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FRW Cosmic Strings with Extra Dimensions in a New

Scalar-tensor Theory of Gravitation

R.Venkateswarlu1, J.Satish2 and K.Pavan Kumar3

Abstract. Explicit field equations of a scalar tensor theory of gravitation

proposed by Sen-Dunn theory obtained with an aid of a five dimensional FRW

metric. Exact solutions of the field equations are derived when the metric

potentials are functions of cosmic time only. The solutions of the field equations

are obtained in (i) = (ii) + = 0 and (iii) = (1+) . Some physical and

geometric properties of the solutions are also discussed.

Key words: Scalar-Tensor, Gravitation, Cosmology, FRW models.

1. Introduction

In view of Kaluza-Klein theories [1 4], the study of higher-dimensional cosmo-

logical models acquired much significance. An interesting possibility known as the

cosmological dimensional reduction process is based on the idea that, at the very

early stage, all dimensions in the universe are comparable. Later, the scale of the

extra dimensions becomes so small as to be unobservable by experiencing contrac-

tion. This process was first proposed by Chodos and Detweiler [5] who showed

that,in the framework of pure gravitational theory of Kaluza-Klein, the extra

dimension contracts to a very small scale, while the other spatial dimensions

expand isotropically. Guth [6] and Alvarez and Gavela [7] observed that during

the contraction process, extra dimensions produce large amount of entropy. Weyl

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[8] proposed a modification of Riemannian manifold in order to geometrize the

whole of gravitation and electromagnetism.

It is well known that a gravitational scalar field, beside the metric of the

space-time must exist in the frame work of the present unified theories. Hence

there has been much interest in scalar tensor theories of gravitation. Several

theories are proposed as an alternatives to Einsteins theory to reveal the nature of

the universe at the early state of evolution .The most important among them being

scalar-tensor theories proposed by Lyra [9], Brans- Dicke[10],Nordtvedt[11] and

Wagoner [12]. Saez and Ballester [13] have developed a new scalar tensor theory

of gravitation in which the metric is coupled with a dimensionless scalar field in a

simple manner. This coupling gives a satisfactory description of weak fields. In

spite of the dimensionless character of the scalar field an antigravity regime

appears. This theory suggests a possible way to solve the missing matter problem

in non-flat FRW cosmologies. Saez[14] , Singh and Agrawal[15] , Shri Ram and

Singh [16] , Shri Ram and Tiwari [17], Reddy and VenkateswaraRao[18],Reddy[19,20],Mohanty and Sahu[21,22,23] ,Adhav et. al. [24] are some

of the authors who have investigated various aspects of scalar tensor theory of

gravitation.

The study of string theory is important in the early stages of the evolution of the

universe before the particle creation. Cosmic strings have received considerable

attention in cosmology as they are believed to give rise to density perturbations

leading to the formation of galaxies [25]. Chatterjee [26] constructed massive

string cosmological model in higher dimensional homogeneous space time. Krori

et al. [27] constructed Bianchi type I string cosmological model in higher

dimension and obtained that matter and strings coexist throughout the evolution of

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the universe. Rahaman et al.[28] obtained exact solutions of the field equations for

a five dimensional space time in Lyra Manifold when the source of gravitation is

massive strings. The work of Venkateswarlu and Reddy [29]has been extended by

Sahu and Panigrahi[30]. Recently, Venkateswarlu, and Pavan kumar [31] have

studied in Higher dimensional F R W cosmological models in Self-creation theory.

Modified theories of gravity have been the subject of study for the last few

decades. As an alternative to Einsteins theory of gravitation, Sen and Dunn [32]

have proposed a new scalar-tensor theory of gravitation in which both the scalar

and tensor fields have intrinsic geometrical significance. The scalar field in this

theory is characterized by the function )( ix= where ix are coordinates in the

four dimensional Lyra manifold and the tensor field is identified with the metric

tensor ijg of the manifold. The field equations given by Sen and Dunn[32] for the

combined scalar and tensor fields are

ij

k

kijjiijij TgRgR2,

,,,

2)

2

1(

2

1 = (1)

where ,

2

3= ijR and R are respectively the usual Ricci-tensor and Riemaan-

curvature scalar (in our units 18 == GC ).

