Inhomogeneous Plane Symmetric Models in a Scalar-Tensor Theory

INHOMOGENEOUS PLANE SYMMETRIC MODELS INA SCALAR-TENSOR THEORY

SHRI RAM and S.K. TIWARIDepartment of Applied Mathematics, Institute of Technology, Banaras Hindu University,

Varanasi, India

(Received 20 August, 1997; accepted 19 May, 1998)

Abstract. For the general plane symmetric metric, some exact solutions of Einstein field equations inthe scalar tensor theory developed by Saez and Ballestev are presented in vaccum and in the presenceof stiff fluid. The physical and kinematical features of the models are also discussed.

1. Introduction

Scalar-tensor theories of gravitation provide the natural generalizations of generalrelativity and provide a convenient set of representations for the observationallimits on possible deviations from general relativity. Several physically acceptablescalar-tensor theories of gravitation have been proposed and widely studied so forby many workers. There are two categories of gravitational theories involving aclassical scalar fieldφ. In the first category the scalar fieldφ has the dimension ofthe inverse of the gravitational constantG among which the Brans-Dicke theory(1961) is of considerable importance and the role of the scalar field is confined toits effects on gravitational field equations. In the second category the theories in-volve a dimensionless scalar field. Saez and Ballester (1985) developed a theory inwhich the metric is coupled with a dimensionless scalar field. This coupling givesa satisfactory description of weak fields. In spite of the dimensionless character ofthe scalar field an anti-gravity regime appears. This theory suggests a possible wayto solve the missing matter problem in non-flat FRW cosmologies. Saez (1985)discussed the initial singularity and inflationary universe and has shown that thereis an anti-gravity regime which would act either at the beginning of the inflation-ary epoch or before. He also obtained a non-singular flat FRW model. Singh andAgrawal (1991) investigated models of Bianchi-types I, III, V, VI0 and KantowskiSachs models in this theory.

The plane symmetric cosmological models in the framework of general rela-tivity are of considerable importance. Taub (1951, 1956) studied inhomogeneouscosmological models of plane symmetry. Singh and Singh (1968) derived a planesymmetric perfect fluid cosmological model from class-one consideration consid-ering the cylindrically-symmetric Marder metric, Singh and Abdussattar (1973)presented a plane symmetric model which is of embedding class higher than one.

Astrophysics and Space Science259: 91–97, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

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92 SHRI RAM AND S.K. TIWARI

Tomimura (1978) studied the evolution in inhomogeneous plane symmetric dustcosmological models. Roy and Tiwari (1983) obtained some plane-symmetric so-lutions of Einstein equations representing inhomogeneous cosmological modelswith viscous fluid and constant bulk viscosity.

In this paper the field equations in the scalar tensor theory developed by Saezand Ballester (1985) are considered for the general plane symmetric metric

ds2 = E(−dt2+ dz2)+G(dx2 + dy2) (1.1)

whereE andG are function ofz and t . Classes of solutions are presented invaccum and in the presence of stiff-matter under the assumptions that the metriccomponentsE andG are separable and scalar functionφ is function of ‘t ’ alone.Their physical and kinematical properties are also discussed.

2. Field Equations

The field equations in Saez and Ballester (1985) theory are

Rµν − 1

2Rgµν −wφn(φ,µφ,ν − 1

2gµνφ,αφ

,α) = K Tµν. (2.1)

The scalar fieldφ satisfies the equation

2φnφ;µ;µ + nφn−1φ,αφ,α = 0 (2.2)

where ‘n is an arbitrary exponent andω is dimensionless coupling constant. In thecase of a perfect fluid distribution, the energy-momentum tensorTµν is given by

Tµν = (ρ + p)vµvν + pgµν (2.3)

whereρ is the matter energy density and ‘p’ the pressure andvµ the fluid fourvelocity vector. As the consequence of Bianchi identities, the equations of motionare

Tµν

;ν = 0. (2.4)

For the metric (1.1), the field equations (2.1)–(2.3) in comoving coordinates, leadto

G′′

G− GG+ G2

2G2− G′2

2G2− EE+ E

2

E2+ E

′′

E− E

′2

E2

= −ωφn(φ′2− φ2)+ 2KpE. (2.5)

−2G

G+ G2

2G2+ G′2

2G2+ EGEG+ E

′G′

EG= wφn(φ′2+ φ2)+ 2kpE (2.6)

