind-
2 9
,=
j.
(11)
fl(p)~Bp+ko+floexp(_4.,J~, 7p),
(17)
Furthermore we must add the Einstein equations in order to determine A, B and C. The energy per unit of length of these solutions is
where B, k0 and R0 are arbitrary constants; we have P0 = — ~ BR0 fl~can also be calculated. However, from eq. (16) we obtain
E=JTt,~gdpd~=2~JTt,eC72dp.
2~JT1tfldP=~(_ 1 ~lim_,j. d$~ dp 0
(12)
(18)
0
As in the flat spacetime [11], for a solution with a winding number n, we can prove that 2 for e2~8)~. (13) E~~itInIi Moreover when e2=8A, the lower bound can be attained by imposing the following first-order equations
Hence the constant B can be expressed in terms of the energy per unit of length and we get B=l—4itInI,i2. (19) We point out that the curvature tensor and the energy—momentumtensor of the solutions go to zero exponentially as p-9c~.By a limit process in which A —~ (and e— such that e2 = 8A), we obtain clearly that the component T’, tends to the Dirac distribution on the two-surface t = const and z = const with -~
=4~eG22(R2~2)
241
Volume 124, number 4,5
PHYSICS LETTERS A
the factor ~ i n I ‘i~.Furthermore, it should be expected that k0 has the same magnitude as ~ ,~for large A. In this sense, we have Justified that metric (1) with source (2) describes an infinite straight cosmic string. The author would like to thank the referee who suggested several corrections.
References [1] T.W.B. Kibble, J. Phys. A 9 (1976) 183.
242
28 September 1987
[2] A. Vilenkin, Phys. Rep. 121(1985) 263. [3] J.R. Gott III, Astrophys. J. 288 (1985) 422. [4] W. Hiscock, Phys. Rev. D 31(1985) 3288. [5] B. Linet, Gen. Rel. Gray. 17 (1985) 1109. [6] D.D. Sokolov and A.A. Starobinskii, Soy. Phys. DokI. 22 (1977) 312. [7] W. Israel, Phys. Rev. D 15 (1977) 935. [8] D. Garfinkle, Phys. Rev. D 32 (1985) 1323. [9] P. Laguna-Castillo and R.A. Matzner, Phys. Rev. D 35 (1987) 2933. [10] H.B. Nielsen and P. Olesen, Nuci. Phys. B 61(1973) 45. [11] E.B. Bogomol’nyi, Soy. J. Nucl. Phys. 24 (1976) 449. [12] H.J. de Vega and F.A. Schaposnik, Phys. Rev. D 14 (1976) 1100.
PHYSICS LETTERS A
28 September 1987
A VORTEX-LINE MODEL FOR INFINITE STRAIGHT COSMIC STRINGS B. LINET UnitéAssociée au CNRS No. 769, Université Pierre et Marie Curie, Inst itut Henri Poincaré, 11, rue Pierreet Marie Curie, 75231 Paris Cedex 05, France Received 12 June 1987; accepted for publication 30 July 1987 Communicated by J.P. Vigier
Within the framework of a simple scalar-gauge theory in general relativity, we consider the vortex-type solutions representing infinite straight cosmic strings. The lower bound of energy can be attained when a particular relation between the coupling constants is satisfied. In this case, we determine exactly the asymptotic metric in terms of the energy of these solutions.
Cosmic strings could be produced at a phase transition in the early universe [1,21. In order to study the gravitational field of an infinite straight cosmic string some simple models have been used [3—51. In particular the spacetime described by the metric (1) ds2 =dt2 —dz2 —dp2 —B2p2 d~ in a coordinate system (t, z, p, q) with p>~Oand 0 ~ ~ <2it, induces on the axis p = 0 a singular line source [6,7] of the Einstein equations having the following energy—momentum tensor,
uates approximately the asymptotic metric (see also ref. [9]). In this letter we start from Garfinkle’s formulas adding a possible winding number n different from 1 and we shall study the solutions when a particular relation between the coupling constants is satisfied. The static metrics with cylindrical symmetry can be written dS2 ~ B d 2 2 C d 2 —e Z —dp —e ~ where A, B and C are functions of p only satisfying
1 —B o~ T’,=T~=——, T~ =T9 =0 4 I’ ~
the following conditions on the axis p_—O, A(0)=B(0)=0, limp2eC=1.
