Static cosmic strings in Brans-Dicke theory

A. A. Sen,* N. Banerjee, and A. BanerjeeRelativity and Cosmology Research Centre, Department of Physics, Jadavpur University, Calcutta 700032, India

~Received 9 January 1997; revised manuscript received 31 March 1997!

The gravitational fields of both local and global static cosmic strings in the context of Brans-Dicke theoryare investigated. It is shown that, for a local cosmic string, Vilenkin’s prescription for the energy-momentumtensor due to the string is inconsistent with the Brans-Dicke theory. For the global string, two kinds ofsolutions for the spacetime outside the string core have been presented, one of which is given in a closed formand may be nonsingular at a finite distance from the string core while the other one cannot be written in aclosed form and moreover contains a singularity at a finite distance from the string core. The motions ofphotons and timelike particles in the field of the global string are also investigated in one of the cases. It hasbeen shown that such a global cosmic string has only a repulsive gravitational effect on a freely movingparticle.@S0556-2821~97!04418-4#

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

I. INTRODUCTION

It was shown by La and Steinhardt@1# that because of theinteraction of the Brans-Dicke~BD! scalar field and theHiggs-type sector, which undergoes a strongly first-orderphase transition, the exponential inflation as in the Guthmodel @2# could be slowed down to power law one. The‘‘graceful exit’’ in inflation is thus resolved as the phasetransition completes via bubble nucleation. At the same time,it is widely accepted that topological defects such as domainwalls, cosmic strings, and monopoles are formed duringinflation and hence must interact with the BD scalar field aswell as the gravitational field produced by them. Cosmicstrings are particularly interesting in view of their capabilityto produce direct observational effects such as gravitationallensing @3# and also because they are possible seeds ofgalaxy formation @4#, which still remains an outstandingproblem in cosmology. Strings are said to be local orglobal depending on their origin from the breakdown oflocal or global U~1! symmetry. While local strings@4–6#are well behaved, having an exterior representing aflat Minkowskian spacetime with a conical defect, globalstrings, however, have strong effects at large distances.The static global string spacetime is found to have asingularity at a finite distance from the core@7–12#. Analyti-cal solutions for the spacetime outside the core were givenby Harari and Sikivie@7# in the linearized approximation,while these are obtained in the full nonlinear theory byCohen and Kaplan@8# and also by Banerjee, Banerjee, andSen@12#.

Owing to the renewed interest in the Brans-Dicke theoryof gravity for the reasons mentioned earlier, it is worthwhileto examine the cosmic strings, both local and global, in thistheory. Very recently Gundlach and Ortiz@13# obtained ana-lytical solutions for a local gauge string in BD theory. Thesesolutions are not quite satisfactory because here the BD

theory is used to solve for the exterior, while in the interiorthe solutions are given for Einstein’s field equations ignoringthe BD scalar field completely. The other solutions in thistheory for a local gauge string have been given by Barrosand Romero@14#, who made the linearized approximationsfor the field equations.

In what follows we present results for the static strings inthe Brans-Dicke theory of gravitation. In Sec. II it is shownthat for a static local gauge string, Vilenkin’s prescription forthe energy momentum tensor due to the stringTr

r5Tuu50

and Ttt5Tz

zÞ0 are indeed inconsistent with the BD theory.The question obviously remains whether a similar situationwill occur in the case of a global string also. The scenariochanges completely in this case and in Sec. III we are reallyable to present two classes of solutions for the spacetimeoutside the core of static global string in the BD theory. Oneof these classes is given in a closed form and may be non-singular depending on the choice of the constant of integra-tion. The other class cannot be written in a closed form andmoreover contains a singularity at finite distance away fromthe string. The solutions we have obtained in the presentpaper are not most general. They do not yield, in the limit,the corresponding global string solutions in Einstein’stheory. But nevertheless the solutions presented here are per-haps the exact analytical solutions obtained for the first time.

In Sec. IV the motions of photons and timelike particlesare discussed. It is interesting to note that like the globalcosmic string in Einstein’s theory, the global string in the BDtheory also has a repulsive effect on a freely moving particlenear the string.

