Int J Theor PhysDOI 10.1007/s10773-014-2088-7

Gravitationally Non-Degenerate Petrov Type–ICosmological Model Filled with Viscous Fluidin Modified Brans-Dicke Cosmology

Sarfaraz Alam ·Priyanka Rai ·L. N. Rai ·Vivek K. Singh

Received: 25 November 2013 / Accepted: 5 March 2014© Springer Science+Business Media New York 2014

Abstract Adding the Cosmological term, which is assumed to be variable in Brans-Dicketheory, we have discussed about a cylindrically symmetric cosmological model filled withviscous fluid with free gravitational field of a non-degenerate petrov type-I. The effect ofviscosity on various kinematical parameters has been discussed. Finally, this model hasbeen transformed to the original form (1961) of Brans-Dicke theory (including a variablecosmological term).

Keywords Non-degenerate · Viscous fluid · Cylindrically symmetric · Cosmologicalmodel · Petrov type-I · Brans-Dicke theory

1 Introduction

After the cosmological constant was first introduced into general relativity by Einstein,its significance was studied by various cosmologists (for example [1]), but no satisfactoryresults of its meaning have been reported as yet. Zeldovich [2] has tried to visualize themeaning of this term from the theory of elementary particles. Further Linde [3] has arguedthat the cosmological term arises from spontaneous symmetry breaking and suggested thatthe term is not a constant but a function of temperature. Also Drietlein [4] connects themass of Higg’s scalar boson with both the cosmological term and the gravitational con-stant. In cosmology, the term may be understood by incorporation with Mach’s principle,

S. AlamDepartment of Mathematics, Maulana Azad College of Engineering and Technology,Neora, Patna, Bihar, India

P. Rai (�) · L. N. RaiDepartment of Mathematics, Patna Science College, Patna 800005, Bihar, Indiae-mail: priyankarai2007@rediffmail.com

V. K. SinghSchool of Physics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, J&K, India

Int J Theor Phys

which suggests the acceptance of Brans-Dicke Lagrangian as a realistic case [5]. A physi-cally, more interesting problem to discuss from the stand point of Mach’s principle is that ofthe cosmological models in Brans-Dicke theory. Brans-Dicke [5] obtained an analogue ofRobertson-Walker model for flat space only, which reduces to the flat Friedmann universein the limit ω → ∞. Morganstern [6, 8] has also discussed the Brans-Dicke Friedmannuniverses. It is found that the scalar field energy density dominates the radiation energydensity as one approaches the initial singularity. The result may be extended to all Fried-mann models in the Brans-Dicke theory. It has been concluded that although the scalarfield may be undetectable at the present epoch, if it exists it may play an important role asone approaches the initial singularity of the cosmology. Morganstern [9, 10] has also dis-cussed in detail about the observational constraints imposed by Brans-Dicke cosmologies.Miyazaki [11] has obtained the Brans-Dicke cosmological solutions for the homogeneousisotropic universes. His solutions are different from those of the Friedmann type in generalrelativity. However, in his solution the geometry of the universe is determined by the cou-pling parameter of the scalar field. The behaviour of the expansion of these universes is thesame as that of Milne universe [12] in the kinematics. The universe expands forever with aconstant velocity and the gravitational constant increases linearly.

The investigation of particle physics within the context of the Brans-Dicke Lagrangian[13] has stimulated the study of the cosmological term with a modified Brans-DickeLagrangian in cosmology and elementary particle physics. Endo and Fukui [14] have stud-ied the variable cosmological term from the point of view of cosmology in Brans-Dicketheory [5] and elementary particle physics especially in the context of Dirac’s large numberhypothesis [15, 16].