The energy momentum tensor for a cloud of massive strings that can be

written as

jijiij xxuu = .

(2)

Here is the rest energy density of the cloud of strings with particles attached to

them, is the tension density of the strings and += p , p being the energy

density of the particles. The velocity iu describes the five velocity which has

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components (1, 0, 0, 0,0) for a cloud of particles and ix represents the direction of

string which will satisfy

1== ii

i

i xxuu and 0=iixu .

(3)

In this paper, we intended to study effect of cosmic strings with extra

dimensions, by considering the five dimensional FRW model, in a new scalar-

tensor theory of gravitation proposed by Sen and Dunn[32]. Section 2 contains the

five dimensional FRW metric and the field equations of this theory. In section 3,

the solution of the field equations is obtained in the context of cosmic strings and

also discussed some properties of the models obtained. Last section contains some

conclusions of the models derived.

2. Metric and Field equations :

Here we consider the five dimensional FRW metric of the form

2222222

2

2222 )(sin

1)( dtAdrdr

kr

drtRdtds +

++

+=

(4)

where R(t) is the scale factor and k=0,-1 or +1is the curvature parameter for flat,

open and closed universe, respectively. The fifth coordinate is also assumed to be

space like coordinate. The direction of the strings is taken to be along X4

axis so

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that we have X

i

= (0,0 , 0, 1/R, 0). Now the field equations for the metric (4) can be

written as

+=++

2

2

2

2

2

2 2333

R

k

R

R

R

R(5)

=++++

2

22

2

2 222

R

k

A

A

RA

AR

R

R

R

R(6)

+=++ 2

2

2

2

2

2 2333

R

k

RA

AR

R

R(7)

Where overhead dot denotes ordinary differentiation with respect to t.

3. Solutions to the field equations

The field equations(5) (7) are a system of three equations with five

unknown parameters RA, , , and . We need two additional conditions to get

a

deterministic solution of the above system of equations. Thus we present the

solutions of the field equations in the following physically meaningful cases:

(i) The simplest relation between and is the proportionality relation,

written as = where is a proportionality constant which gives rise to thefollowing three cases:

(a)For 1= , we get geometric strings(or) Nambu strings

(b)For 1= ,we get massive strings

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(c)For )1( += , 0 ,we get p-string or Takabayaski strings

(ii) a functional relationship [33] between the metric functions A and R of the

form

mRA = (8)

Where m is any arbitrary constant.

3.1 Geometric strings (or) Nambu strings( = ) i.e, when

1=

Now the field equations (5) - (7) together with (8) reduces to

0

)2(

2)12(

2

2

2=

+

+++Rm

k

R

Rm

R

R

(9)

Case 3.1.1: Flat model (i.e., 0=K )

In this case the field equations admit the solution

)22(

1

21 )])(22[()(+++= mctcmtR (10)

Where c1and c2 are integrating constants. Thus the general solution of the field

equations (5)-(7) is given by

)22(

1

21 )])(22[()(+++= mctcmtR

)22(

21 )])(22[()(+++= mm

ctcmtA (11)

and 1210 )(p

ctc += where 0 is arbitrary constant (12)

where2

1

2

2

1)22(

)14(2

+

++=m

mmp

, )32()32( +

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Since )32()32( +

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The kinematical parameters viz., scalar expansion , the shear scalar , spatial

volume Vand the deceleration parameterq are given by

)(25

21

1

ctcc+

=

)(18

67

21

1

ctc

c

+= (17)

[ ] 25

21 )( ctcgV +==

5

1=q

The scale factor R expands indefinitely while the other parameter, namely A(t),

gradually diminishes with increase in time. Thus the extra dimension becomes

insignificant as time proceeds after the creation and we are left with the real four-

dimensional world. The declaration parameter q acts as an indicator of the

existence of inflation. Ifq > 0, the model decelerates in the standard way while q

< 0 the model inflates. Also the model is non-inflationary in nature ,since

).0(5

1>=q

ARV ,0,0, as 0t hence we have a line singularity .The volume is

increasing indefinitely with increasing time..