Lastly we must stress that unlike in Einstein’s theory it ispossible to construct a nonsingular solution for a static globalstring in the BD theory.

II. LOCAL COSMIC STRING IN BRANS-DICKE THEORY

The gravitational field equations in Brans-Dicke theoryare given by*Electronic address: anjan@juphys.ernet.in

PHYSICAL REVIEW D 15 SEPTEMBER 1997VOLUME 56, NUMBER 6

560556-2821/97/56~6!/3706~5!/$10.00 3706 © 1997 The American Physical Society

Gmn5Tmn

f1

v

f2 S f ,mf ,n21

2gmnf ,af ,aD

11

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

in units where 8pG5c51. Tmn is the energy momentumtensor representing the contribution from any field except theBrans-Dicke scalar fieldf andv is a constant parameter. Todescribe the spacetime geometry due to an infinitely longstatic cosmic string, the line element is taken to be the gen-eral static cylindrically symmetric one given by

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

whereK, U, W are all functions of the radial coordinateralone. For a local gauge string, which is infinitely long butinfinitesimally thin, the energy momentum tensor compo-nents are taken to be@3#

Ttt5Tz

z52s, Trr5Tu

u50, ~3!

s being the energy density of the string. With Eqs.~2! and~3!, the nontrivial components from Eq.~1! become

2W9

W1

K8W8

W2U82

5s

fe2~K2U !1

v

2f2 f822~K82U8!f8

f

1f9

f1

W8f8

Wf, ~4!

K8W8

W2U825

v

2f2 f822~K82U8!f8

f2

W8f8

Wf, ~5!

K91U8252v

2f2 f822U8f8

f2

f9

f, ~6!

2W9

W12U912U8

W8

W2K92U82

5s

fe2~K2U !1

v

2f2 f821f9

f1

W8f8

Wf2U8

f8

f. ~7!

The wave equation for the scalar fieldf follows from thevariation with respect tof and is given by

f91W8

Wf852

2se2~K2U !

~2v13!. ~8!

A prime represents differentiation with respect to the radialcoordinater . The relationT;n

mn50 follows from Eqs.~4!–~8!,which in turn yield a relation like

K8s50. ~9!

In the presence of the string field,s is necessarily nonzero,and thus one ought to have

K5const. ~10!

If one adds Eqs.~5! and ~6! together and use Eq.~10!,

f9

f1

W8f8

Wf50 ~11!

is obtained. In view of the wave equation~8!, this yields theresult

s

~2v13!50. ~12!

This indicates thats50, i.e., the string energy vanishes. Ifone has a nontrivial string, i.e.,sÞ0 in Brans-Dicke theory,one must have an infinitely large value of the parametervsuch that the left-hand side of Eq.~12! goes to zero even fornonzeros. But it is quite well known that forv→`, BDtheory is indistinguishable from general relativity~GR!.Thus the obvious conclusion is that a straight thin gaugestring, treated previously by Vilenkin in GR, is incompatiblewith the Brans-Dicke theory.

III. GLOBAL COSMIC STRINGIN BRANS-DICKE THEORY

For a global cosmic string, the energy-momentum tensorcomponents are calculated from the action density for a com-plex scalar fieldc along with a Mexican hat potential:

L51

2gmnc ,m* c ,n2

l

4~c* c2v2!, ~13!

wherel andv are constants andd5(vAl)21 is a measureof the core radius of the string. It has been shown that thefield configuration can be chosen as

c~r !5v f ~r !eiu ~14!

in cylindrical polar coordinates without any loss of general-ity ~for a discussion, see@9#!. In what follows, we shall lookfor the solutions of the global string in BD theory outside thecore radius of the string. The line element is taken as thegeneral static cylindrically symmetric one given by Eq.~2!.The usual boundary condition onf (r ) is f (0)50 andf (r )→1 as r→d. For our purpose, then, it is a good ap-proximation to take

f ~r !51, f 8~r !50. ~15!