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INHOMOGENEOUS PLANE SYMMETRIC MODELS IN A SCALAR-TENSOR THEORY 93

−2G′′

G+ G2

2G2+ G′2

2G2+ EGEG+ E

′G′

EG= wφn(φ′2+ φ2)+ 2kρE (2.7)

−G′

G+ GG

′

2G2+ E′G

2EG+ EG′

2EG= wφnφ′φ (2.8)

φ′′

φ− φφ+ G

′φ′

Gφ− GφGφ+ n

2

(φ′2

φ2− φ

2

φ2

)= 0 (2.9)

p′ + 1

2(ρ + p)E

′

E= 0 (2.10)

ρ + 1

2(ρ + p)

(2G

G+ EE

)= 0 (2.11)

where a dash and dot denote derivatives with respect to ‘z’ and ‘t ’ respectively.

3. Solutions of Field Equations

Equations (2.5)–(2.11) are highly non-linear partial differential equations and henceit is very difficult to solve them, as there exist no standard method for their solution.One has to make certain simplifying assumptions either mathematical or at the costof physics of the problem. Here we make the mathematical assumption that themetric componentsE andG are separable as products of functions of ‘z’ and ‘t ’,and they can be written as

G = GtGz (3.1)

E = EtEz (3.2)

whereGt = G(t) etc. We also assume that the scalar fieldφ is function of time ‘t ’only. Under these assumptions, Equation (2.5)–(2.11) reduce to

G′′zGz

− Gt

Gt

+ G2t

2G2t

− G′2z2G2

z

− EtEt+ E

2t

E2t

+ E′′z

Ez− E

′2z

E2z

= wφnφ2+ 2kp Et Ez (3.3)

−2Gt

Gt

+ G2t

2G2t

+ G′2z2G2

z

+ Et Gt

EtGt

+ E′zG′z

EzGz

= wφnφ2+ 2kpEtEz (3.4)

−2G′′zGz

+ G2t

2G2t

+ G′2z2G2

z

+ Et Gt

EtGt

+ E′zG′z

EzGz

= wφnφ2+ 2kρEtEz (3.5)

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−G′zGz

GzGt

+ E′zEt

EzEt+ EtG

′z

EtGz

= 0 (3.6)

φ

φ+ Gt φ

Gtφ+ n

2

φ2

φ2= 0 (3.7)

p′ + 1

2(ρ + p)

(E′zEz

)= 0 (3.8)

ρ + 12(ρ + p)

(2GtGt+ Et

Et

)= 0. (3.9)

Subtracting Equation (3.5) and (3.4), we get

Gt

Gt

= G′′zGz

= εa2; ε = 0,1,−1. (3.10)

It is difficult to solve these equations in general so we consider some cases ofphysical interest.

Case I. Linear case(ε = 0)In vaccum(ρ = p = 0), Equations (3.3)–(3.10) have solutions

G = (a1t + a2)(a3z+ a4) (3.11)

E = (a1t + a2)3/2(a3z+ a4)

−1/2 (3.12)

φn2+1 = (n+ 2)m

2a1log(a1t + a2) (3.13)

w = 2a21

m2(3.14)

wherea′i s andm are arbitrary constants. The metric of the corresponding solutionscan be written in the form

ds2 = (a1t + a2)3/2(a3z+ a4)

−1/2(−dt2+ dz2)

+(a1t + a2)(a3z+ a4)(dx2 + dy2). (3.15)

In the presence of stiff fluid(ρ = p), Equations (3.3)–(3.10) have the same solu-tions as given by (3.11)–(3.13) wherew remains arbitrary. The energy density andpressure are given by

ρ = p =(

2a21 −wm2

2k

)(a1t + a2)

−7/2(a3z+ a4)12 . (3.16)

The spatial volumeV is given by

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V 3 = (a1t + a2)7/2(a3z+ a4)

3/2. (3.17)

The metric (3.15) has the time singularity att = −a2/a1, the energy-density andpressure being infinite there. Ast → ∞, the spatial volume becomes infinite andthe energy-density tends to zero and so the model essentially gives an empty spacefor large ‘t ’.

The scalar expansionθ and the shear scalarσ are given by (Collins and WainWright, 1983)

θ =(

7a1

4

)(a3z+ a4)

14

(a1t + a2)7/4(3.18)

σ = (a3z+ a4)14

4√

2(a1t + a2)34

[a2

1

3(a1t + a2)+ a2

3

(a3z+ a4)2

] 12

. (3.19)

These scalar are decreasing function of time and tends to zero ast → ∞. Thedeceleration parameterq has the value−36

49 which indicates inflation.