Further, astronomical observations of the large-scale distribution of galaxies in our uni-verse have shown that the distribution of matter can be satisfactorily described by a perfectfluid. It has, however, been conjectured that some time during an earlier phase in the evo-lution of the universe when galaxies were formed, the material distribution behaved like aviscous fluid ([17], page 124). It is therefore of interest to obtain cosmological models forsuch distributions. It is also well known that there is a certain degree of anisotropy in theactual universe. Therefore, we have chosen the metric for the cosmological model to becylindrically symmetric. Thus, in this paper, we have considered a cylindrically symmetricmodel filled with viscous fluid in a modified Brans-Dicke theory in which the variable cos-mological term Q is an explicit function of a scalar field φ as proposed by Bergmann [18]and Wagoner [19] and discussed in detail by Endo and Fukui [14].

The Brans-Dicke field equations with cosmological term Q are [14]:

Gij + gijQ = 8π

φTij + ω

φ2

(φ,iφ,j − 1

2gijφ,kφ

,k

)+ 1

φ(φi;j − gij φ), (1.1)

φ = 8π

(2ω + 3)μT , (1.2)

Q = (2ω+ 3)

4.(1 − μ)

μ

φ

φ= 8π (1 − μ)

4φT , (1.3)

where the constant μ shows how much our theory including Q(φ) deviates from that ofBrans and Dicke, ω is coupling constant and Tij is energy-momentum tensor for a viscousdistribution [20]. Semicolons denote covariant differentiation with respect to the metric gijand commas mean partial differentiation with respect to the coordinate xi . The theory canalso be represented in a different form under a unit transformation (UT) [21] in which

Int J Theor Phys

length, time and reciprocal mass are scaled by the function λ12 (x). Then under the conformal

transformationgij → gij = φgij (1.4)

Equations (1.1)–(1.3) go to the form

Gij + gijQ = (8π) T ij + 1

2(2ω + 3)

(∧,i ∧,j −1

2gij ∧,k ∧,k

)(1.5)

�∧ = 8π

(2ω + 3)μT ,∧ = logφ, (1.6)

Q = (2ω + 3)

4.(1 − μ)

μ�∧ = 8π (1 − μ)

4T , (1.7)

where the barred quantities are defined in terms of gij as their unbarred counterparts aredefined in terms of the unbarred metric gij and all barred operations are performed withrespect to the barred metric and barred Christoffel symbols.

In Section 2, a cylindrically symmetric metric is considered and the energy-momentumtensor is taken to be that of viscous fluid [20]. In Section 3, we have obtained pressure,density expressions for this model which is also of petrov type-I. The effect of viscosityon various kinematical parameters has been also discussed. It is found that the kinematicviscosity prevents shear, expansion and the free gravitational field from withering away.Finally, in Section 4 we have transformed this model to the 1961 form of Brans-Dicketheory.

2 Derivation of the Line-Element

We consider a cylindrically-symmetric metric in the form [22]

ds2 = A2(dx2 − dt2)+ B2dy2 + C2dz2 (2.1)

where A, B, C are functions of x4 ≡ t only. This ensures that the model is spatially homo-geneous. The energy-momentum tensor for a viscous fluid distribution is given by Landauand Lifschitz [20]

Tj

i = (ε + p)vivj + pg

ji − η

(vji; + vj;i + vjvlvi;l + viv

lvj;l)

−(ξ − 2

3η

)vl;l

(gji + vivj

)(2.2)

together withgijvivj = −1 (2.3)

where p and ∈ are the proper pressure and density respectively, η and ξ are the two coef-ficients of viscosity and semicolons indicate covariant differentiation. vi is the flow vectorsatisfying (2.3). We assume the coordinates to be comoving so that v1 = v2 = v3 = 0 andv4 = 1

A. Scalar field ∧ is also taken to be a function of t only. The field equations (1.5) and

(1.6) for the line element (2.1) turn into

1

A2

[B44

B+ C44

C+ B4C4

BC− A4B4

AB− A4C4

AC

]+Q

= 8π

[p − 2η

A4

A2−

(ξ − 2

3η

)vl;l

]+ (2ω+ 3)

4A2∧2

4 (2.4)

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1

A2

[(A4

A

)4+ C44

C

]+Q = 8π

[p − 2η

B4

AB−

(ξ − 2

3η

)vl;l

]+ (2ω+ 3)

4A2∧2,4 (2.5)