Case3.1. 2:Closed model i.e. 1=K :

In this case the field equations (5) - (7) together with (8) reduces to

0)2(

2)12(

2

2

2=

++++

RmR

Rm

R

R (18)

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)24(2)24)(2(

4 ++++

= mRdmm

R

(19)

Where d is an integral constant. Equation (19) admits a closed form solution

only if d=0,and is given by

n

ncttR 3

2)(

+=

(20)

and consequently we have

m

n

ncttA

+= 3

2)(

(21)

together with the scalar field

330 )2(p

ct+= where m

p3

3

=

. (22)

and )24)(2( ++= mmn , 21

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2

2

32

dn

nctm

+

(23)

The string energy density, tension density ,

)22(

3

2

03)2)(2(12

++==p

nctmm (24)

Since.

2

12

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2

1=q

It is noticed that the extra dimension A(t) contracts while R(t) expands with

increase in time.Since q > 0 for 1=m which shows the nonexistence of inflationin higher-dimensional FRW closed models in Sen-Dunns theory of gravitation.

since 0)(,)(,,0 tAtRV as t indicates that the closed FRW model

possess a line singularity as t .The volume is increasing indefinitely with

increasing time.

Case 3.1.2:Open model i.e. 1=K :

In this case the field equations (5) - (7) together with (8) reduces to

0)2(

2)12(

2

2

2=

++++

RmR

Rm

R

R (29)

The first integral of above equation is

)24(2

)24)(2(

4 ++++

= meRmm

R

(30)

Since we are looking for a closed form solution ,we take e =0.and hence equation

(30),on integration yields

)24)(2(

)24)(2(2

)(4

++

+++= mm

mmct

tR

(31)

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m

mm

mmcttA

++

+++=

)24)(2(

)24)(2(2)(

4

(32)

The scalar field is obtained as

( ) 440 )24)(2(2p

mmct +++= (33)

where m

p3

4

= . 2 and 0> are identically satisfied for

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Scalar expansion

+

=

4

724

21

4ct

Shear scalar

+= ]

4

72

1

72

199

4ct

(36)

Spatial volume

)3(

4

7

)4

72(2

+

+

==

m

ct

gV

Deceleration parameter)3(2 +

=

=m

m

a

aaq

3.2 Massive strings ( 0=+ ) i.e, when 1=

In this case we consider the equations (5)-(7) together with (8) yields

02

)2(2

2

2=+++

R

k

R

Rm

R

R (34)

Case 1: 0=K (Flat model):

Which admit the solution

)3(

1

65 )])(3[()(+++= mctcmtR (35)

)3(

65 )])(3[()(+++= mm

ctcmtA (36)

5650 )(p

ctc += ,where2

1

25 )3(

)1(6

+

+=

m

mp

(37)

Where c5 , c6 are arbitrary constants.

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By making use of equations (35) and (36) in equations (5) - (7), the string energy

density and the tension density become zero. Hence it is observed that cosmic

massive strings do not co-exist with the scalar field in this theory.

Case 2: 1=K (Closed model):

In this case equation (34) reduces to the form

02

)2(2

2

2=+++

RR

Rm

R

R

(38)

which admits the solution

+=

n

ncttR 7

2)(

(39)

m

n

ncttA

+= 7

2)( (40)

where )2( += mn and 2

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and the corresponding scalar field is given by equation (41).

The string energy density, tension density are given by

)22(

7

2

06)2)(2(2

++==p

nctmm (43)

and the particle density p , the scalar expansion , the shear scalar, spatial

volume Vand the deceleration parameterq are given by

)22(

7

2

06)2)(2(4

++=p

p nctmm

)2()3(2

7 nctm+

+=

( ) 21

2

7

2

)2(

9127

9

2

++

=nct

mm (44)

3

72+

+==

m

n

nctgV

)3(2 +=

= mm

a

aa

q

and if 3

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particles dominate over the strings in this model. This model has no singularity at t

= 0 and undergoes expansion with time.