Case II. Hyperbolic case(ε = 1).In vaccum(ρ = p = 0), Equations (3.3)–(3.10) have solutions,

G = cosh(at + b1) cosh(az+ b2) (3.20)

E = cosh3/2(at + b1) cosh−1/2(az+ b2) (3.21)

φn2+1 = (n+ 2)k2 tan−1(exp(at + b1)) (3.22)

ω = 2a2

k2(3.23)

whereb1, b2 andk are constants. The metric of the corresponding solutions can bewritten in the form

ds2 = cosh3/2(at + b1) cosh−1/2(az+ b2)(−dt2+ dz2) (3.24)

+ cosh(at + b1) cosh(az+ b2)(dx2 + dy2).

In the presence of stiff matter, Equations (3.3)–(3.10) have the solutions given by(3.20)–(3.22) where ‘w’ remains arbitrary. Making use of (3.8)–(3.9), we calculatethe expression for energy density and pressure as

ρ = p = 2a2 −wk2

2ksech7/2(at + b1) cosh

12 (az+ b2). (3.25)


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V 3 = cosh7/2(at + b1) cosh3/2(az+ b2). (3.26)

The expression for the scalar expansion, shear scalar and the deceleration parame-ter are

θ =(

7a

4

)sinh(at + b1) cosh

14 (az+ b2)

cosh7/4(at + b1)(3.27)

σ = a cosh14 (az+ b2)

4√

2 cosh3/4(at + C1)

[1

3tanh2(at + b1)− tanh2(az+ b2)

] 12

(3.28)

q = −84

49coth2(at + b1)− 36

49. (3.29)

The model (3.24) has no finite singularity. Ast → ∞, the spatial volume tendsto infinity andθ , σ tend to zero. The negative value of the deceleration parametershows that the model inflates.

Case III. Trigonometric case(ε = −1).In vaccum, Equations (3.3)–(3.10) have solutions given by

G = cos(at + C1) cos(az+ C2) (3.50)

E = cos3/2(at + C1) cos−12 (az+ C2) (3.31)

φn2+1 = (n+ 2)`2

2alog

(sec(at + C1)+ tan(at + C2)

)(3.32)

ω = 2a2

`2(3.33)

whereC1, C2 and` are constants. The metric of the solution can be written in theform

ds2 = cos3/2(at + C1) cos− 12 (az+ C2)(−dt2+ dz2)

+ cos(at + C1) cos(az+ C2)(dx2 + dy2). (3.34)

This metric has no finite time singularity. In the presence of stiff-matter, solutionsof the field equations (3.3)–(3.10) are same as in case of vaccum, except that ‘ω’remains arbitrary. The energy-density and pressure have the expressions.

ρ = p = 2a2 − ω`2

ksec7/2(at + c1) cos

12 (az+ C2). (3.35)


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V 3 = cos7/2(at + C1) cos3/2(az+ C2). (3.36)

As t →∞, metric (3.34) becomes static and the energy density depends onz.The scalar expansion, shear scalar and the deceleration parameter are,

θ =(−7a

4

)sin(at + C1) cos

14 (az+ C2)

cos7/4(at + C1)(3.37)

σ = a cos14 (az+ C2)

4√

2 cos3/4(at + C1)

[1

3tan2(at + C1)− tan2(az+ C2)

] 12

(3.38)

q = 84

49cot2(at + C1)+ 36

49. (3.39)

The positive value ofq indicates that the model decelerates in the standard way.

References

Taub, A.H.: 1951,Ann. Maths53, 472.Brans, C. and Dicke, R.H.: 1961,Phys. Rev.124, 925.Singh, K.P. and Singh, D.N.: 1968,Mon. Not. R. Astron. Soc.140, 453.Tomimura, N.: 1978,Nuovo Cimento44B, 372.Collins, C.B. and Wainwright, J.: 1983,Phys. Rev. D27, 1209.Roy, S.R. and Tiwari, O.P.: 1983,Ind. J. Pur. Appl. Maths14(2), 233.Saez, D.: 1985,A Simple coupling with cosmological implications, preprint.Saez, D. and Ballester, V.J.: 1985,Phys. Lett. A113, 467.Singh, T. and Agrawal, A.K.: 1991,Astrophys. Space Sci.182, 289.

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Inhomogeneous Plane Symmetric Models in a Scalar-Tensor Theory