1A2

[(A4A

)4+ B44

B

]+Q = 8π

[p − 2η C4

AC−

(ξ − 2

3η)

vl;l]+ (2ω+3)

4A2 ∧2,4 (2.6)

1

A2

[A4

A

(B4

B+ C4

C

)+ B4C4

BC

]+Q = −8πε + (2ω+ 3)

4A2∧2,4 (2.7)

[∧44 +∧4

(B4

B+ C4

C

)]= 8πμA2

(2ω+ 3)

[(ε − 3p)+ 3ξ

A

d

dtlog (ABC)

](2.8)

The suffix 4 after the symbols A, B, C denotes ordinary differentiation with respect to ‘t’.Equations (2.4)–(2.8) are five equations in six unknowns A, B, C, p,∈ and ∧. The coeffi-cients of viscosity η and ξ are taken as constants. Equations (2.4)–(2.8) are not independent,but they are related by the contracted Bianchi identities. In the present case they lead to thesingle condition

dε

dt+ (ε + p)

d

dtlog (ABC)−

(ξ − 2

3η

)1

A

(d

dtlog(ABC)

)2

−2η

A

(A2

4

A2+ B2

4

B2+ C2

4

C2

)= 0

For complete determinacy of the system one extra condition is needed. An obvious oneis imposition of an equation of state. However, we proceed from a different consideration.Although the distribution of matter at each point determines the nature of expansion inthe model, the later is also affected by the free gravitational field through its effect on theexpansion, vorticity and shear in the fluid flow. A prescription of such a field may thereforebe made on a priori basis. The cosmological models of Robertson and Walker, as well as theuniverses of Einstein and DeSitter, have vanishing free gravitational fields. Here, we assumethat C14

14 = C2323 = 0. The resulting space-time will obviously be of non-degenerate Petrov

type-1. If we take any of the two metric potentials to be equal, the space-time becomesconformal to flat and the pressure and density terms arise only due to viscosity, which doesnot correspond to a realistic distribution. Therefore we consider A, B, C to be unequal.

From (2.4) and (2.5), we get

B44

B+ B4C4

BC− A4

A

(B4

B+ C4

C

)−

(A4

A

)4= 16πηA

[B4

B− A4

A

](2.9)

From (2.5) and (2.6), we get

B44

B− C44

C= 16πηA

[B4

B− C4

C

](2.10)

The condition C1414 = C23

23 = 0 leads to

B44

B+ C44

C− 2B4C4

BC− 2

(A4

A

)4= 0. (2.11)

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From (2.9) and (2.10), we get

A = LB

(B

C

)K

(2.12)

where K and L are constants of integration.From (2.11) and (2.12), we get

K

[B4

B− C4

C

]4+ B4C4

BC+ 1

2

[B44

B− C44

C

]− B2

4

B2= 0. (2.13)

From (2.10) and (2.12), we get

B44

B− C44

C= 16πηLB

(B

C

)K [B4

B− C4

C

](2.14)

From (2.13) and (2.14), we get

B = N1

(2K+1) (S − 1)(α − t)−K

8πηLN(α − t)(1−s) + β(s − 1)(2.15)

and

C = (S − 1)(α − t)(1+K)

N1

(2K+1)[8πηLN(α − t)(1−s) + β(s − 1)

] , (2.16)

where α, β and N are constants of integration and S = 2K(K + 1).From (2.12), (2.15) and (2.16), we get

A = LN(S − 1)(α − t)−s[8πηLN(α − t)(1−s) + β(s − 1)

] (2.17)

Hence the line-element (2.1) becomes

ds2 =[

(s − 1)

8πηLN(α − t)(1−s) + β(s − 1)

]2 [L2N2(α − t)−2s × (dx2 − dt2)+N

22K+1 (α − t)−2Kdy2

+ 1

N2

(2K+1)

(α − t)2(1+K)dz2

](2.18)

By a suitable transformation of coordinates, the metric (2.18) is reduced to the form [23]:

ds2 =[

(s − 1)T −S

8πηT (1−s) + a(s − 1)

]2

[dX2 − dT 2 + T 2K(2K+1)dY 2 + T 2(K+1)(2K+1)dZ2] (2.19)

where a is an arbitrary constant.