Case 3: 1=K (Open model):


0

2)2(

2

2

2=++

RR

Rm

R

R

(45)

Which admits the solution

+

++=

)2(

)2(2)(

8

m

mcttR

(46)

m

m

mcttA

+

++=

)2(

)2(2)(

8

(47)

780 ))2(2(p

mct ++= where2

1

7 )12( += mmp (48)

where c8 is arbitrary constant.

Thus the metric (4) takes the form

+

++

+

+

+++= 22222

2

22

822 sin

1)2(

)2(2 drdr

r

dr

m

mctdtds

2

2

8

)2(

)2(2d

m

mctm

+

++(49)


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)22(

8

2

07)2(2)(2(2

+++==p

mctmm (50)

As time increases the scale factorsR andA increase indefinitely. So in the above

open model there is no compactification of extra dimension for m >0 . The extradimension A contracts if .2

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Case 1: 0=K (Flat model)

)3(1

109)])(3[()( +++= mctcmtR (53)

)3(

109)])(3[()( +++= m

m

ctcmtA (54)

81090 )(p

ctc += where2

1

28 )3(

)1(6

++=

m

mp

(55)

where c9 and c10 are arbitrary constant.

By making use of equations (52) and (53) in equations (5) - (7), ==0 . Hence

it is observed that cosmic p- strings do not co-exist with the scalar field in this

theory.

Case 2: 1=K (Closed model):


0

)12(

)1(2)2(

2

2

2=

++

+++RmmR

Rm

R

R

(56)

which admit the solution of the form

)42)(12(

)42)(12()1(2)(

11

+++

++++=

mmm

mmmcttR

(57)

m

mmm

mmmcttA

+++

++++=

)42)(12(

)42)(12()1(2)(

11

(58)

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( 9110 )42)(12()1(2)( pmmmctt ++++= (59)

where2

1

9)1(2

1)1(6

++

=

mmp


Thus the metric (4) takes the form

+

++

+++

+++++= 22222

2

22

1122 sin1)42)(12(

)42)(12()1(2

drdr

r

dr

mmm

mmmctdtds

2

2

11

)42)(12(

)42)(12()1(2

d

mmm

mmmctm

+++

++++ (60)


)2(12)1( 20 +=+= mm

( )22(11 9)42)(12()1(2 ++++ pmmmct

(61)

the scalar expansion , the shear scalar, spatial volume Vand the deceleration

parameterq are given by

)42)(12()1(2

)3(12

11 ++++

+

= mmmctm

( ) 21

11

2

)42)(12()1(2

91271

9

2

++++

+=

mmmct

mm

(62)

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3

11

)42)(12(

)42)(12()1(2+

+++

++++==

m

mmm

mmmctgV

)3(2 +==

m

m

a

aaq

As time increases the scale factorsR andA increase indefinitely. So in the above

closed model there is no compactification of extra dimension for m >0 . The extra

dimension A contracts if .2

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where2

1

10)1(2

1)1(6

++

=

mmp


+

++

+

+++

+++++= 22222

2

22

1222 sin1)42)(12(

)42)(12()1(2

drdr

r

dr

mmm

mmmctdtds

2

2

12

)42)(12(

)42)(12()1(2

d

mmm

mmmctm

+++

++++ (67)


)2(12)1( 20 +=+= mm ( )22(12 10)42)(12()1(2 ++++ pmmmct

(68)

the scalar expansion , the shear scalar, spatial volume Vand the deceleration

parameterq are given by

)42)(12()1(2

)3(12

12 +++++

=mmmct

m

( ) 21

12

2

)42)(12()1(2

91271

9

2

++++

+=

mmmct

mm

(69)

3

12

)42)(12(

)42)(12()1(2+

+++

++++==

m

mmm

mmmctgV

)3(2 +==

m

m

a

aaq

The extra dimension A contracts if .2

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