PHYSICAL REVIEW D 15 MAY 1998VOLUME 57, NUMBER 10

Local cosmic string in generalized scalar tensor theory

A. A. Sen* and N. Banerjee†

Relativity and Cosmology Research Centre, Department of Physics, Jadavpur University, Calcutta 700032, India~Received 11 November 1997; published 10 April 1998!

A recent investigation shows that a local gauge string with a phenomenological energy momentum tensor, asprescribed by Vilenkin, is inconsistent in Brans-Dicke theory. In this work it is shown that such a string isconsistent in a more general scalar tensor theory wherev is a function of the scalar field. A set of solutions offull nonlinear Einstein equations for the interior region of such a string is presented.@S0556-2821~98!03910-1#

PACS number~s!: 11.27.1d, 04.20.Jb, 04.50.1h

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In a very recent communication@1#, the present authorhave shown that an infinitely long straight static local gaustring, given by the energy momentum tensor componeTt

t5TzzÞ0 and all otherTn

m50 @2#, is inconsistent in BransDicke ~BD! theory of gravity. Because of the relevencethe BD-type scalar field in the context of a cosmic string~seeGundlach and Ortiz@3# or Romero and Barros@4# for de-tailed discussions!, it becomes necessary to investigawhether a local gauge string can give rise to consistent stions of the gravitational field equations for a more genescalar tensor theory. In this paper, we show that such a sis indeed consistent with Nordtvedt’s generalized scalarsor theory@5#.

The gravitational field equations in this theory are givby

Gmn5Tmn

f1

v~f!

f2 S f ,mf ,n21

2gmnf ,af ,aD

11

f~f ,m;n2gmnhf!, ~1!

where the dimensionless parameterv is now a function ofthe scalar fieldf. The wave equation for the scalar fieldf is

hf51

~2v13!FT2f ,af ,a

dv

dfG . ~2!

In these equations,Tmn represents the energy momentutensor components for all the fields except the scalar fielfandT is the trace ofTmn . Brans-Dicke theory is evidently aspecial case of this theory whenv is constant. The conservation of matter is represented by the equation

T;nmn50. ~3!

It should be noted, however, that Eqs.~2! and ~3! are notindependent, as in view of Eq.~1! and the Bianchi identity,one yields the other.

The general static cylindrically symmetric metric

ds25e2~K2U !~2dt21dr2!1e2Udz21e22UW2du2 ~4!

*Email address: anjan@juphys.ernet.in†Email address: narayan@ juphys.ernet.in

570556-2821/98/57~10!/6558~3!/$15.00

ets

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is taken to describe the spacetime given by an infinitely lostatic local string with the axis of symmetry being thez axis.K,U,W are all functions of the radial coordinater alone. Thelocal gauge string is characterized by an energy density astress along the symmetry axis given by

Ttt5Tz

z52s~r !, ~5!

and all other components are zero@2#. The field equationscan be written as

2W9

W1

K8W8

W2U825

se2~K2U !

f1

v

2

f82

f2 2~K82U8!f8

f

1S f9

f1

W8f8

Wf D , ~6!

K8W8

W2U825

v

2

f82

f2 2~K82U8!f8

f2

W8f8

Wf, ~7!

K91U8252v

2

f82

f2 2f9

f2

U8f8

f, ~8!

2W9

W2U8212U912

U8W8

W2K9

5se2~K2U !

f1

v

2

f82

f2 2U8f8

f1S f9

f1

W8f8

Wf D ,

~9!

where a prime represents differentiation with respect tor .The wave equation for the scalar field now looks like

f91f8W8

W52

2se2~K2U !

~2v13!2

f82

~2v13!

dv

df. ~10!

From the conservation equation~3! one can write

K8s50. ~11!

For a nontrivial existence of the cosmic string,sÞ0, andhence

K850, ~12!

for the interior of the string. SoK is a constant and in whafollows we shall takee2K51 which only leads to a rescalin

6558 © 1998 The American Physical Society

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q

e

r

ve

la

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57 6559BRIEF REPORTS

of the coordinates and no loss of generality. WithK850,Eqs.~7! and ~8! now combine to yield the equation

f9

f1

W8f8

Wf50. ~13!

In case of BD theory,dv/df 50 and thus Eqs.~13! and~10! together yields50, indicating the nonexistence of thstring @1#. For a varyingv theory, however,s does not haveto vanish as evident from Eq.~10!. For f8Þ0, Eq. ~13!readily integrates to yield

f85a

W, ~14!

a being a constant of integration and should be nonzerohave a nontrivial scalar field. In view of Eqs.~12! and ~13!,the field equations~6! and ~9! combine to form

U91U8W8

W1

U8f8

f50. ~15!

This equation has a first integral

U8Wf5a1 , ~16!

whena1 is an arbitary constant. This equation along with E~14! yields

U85bf8

f, ~17!

whereb5a1 /a.In what follows, we shall try to find exact solutions for th

interior spacetime metric~i.e., for sÞ0).Interior solution.As a consequence of Eqs.~12! and~13!,

Eqs. ~7! and ~8! become identical and we are left with fouindependent equations:

s521

2~f8fb!2

dv

df, ~18a!