The nonzero components of the energy momentum tensornow become@15#

Ttt5Tr

r5Tzz52Tu

u52v2

2

e2U

W2 . ~16!

The gravitational field equations~1! for a global string in BDtheory look like

56 3707STATIC COSMIC STRINGS IN BRANS-DICKE THEORY

2W9

W1

K8W8

W2U825

v2

2W2fe2K1

v

2f2 f82

2~K82U8!f8

f1

f9

f1

W8f8

Wf,

~17!

K8W8

W2U8252

v2

2W2fe2K1

v

2f2 f822~K82U8!f8

f

2W8f8

Wf, ~18!

K91U825v2

2W2fe2K2

v

2f2 f822U8f8

f2

f9

f, ~19!

2W9

W12U912U8

W8

W2K92U82

5v2

2W2fe2K1

v

2f2 f821f9

f1

W8f8

Wf2U8

f8

f.

~20!

The wave equation for the BD scalar field is

f91W8

Wf852

v2

~2v13!

e2K

W2 . ~21!

The Euler-Lagrange equation for the complex scalar fieldcis now irrelevant because with the choicef 8(r )50 it be-comes an identity. By subtracting Eq.~20! from Eq.~17! oneobtains

~K922U9!1~K822U8!S W8

W1

f8

f D50, ~22!

which yields either

K852U8 ~23!

or

K822U85b

Wf, ~24!

b being constant of integration. Equation~23! yields, on in-tegration, the result

K52U, ~25!

where a constant of integration has been put equal to zero,which can be done without any loss generality. In the subse-quent discussion, we shall treat this case and not the onegiven by Eq.~24!. It should be noted that with Eq.~25! thespacetime admits a Lorentz boost along the symmetry axis ofthe string.

Now by using Eqs.~18!, ~19!, and~25!, one can write

~K8Wf!81~Wf8!850. ~26!

This readily integrates to

K852f8

f1

a1

Wf, ~27!

wherea1 is a constant of integration. In what follows, weseek a solution in the special case wherea150: i.e.,

eK5a2

f, ~28!

a2 being another constant of integration. By subtracting Eq.~18! from Eq. ~17! and using Eq.~21!, one obtains

2W9

W52AS f9

f1

W8f8

Wf D1W8f8

Wf, ~29!

whereA52v12. With the help of Eqs.~26!, ~28!, and~29!,one can obtain, after a long but straightforward calculation,the first integral

AW~e2K!85W8e2K1a3 , ~30!

a3 being a constant of integration. For the special case wherea350, this equation can be easily integrated to yield

e2K5a4W1/A, ~31!

wherea4 is another constant of integration. Using Eqs.~31!and ~28! in Eq. ~21! one can write

f2A12f91Af2A11f821v2

A1150, ~32!

where the constantsa2 ,a4 are chosen to be unity. This, how-ever, does not lead to any loss of generality. The equation~32! yields a first integral of the form

f851

fA11/2 F 2v2

A111a5f G1/2

, ~33!

a5 being a constant of integration. We shall integrate thisequation in two cases, fora550 and fora5Þ0.

Case„i…: a550

In this case, one has a very simple solution forf:

f5B1/m~r 2r 0!1/m, ~34!

whereB5 12 v(4v17)A2/(2v13), andm5(4v17)/2, and

r 0 is a constant of integration.In view of Eqs.~25!, ~28!, ~31!, and~34!, the metric can

now be written as

ds25B21/m~r 2r 0!21/m~2dt21dr21dz2!

1B~2A11!/m~r 2r 0!~2A11!/mdu2. ~35!

For r 050, from the metric itself, it is quite apparent thatthere is no singularity at any finite distance from the core ofthe string. This is unlike the situation for a global string ingeneral relativity. One can also calculate the energy densityfor the global string:

3708 56A. A. SEN, N. BANERJEE, AND A. BANERJEE

Ttt52

v2

2B2~2A11!/mr 2~2A11!/m for r 050. ~36!

One can see from the above equation thatTtt→` only for

r→0, if one assumesv.2(3/2). As the proper distanceRis given byR5*eUdr5*B21/2mr 21/2mdr, it is readily seenthat r→0 implies R→0, and hence it will be illogical toconsider this singularity as the spacetime considered here isoutside the core.

