Int J Theor Phys (2013) 52:793–797DOI 10.1007/s10773-012-1388-z

Tilted Bianchi Type-I Cosmological Model in LyraGeometry

S.K. Sahu · Tapas Kumar

Received: 17 February 2012 / Accepted: 12 October 2012 / Published online: 30 October 2012© Springer Science+Business Media New York 2012

Abstract Tilted Bianchi-I cosmological model is investigated in the frame work of Lyra(in Math. Z. 54:52, 1951) Geometry. Exact solutions to the field equations are derived whenthe metric potentials are functions of cosmic time only. Some physical and geometricalproperties of the solutions are also discussed.

Keywords Stiff fluid · Tilted cosmological model · Lyra Geometry

1 Introduction

The Bianchi type-I cosmological model is said to be tilted if the fluid velocity vector isnot orthogonal to the group of orbits. This model is spatially homogeneous relative to theobservers whose world line are orthogonally relative to the group of orbits, but is spatiallyinhomogeneous relative to the observers comoving with the fluid. In the tilted Bianchi type-Icosmological model, the tilt can become extreme in a finite time as measured along the fluidcongruence, which implies that the group of orbits become time-like. The Tilted Bianchitype-I cosmological model has been widely studied in the frame work of Lyra [1] Geometryin the search of realistic picture of the universe during the early stages of evolution. It alsohelps to study the details regarding the cosmic microwave background radiation, nucleosyn-thesis as well as the isotropization of the universe.

The general dynamics of tilted cosmological models have been studied by Kings andEllis [2], Ellis and King [3], Collins and Ellis [4], Bali and Sharma [5, 6], Bali and Meena[7], Sahu [8, 9].

Lyra [1] proposed a modification of Riemannian geometry by introducing gauge func-tion into the structure less manifold, as a result of which the cosmological constant arises

S.K. Sahu (�)Department of Mathematics, Lingaya’s University, Faridabad, 121002 Delhi NCR, Indiae-mail: subat_sahoo2002@rediffmail.com

T. KumarDepartment of Information and Technology, Lingaya’s University, Faridabad, 121002 Delhi NCR, India

794 Int J Theor Phys (2013) 52:793–797

naturally form the geometry. This bears a remarkable resemblance to Weyl’s [10] geometry.Further Sen [11] and Sen and Dunn [12] formulated a new scalar-tensor theory of gravita-tion and constructed an analog of the Einstein field equations based on Lyra [1] geometry.Many authors studied cosmological model with constant and time dependent displacementfield [13–17]. Soleng [18] has pointed out that the cosmologies based on Lyra [1] geometrywith constant gauge vector will either include a creation field and be equal to Hoyle andNalikar [19] or contain a special vacuum field which together with the gauge vector formmay be considered as cosmological term. Further Mohanty and Sahu [20–22], Sahu [23],Tripathy et al. [24–28] have studied the various aspects of Bianchi type cosmological mod-els. Recently Bianchi-I universe with anisotropic dark energy model within the frame workof Lyra [1] geometry has been studied with time varying displacement field by Adhav [29].

In the present paper, the tilted Bianchi-I cosmological model in Lyra’s [1] geometry hasbeen studied for the time varying displacement field vector. In Sect. 2, we have derived thefield equations for the tilted Bianchi type-I cosmological model in Lyra [1] geometry. InSect. 3, we have obtained the solutions of the field equations. In Sect. 4, we have mentionedsome physical and geometrical properties of the solution as obtained in the preceding sectionand we have given the concluding remark in Sect. 5.

2 Field Equations

The field equations in normal gauge for Lyra [1] geometry as obtained by Sen [11] are

Rij − 1

2g

j

i R + 3

2φiφ

i − 3

2g

j

i φkφk = −8πT

j

i (1)

where φi is a time varying displacement field vector defined as

φi = (β(t),0,0,0

)(2)

and other symbols have their usual meaning in the Riemannian geometry.We consider here the Bianchi type-I metric in the form

ds2 = −dt2 + A2dx2 + B2(dy2 + dz2

)(3)

where A and B are function of cosmic time t only.The energy momentum tensor for perfect fluid distribution with heat conduction is given

by Ellis [30] as

Tj

i = (ρ + p)uiuj + pg

j

i + qiui + uiq

j (4)

together with

gijuiuj = −1 (5)

qiqj > 0 (6)

and

qiui = 0 (7)

Int J Theor Phys (2013) 52:793–797 795

where p is the pressure, ρ is the energy density, qi is the heat conduction vector orthogonalto ui . The fluid flow vector ui has the components ( sinhλ

A,0,0, coshλ) satisfying (5) where

λ is the tilt angle.With the help of (4)–(7), the field equations (1) for the metric (3) in the co-moving co-

ordinate system take the following explicit forms:

2B44

B+ B2

4

B2= −

[8π

{(ρ + p) sinh2 λ + p + 2q1 sinhλ

A

}+ 3

4β2

](8)

B44

B+ A4B4

AB+ A44

A= −

(8πp + 3

4β2

)(9)

2A4B4

AB+ B2

4

B2= −

[8π

{−(ρ + p) cosh2 λ + p − 2q1 sinhλ

A

}− 3

4β2

](10)

and

(ρ + p)A sinhλ coshλ + q1 coshλ + q1sinh2 λ

coshλ= 0 (11)

where the suffixes 1 and 4 after a field variable represent differentiation with respect to x

and t respectively.

3 Solutions

Equations (8)–(11) are four equations with seven unknowns A, B , p, ρ,φ,λ and q1, there-fore here we consider the following relations between the scale factors.

p = ρ (Mohanty et al. [20]) (12)

β = β0Tα (Adhav [29]) (13)

where β0 and α are constants and

A = Bn (Bali and Meena [7]) (14)

In order to derive the exact solutions of the field equations (8)–(11) easily, we use the fol-lowing scale transformation

A = enγ , B = eγ , dt = AB2dT (15)

With the help of (15), the field equations (8)–(11) reduce to

2γ ′′ − (2n + 1)γ ′2 = −[

8π

{(ρ + p) sinh2 λ + p + 2q1 sinhλ

enγ

}+ 3

4β2

]e(2n+4)γ (16)

(1 + n)γ ′′ + (1 − 2n)γ ′2 = −(

8πp + 3

4β2

)e(2n+4)γ (17)

796 Int J Theor Phys (2013) 52:793–797

(2n + 1)γ ′2 = −[

8π

{−(ρ + p) cosh2 λ + p − 2q1 sinhλ

enγ

}− 3

4β2

]e(2n+4)γ (18)

and

(ρ + p)enγ sinhλ coshλ + q1 coshλ + q1sinh2 λ

coshλ= 0 (19)

Here afterwards the prime stands for ddT

.From (16) and (18), we get

γ ′′ = 0 (20)

which yields

γ = K3T + K4 (21)

where K3(�= 0), K4 are arbitrary constants.Thus the corresponding metric for the model can be written as

ds2 = −e(2n+4)T dT 2 + e2nT dX2 + e2T(dY 2 + dZ2

)(22)

4 Some Physical and Geometrical Properties of the Solutions

Substituting (13) and (21) in (17), we get

p(= ρ) = K5T2α + K6e

T (23)

where

K5

(= − 3β2

0

32π

), K6

(= − (1 − 2n)

8πK2

3 e− (2n+4)K3+K4

K3

)

are constants.With the help of (13), (19), (21) and (23), the tilt angle can be calculated from (16) as

sinh2 λ = 1

2

(K5T

2α + K6eT){(

K5T2α + K6e

T)−1 − 1

}(24)

The heat conduction vector can now be written as

q1 =(K5T

2α + K6eT )

32 en(K3T +K4){(K5T

2α + K6eT )−1 − 1} 1

2

×{(K5T2α + K6e

T ){[K5T2α + K6e

T ]−1 − 1} 12 + 1}√

2{(K5T 2α + K6eT ){1 − (K5T 2α + K6eT )−1} − 1} (25)

The spatial volume for the model (22) is given by

V ol. = K7eT (26)

where K7(= eK4+(n+2)K3

K3 ) is a constant.From (23)–(25) and (13), we find that the pressure, energy density, tilt angle, heat con-

duction vector of the fluid distribution and the time varying displacement field vector areconstants at time T = 0 and gradually increases in the course of evolution. Equation (26)shows the isotropic expansion of the universe with time which corresponds to a radius scalefactor a = a0e

T/3 where the constant a0 is related to K7. This is obviously a de Sitter kindof solution.

Int J Theor Phys (2013) 52:793–797 797

5 Conclusion

In this paper, we have investigated tilted Bianchi type-I stiff fluid cosmological model inLyra’s [1] geometry for time varying displacement field vector. We find that the pressure,energy density, tilt angle, heat conduction vector of the fluid distribution and the time varyingdisplacement field vector are constants at time T = 0 and gradually increases as the increaseof the age of the Universe. Further we also find that the role of time varying displacementfield vector plays a crucial role in the dynamics of the universe.

Acknowledgements The authors would like to convey their sincere thanks and gratitude to the anonymousreferees for their valuable comments and kind suggestions for the improvement of this paper. The authorsare also thankful to Dr. B.L. Raina, Professor, Lingaya’s University for his valuable discussion related to theabove work.

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