2f9

f5S D1

v

2 Df82

f2 , ~18b!

f85a

W, ~18c!

U85bf8

f, ~178!

where D5b21b, a constant. Equation~18a! follows fromEqs. ~10! and ~13! whereas Eq.~18b! follows from Eq. ~8!.Now one has four equations and five unknowns to be solfrom them. But in the generalized scalar tensor theoriesv isa function of f and thus for a particular choice ofv5v(f), the system of equations can be solved.

A number of different choices ofv as a function off arealready available in the literature, depending on particuphysical interests. Barkar’s choice ofv @6#, given by

to

.

d

r

v5423f

2~f21!, ~19!

will be used for further analysis in the present work. Tphysical motivation for the choice of Barkar is thatG, theNewtonian constant of gravitation, remains a constant incase in spite of the nonminimal coupling between the scfield and geometry.

From Eq.~19! one obtains

dv

df52

1

2

1

~f21!2 ~20!

and Eq.~18a! yields

s51

4F f8fb

~f21!G2

, ~21!

which evidently ensures thats is positive. It deserves mention that for some other choices ofv, already available in theliterature,s turns out to be negative. One such choice isv5 3f/2(12f) which is called the model with curvaturcoupling ~see Van den Bergh@7# and references therein!.Obviously these theories do not incorporate a cosmic stof this type.

In what follows, we will try to solve the system of equations in Barkar’s theory for two choices of the constant,bandD, namely,~1! b50, D50 and~2! D51.

Case 1: D50, b50. In this case, from Eq.~17!, one canfind U850 — i.e., U is a constant — and one can chooU50 ~i.e., e2U51) without any loss of generality, by asimple rescaling of the coordinates. Physically this choallows a Lorentz boost along the symmetry axis of the stri

Equation~21! yields

s51

4

f82

~f21!2 . ~22!

With the help of Eq.~19!, Eq. ~18b! can be written as

f9

f85F 1

f2

1

4~f21!Gf8,

which readily yields a first integral

ln f85 ln f0@f/~f21!1/4#, ~23!

wheref0 is a constant of integration. A series solutionthis equation is possible, expressingr as a power series off,which, however, is not invertible to expressf5f(r ). But ass,v,W are known functions off and its derivatives, thecomplete solution can be obtained in principle.

Case 2: D51. For this case Eq.~18b! together with Eq.~19! yields a solution forf in closed form:

f511~mr1n!4/5, ~24!

wherem andn are arbitary integration constants. Equatio~18c! and~17! yield the solutions forW andU, respectively,as

W25W02~mr1n!2/5 ~25!

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6560 57BRIEF REPORTS

and

e2U5f2b5@11~mr1n!4/5#2b. ~26!

HereW055a/4m.The complete solution for the metric for the interior of th

string is then

ds25@11~mr1n!4/5#22b@2dt21dr21W02~mr1n!2/5du2#

1@11~mr1n!4/5#2bdz2, ~27!

and the string energy densitys(r ) is given by

s54m2@11~mr1n!4/5#2b

25~mr1n!2 . ~28!

As a conclusion, one can say that although the solutiobtained in the present work are by no means general onexplicitly exhibits a consistent set of interior solutions of tnonlinear Einstein’s equations for a local gauge string i

s, it

a

varyingv theory. So although a local cosmic string is incosistent in Brans-Dicke theory, it is quite consistent in a mogeneral scalar tensor theory of gravity.

It deserves mention that in a recent work by Guimara˜es@8#, the solutions of Einstein’s equations for a gauge strhave been presented in a weak field approximation offield equations in a similarv-varying scalar tensor theoryGuimaraes’ work does not consider the phenomenologiexpression forTn

m , i.e., T005Tz

zÞ0, and all otherTnm50 as

prescribed by Vilenkin, but rather has all the diagonal coponents ofTn

m to be nonvanishing. This type of a cosmstring is consistent in BD theory as well, even in the funonlinear version of the theory, as shown by the presauthors@1#.

The authors are grateful to Professor A. Banerjee forgenerous help and suggestions. One of the authers~A.A.S.!is grateful to the University Grants Commission, India, ffinancial support.

@1# A. A. Sen, N. Banerjee, and A. Banerjee, Phys. Rev. D56,3706 ~1997!.

@2# A. Vilenkin, Phys. Rev. D23, 852 ~1981!.@3# C. Gundlach and M. E. Ortiz, Phys. Rev. D42, 2521~1990!.@4# A. Barros and C. Romero, J. Math. Phys.36, 5800~1995!.

@5# K. Nordvedt, Jr., Astrophys. J.161, 1059~1970!.@6# B. M. Barkar, Astrophys. J.219, 5 ~1978!.@7# N. Van den Bergh, Gen. Relativ. Gravit.14, 17 ~1982!.@8# M. E. X. Guimaraes, Class. Quantum Grav.14, 435 ~1997!.