For r 0.0, one has a singularity at a finite distance fromthe axis atr 5r 0 . This is apparent from the metric~35! andis confirmed by the expressionTt

t52(v2/2)B2(2A11)/m(r2r 0)2(2A11)/m.

For r 0,0 again there is no singularity, either atr 50 or atany other value ofr . Furthermore, it is interesting to notethat for v→`, m→`, and from Eq.~34! one obtainsf850, f→1, a constant. But in this limitB→` and the metricbecomes

ds252dt21dr21dz21B2r 2du2

whereB→`. Thus forv→` limit, althoughf becomes aconstant, the metric cannot be properly reduced to the corre-sponding GR case.

Case„ii …: a5Þ0

In this case there is no simple solution forf. A seriessolution can be obtained ifA11/252v15/2 is assumed tobe an integer. Equation~33! can be written in the form

aE X21/2~X2b!ndX5E dr, ~37!

where

X5b1a5f,

b52v2/~A11!,

a5a52~n11! ,

n5A11/2.

If n is a positive integer, (X2b)n can be written in the form

~X2b!n5(l 50

nn!

l ! ~n2 l !!Xn2 l~2b! l . ~38!

With this expression, Eq.~37! can be integrated to yield

aX1/2(l 50

nn!

l ! ~n2 l !! S 1

n2 l 11/2D ~2b! lXn2 l5r , ~39!

i.e.,

a~b1a5f!1/2(l 50

nn!

l ! ~n2 l !!

2~2b! l

2n22l 11~b1a5f!n2 l5r .

~40!

So along with Eqs.~25!, ~28!, and~31! this equation gives acomplete set of solutions for the metric. It is, however, not

possible to invert this equation to writef as a function ofr ,and thus further analytical study of the spacetime given bythis metric is not possible. But the important thing, apparentfrom Eq. ~40! itself, is thatf50 at a finite value ofr andthus in view of Eq.~28!, it is clear that the spacetime has asingularity at a finite distance from the string core. This fea-ture is the same as that in the case of a global string in GR@7#.

IV. MOTION OF PARTICLES AROUNDA GLOBAL STRING IN BD THEORY

We shall use the metric given by Eq.~35! in the casea550 andr 050 to investigate the motion of particles around aglobal string in thez5const plane. The geodesic equation forthis metric~with z5const! are given by

r 21

2mrr 22

1

2mrt22

2A11

2mB~2A12!/mr ~2A12!/m21u250,

~41!

t21

rmr t50, ~42!

u1~2A11!

rmr u50, ~43!

along with

B21/mr 21/m~2 t21 r 2!1B~2A11!/mr ~2A11!/mu250 ~44!

for photons and

B21/mr 21/m~2 t21 r 2!1B~2A11!/mr ~2A11!/mu2521~45!

for timelike particles. In this section, an overdot representsdifferentiation with respect to some affine parameterl. In thelatter case, i.e., for Eq.~45! l is chosen to be equal tos sothat we have an equation instead of an inequality. Equations~42! and ~43! readily integrate to yield

t5ar 1/m, ~46!

u5br 2~2A11!/m, ~47!

respectively, wherea andb are constants of integration. Us-ing these expressions fort and u in Eq. ~44!, one obtains,after a long but straightforward calculation, the result

S dr

duD 2

5r 2

u25r 4~A11!/m@ l 22Q~r !# ~48!

for photons. Here l 5a/b, a constant, and Q(r )5B2(A11)/mr 22(A11)/m.

In view of Eq. ~48!, the inequality

l 2>Q~r ! ~49!

should always be satisfied so that (dr/du)2, a perfect square,always remains positive. Forv.23/2, Q(r ) is a monotoni-cally decreasing function ofr . So the outgoing photons, once

56 3709STATIC COSMIC STRINGS IN BRANS-DICKE THEORY

emitted will travel up to infinity. The incoming photons,however, will face a barrier atr 5r 0 for which l 25Q(r 0),and will not be able to penetrate any further.