3 Some Physical and Geometrical Features

The pressure p and density ∈ in the model (2.19) are given by

8πp = a2

T 2(1−S)

[(4k4 + 4k3 − 5k2 − 3k + 3)+ (16k4+20k3−18k2−16k+10)

4

tan2{(−(16k4+20k3−18k2−16k+10)

(2ω+3)(4k2+4k−1)2

) 12

log(mT 4k2+4k−1

)}]+ 8πaξ

T 1−S (4k2 + 4k − 2)+Q

(3.1)

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−8π∈ = a2

T 2(1−S)

[(4k4 + 8k3 − 3k2 − 7k + 1)+ (16k4+20k3−18k2−16k+10)

4

tan2{(−(16k4+20k3−18k2−6k+10)

(2ω+3)(4k2+4k−1)2

) 12

log(mT 4k2+4k−1

)}]+Q

(3.2)also scalar field ∧ is given by

∧ = log sec

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log(mT 4k2+4k−1

)⎤⎦ (3.3)

and

Q = (1−u)4μ

a2(16k4+20k3−18k2−16k+10)T 2(1−S)

sec2[{−(16k4+20k3−18k2−16k+10)

(2ω+3)(4k2+4k−1)2

} 12

log(mT 4k2+4k−1

)]

(3.4)The model has to satisfy the reality conditions [24]:

(i) ε + p > 0

(ii) ε + 3p > 0

which requires thata2 > 0, ω < − 3

2 ,Q > 0 (i.e. μ < 1)

Q >−a2

(16k4+20k3−18k2−16k+10

)T 2(1−S)

(3.5)

sec2

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log(mT 4k2+4k−1

)⎤⎦ (3.6)

The flow vector viof the distribution for the model (2.19) given by

v1 = v

2 = v3 = 0 and v

4 = 8πηT

(S − 1)+ aT S (3.7)

Obviously vi;jvj = 0.Hence the flow is geodesic.The rotation tensor

ωij = vi;j − vj ;i = 0.Thus, the fluid filling the universe is non-rotational.

The expansion scalar θ = 13 vi;i is given by

θ = a(s − 1)T S−1 − 16πη (3.8)

Shear tensor σij = 12

(vi;j + vj ;i

) − θ(gi;j + vivj

)is given by

σ11 =13 (s − 1)(5s − s2 − 1)T −(s+1)

8πηT (1−s) + a(s − 1),

σ22 =13 (s − 1)(3k − s + 1)T (s−2k−1)

8πηT (1−s) + a(s − 1),

σ33 = − 13 (s − 1)(3k + s + 2)T (s+2k+1)

8πηT (1−s) + a(s − 1),

σ44 = 0., (3.9)

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Also the shear σ is

σ 2 = 12σijσ

ij = 118

[8πηs−1 + aT (s−1)

]2

[(5s − s2 − 1)2 + (3k − s + 1)2 + (3k + s + 2)2

](3.10)

The nonvanishing components of conformal curvature tensor Cjkhi are

C1212 = −C13

13 = −k (k + 1) (2k + 1)

[8πη

s − 1+ aT (s−1)

]2

(3.11)

Thus, the viscosity prevents the free gravitational field as well as the shear from witheringaway. It is also clear from equation (3.8) that the effect of viscosity is to retard expansion ofthe model.

The pressure, density, scalar field and cosmological constant are singular at

T 4k2+4k−1 =[

1

m

]exp

⎡⎢⎣π

{(2ω+ 3)

(4k2 + 4k − 1

)2

−4(16k4 + 20k3 − 18k2 − 16k + 10

} 12

⎤⎥⎦ (3.12)

The model exists for a finite time

[1m

]≤ T 4k2+4k−1

<[

1m

]exp

[π

{(2ω+3)

(4k2+4k−1

)2

−4(16k4+20k3−18k2−16k+10

} 12]

(3.13)

When μ= 1, the cosmological term Q vanishes and the model (2.19) reduces into a Brans-Dicke analogue of one of the viscous models due to Roy and Prakash [25] in generalrelativity.