(2)
-
—
—
where (5(2) i,/j is the Dirac distribution at the origin on the two-surface t = const and z = const, being the
The complex scalar field CP and the potential A
determinant of the induced metric. The linear mass density ~ of the cosmic string is obtained thereby [5]
the form c1 = Re~’, with neZ,
(6)
,~
9 take
p=~(l—B), with 0
(3)
We have chosen units in which G=c=h= 1. Nevertheless the only consistent way to find the metric describing an infinite straight cosmic string is to solve the coupled Einstein-scalar-gauge field equations of a simple theory, for example by considering an abelian gauge field A9 and a charged scalar field cb with a coupling constant e and a potential (4) This has been done by Garfinkle [8] and he eval240
A~=e~(P—n), A~=A~.=A~=0,
(7)
where R and P are functions of p only. We look for vortex-type solutions [10] such that the scalar field and the magnetic field are finite everywhere. To get solutions with a finite energy, the appropriate boundary conditions are R(p)’-.’~, P(p)-~0 as p—~cx. (8) The magnetic flux is quantized and it is easy to see that
0375-960l/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 124, number 4,5
PHYSICS LETFERS A
P(0)=n.
(9)
The field equations for R and Pare d2R ld + ~—~(A+B+C)~ —R[4A(R2 d2P
+
ld
-~—~-
+e~’P2] 0,
~2)
(A+B—C)
—e2R2P=0.
The non-vanishing components energy—momentum tensor are
(10) of
the
dR
28 September 1987
~~C12RP~
(14)
where ~=l for n>0 and ~=—l for n<0. Putting A = B = 0, the field equations (10) are automatically satisfied by the solutions to eqs. (14). As in the flat spacetime [121, for such solutions we can verify that — T~° =0. Therefore it is consistent to consider ~ only the gravitational potential C. Introducing the notation $2 = eC (15) the Einstein equations reduce to
T’,=T~=
[(~)2+e_CR2P2+2i~(R2
~
~2)2
—
T +
j
e_C /dP\21 (.....)
T9
+
22(R2
j
e_C/dP\21
i
r
p~0. We can establish for eqs. (14) and (16) the asymptotic behavior for large p,
(dR\2 +e~’R2P2—2t(R2
R(p)_7+Rop_i~2exp(_2~Ji.~,1p),
~j2)2
= ~ L—
P(p)’—’P
2 exp(—2\/~,7p), 0p”
e~~’ fdp\21
~
(16)
conditions ing number(6) which andsatisfies (8) andthe which prescribed is regular boundary everywhere, in particular R(0)=0 and fl~dP/dpfinite as
2) 2
[(dR) (s,)
8irfl dp2
where Tt~is given by (11). There exists surely a unique solution to eqs. (14) and (16) for each wind-
2 9
,=
j.
(11)
fl(p)~Bp+ko+floexp(_4.,J~, 7p),
(17)
Furthermore we must add the Einstein equations in order to determine A, B and C. The energy per unit of length of these solutions is
where B, k0 and R0 are arbitrary constants; we have P0 = — ~ BR0 fl~can also be calculated. However, from eq. (16) we obtain
E=JTt,~gdpd~=2~JTt,eC72dp.
2~JT1tfldP=~(_ 1 ~lim_,j. d$~ dp 0
(12)
(18)
0
As in the flat spacetime [11], for a solution with a winding number n, we can prove that 2 for e2~8)~. (13) E~~itInIi Moreover when e2=8A, the lower bound can be attained by imposing the following first-order equations
Hence the constant B can be expressed in terms of the energy per unit of length and we get B=l—4itInI,i2. (19) We point out that the curvature tensor and the energy—momentumtensor of the solutions go to zero exponentially as p-9c~.By a limit process in which A —~ (and e— such that e2 = 8A), we obtain clearly that the component T’, tends to the Dirac distribution on the two-surface t = const and z = const with -~
=4~eG22(R2~2)
241
Volume 124, number 4,5
PHYSICS LETTERS A
the factor ~ i n I ‘i~.Furthermore, it should be expected that k0 has the same magnitude as ~ ,~for large A. In this sense, we have Justified that metric (1) with source (2) describes an infinite straight cosmic string. The author would like to thank the referee who suggested several corrections.
References [1] T.W.B. Kibble, J. Phys. A 9 (1976) 183.
242
28 September 1987
[2] A. Vilenkin, Phys. Rep. 121(1985) 263. [3] J.R. Gott III, Astrophys. J. 288 (1985) 422. [4] W. Hiscock, Phys. Rev. D 31(1985) 3288. [5] B. Linet, Gen. Rel. Gray. 17 (1985) 1109. [6] D.D. Sokolov and A.A. Starobinskii, Soy. Phys. DokI. 22 (1977) 312. [7] W. Israel, Phys. Rev. D 15 (1977) 935. [8] D. Garfinkle, Phys. Rev. D 32 (1985) 1323. [9] P. Laguna-Castillo and R.A. Matzner, Phys. Rev. D 35 (1987) 2933. [10] H.B. Nielsen and P. Olesen, Nuci. Phys. B 61(1973) 45. [11] E.B. Bogomol’nyi, Soy. J. Nucl. Phys. 24 (1976) 449. [12] H.J. de Vega and F.A. Schaposnik, Phys. Rev. D 14 (1976) 1100.