For timelike particles, using Eqs.~46! and ~47! in Eq.~45!, one obtains after a similar calculation, the equation

S dr

duD 2

5r 2

u25r 4~A11!/m$ l 22@Q~r !1P~r !#%, ~50!

whereP(r )5(B1/m/b2)r 21/m.With similar arguments as in the case of photons, one can

conclude that the outgoing timelike particles will be able toescape to infinity, whereas an incoming particle will be re-pelled at some distance from the core. As bothP(r ) andQ(r ) are independent ofu, this analysis holds irrespective ofthe angle of emission or incidence of the particles. In theabove analysis, we have assumedv.23/2. If, however,27/4,v,23/2, the result could have been different. Butnormallyv is taken to be positive, and moreover, ifv is lessthan23/2, it should be less than22 so as to makeG, theNewtonian constant of gravitation, positive@16#. So we donot include that range of values forv, i.e., 27/4,v,23/2.

The study of the motion of particles reveals that the globalstring in BD theory, given by the metric~35!, has a repulsiveeffect. This can also be shown by calculating the radial ac-celerationv1 of a particle that remains stationary~i.e., v1

5v25v350! in the field of the string. Nowv15v ;01 v0

5v0G001 v052(1/2m)B1/mr (12m)/m,0. So the particle has to

accelerate towards the string, which implies that the gravita-tional force due to the string itself is repulsive. It is similar tothe case of a global string in GR, which also gives rise to arepulsive effect at a certain distance from the core@7#.

V. DISCUSSION

We have shown that in the context of the full nonlinearBrans-Dicke theory of gravity, an infinitely long straightstatic local string, given by thead hoc energy momentumtensor componentsTt

t5Tzz52s and Tr

r5Tuu50, does not

exist. Gundlach and Ortiz@13# obtained solutions for such astring, but they assumed that the interior was dominated bythe string energy and neglected the contribution from the BD

scalar field to the energy momentum tensor completely andmatched this solution to an exterior BD solution. As theyworked in a conformally transformed version of BD theory,where the nonminimal coupling of the scalar field with thegeometry was broken and the contribution of the scalar fieldwould manifest itself only in the energy momentum tensorcomponents, they effectively obtained the interior solution inGR. For this reason they failed to detect the inconsistency inthe field equations. Barros and Romero@14#, however,solved the field equations for the interior along with the BDscalar field, but their work was in the linearized approxima-tion of the field equations. Without these approximations thelocal gauge string does not exist in BD theory, which hasbeen shown explicitly above.

A global static cosmic string, however, is quite consistentin Brans-Dicke theory and we obtained the exact solutionsfor the spacetime metric in some special cases. One of thesesolutions, represented by Eq.~35! exhibits no singularity at afinite distance from the string axis forr 050, or for r 0,0.These examples are important as in GR all the solutions fora global string have singularities at finite values ofr ~seeRef. @12# for a discussion!. The other solution, which is cer-tainly a bit more general one, exhibits a singularity at a finitedistance from the axis. The solutions presented here are,however, not the most general ones, as some of the arbitraryconstants of integrations are chosen to be equal to zero orunity, etc. Some of the choices, such as the constant of inte-gration in Eq.~25!, which has been put equal to zero, ora25a451 in Eq. ~32!, lead only to a rescaling of the coor-dinates and do not affect the nature of solutions. Fora1Þ0@Eq. ~28!# or a3Þ0 @Eq. ~31!#, however, the solutions willindeed be different and more general. We had to choosea150 anda350 so that the equations could be integrated ana-lytically. The physical meaning of these choices is not quiteapparent, but these are valid assumptions asa1 and a3 arearbitary integration constants. The solutions obtained hereare important as they are perhaps the first exact analyticalsolutions for a global string spacetime in BD theory of grav-ity.

ACKNOWLEDGMENTS

One of the authors~A.A.S.! is grateful to UniversityGrants Commission, India, for financial support.

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3710 56A. A. SEN, N. BANERJEE, AND A. BANERJEE