4 Transformations of the Solutions

Under the transformations

gij → gij = 1φgij ; T ij → Tij = φT ij ;

T → T = φ2T ; p → p = φ2p;ε → ε = φ2ε; φ → φ = e∧;

Q → Q = φQ; vi → vi = φ12 vi .

(4.1)

The field equations (1.5)–(1.7) are changed into (1.1)–(1.3).

φ = sec

⎡⎣

{− (

16k4 + 20k3 − 18k2 − 16k + 10)

(2ω + 3)(4k2 + 4k − 1

)2

} 12

log(mT 4k2+4k−1

)⎤⎦ (4.2)

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gij = cos

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log(mT 4k2+4k−1

)⎤⎦ gij (4.3)

i.e. g11 = cos

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log{mT 4k2+4k−1}⎤⎦

×[

(s − 1)T −s

8πηT (1−s) + a(s − 1)

]2

g22 = cos

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω + 3)(4k2 + 4k − 1)2

} 12

log{mT 4k2+4k−1}⎤⎦

×[

(s − 1)T −s

8πηT (1−s) + a(s − 1)

]2

T 2k(2k+1)

g33 = cos

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω + 3)(4k2 + 4k − 1)2

} 12

log{mT 4k2+4k−1}⎤⎦

×[

(s − 1)T −s

8πηT (1−s) + a(s − 1)

]2

T 2(k+1)(2k+1)

g44 = − cos

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log{mT (4k2+4k−1)}⎤⎦

×[

(s − 1)T −s

8πηT (1−s) + a(s − 1)

]2

v4 = sec12

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log{mT (4k2+4k−1)}⎤⎦

×[

8πηT (1−s) + a(s − 1)

(s − 1)T −s

]

8πp = a2(16k4 + 20k3 − 18k2 − 16k + 10)

4μT 2(1−s)

sec4

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log{mT (4k2+4k−1)}⎤⎦

+ a2(−2k3 − k2 + 2k + 1)+ 16πaξ(4k2 + 4k − 2)

2T 2(1−s)(4.4)

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sec2

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log{mT (4k2+4k−1)}⎤⎦

−8πξ = a2(16k4 + 20k3 − 18k2 − 16k + 10)

4μT 2(1−s)

sec4

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log{mT (4k2+4k−1)}⎤⎦

+a2(6k3 + 3k2 − 6k − 3)

2T 2(1−s)×

× sec2[{−(16k4+20k3−18k2−16k+10)

(2ω+3)(4k2+4k−1)2

} 12

log{mT (4k2+4k−1)}]

Q = (1−μ)4μ

a2(16k4+20k3−18k2−16k+10)T 2(1−s)

(4.5)

sec3

⎡⎣

{−(16k4 + 20k3 − 18k2 − 16k + 10)

(2ω+ 3)(4k2 + 4k − 1)2

} 12

log{mT (4k2+4k−1)}⎤⎦ (4.6)

The reality condition should also be imposed on the solutions in (4.1)–(4.6) similar tothose in Section 3.

5 Discussion and Conclusion

We have successfully discussed an exact solution of a cylindrically symmetric cosmologi-cal model filled with viscous fluid with free gravitational field of a non-degenerate petrovtype-I. The effect of viscosity on various kinematical parameters has been discussed and itwas found that the kinematic viscosity prevents shear, expansion and the free gravitationalfield from withering away. It is also clear from (3.8) that the effect of viscosity is to retardexpansion of the model. When μ = 1, the cosmological term Q vanishes and the model(2.19) reduces into a Brans-Dicke analogue of one of the viscous models due to Roy andPrakash [25] in general relativity. Finally, this model has been transformed to the originalform (1961) of Brans-Dicke theory. All the solutions obtained in this paper are new and theyare expected to reveal new features of viscous models in Brans-Dicke cosmology.

Acknowledgments We are grateful to learned reviewers for their valuable suggestions to improve thequality of the present paper.

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