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Astrophys Space Sci (2009) 323: 87–90

DOI 10.1007/s10509-009-0042-6

O R I G I N A L A R T I C L E

Zero mass scalar field with bulk viscous cosmological solutionsin Lyra geometry

K.S. Adhav · S.D. Katore · R.S. Rane · K.S. Wankhade

Received: 2 October 2008 / Accepted: 16 May 2009 / Published online: 9 June 2009

© Springer Science+Business Media B.V. 2009

Abstract In this paper, we have investigated spatially ho-

mogeneous isotropic Friedmann cosmological model withbulk viscosity and zero-mass scalar field in Lyra manifold.

The cosmological models are obtained with the help of the

special law of variation for Hubble’s parameter proposed by

Bermann (Nuovo Cimento 74B:182, 1983) and power law

relation. Some physical properties of the models are dis-

cussed.

Keywords Friedmann cosmological model · Zero-mass

scalar field · Bulk viscosity · Hubble’s parameter · Lyra

geometry

1 Introduction

Lyra (1951) proposed a modification of Riemannian geom-

etry by introducing gauge function into the structure-less

manifold, as a result of which the cosmological constant

arises naturally from the geometry. This bears a remarkable

resemblance to Weyl’s (1918) geometry. In subsequent in-

vestigations Sen (1957), Sen and Dunn (1971) formulated

K.S. Adhav · S.D. Katore · R.S. Rane · K.S. Wankhade ()

Sant Gadge Baba Amravati University, Amravati 444602, Indiae-mail: [email protected]

K.S. Adhav

e-mail: [email protected]

S.D. Katore


R.S. Rane


R.S. Rane · K.S. Wankhade

Y.C. Science and Arts College, Mangrulpir, India

a new scalar-tensor theory of gravitation and constructed

an analog of the Einstein’s field equations based on Lyra’s

geometry. Halford (1972) has shown that the scalar-tensor

treatment based on Lyra’s geometry predicts the same ef-

fects as in general relativity.

The field equations in the normal gauge in Lyra manifold

as obtained by Sen (1957) are

Rij −1

2gij R +

3

2ϕi ϕj −

3

4gij ϕαϕα

= −8πGT ij (1)

where ϕi is the displacement field and other symbols have

their usual meaning as in Riemannian geometry. The dis-placement field ϕi can be written as

ϕi = (0, 0, 0,β),

where β is a constant (we use the gravitational units 8πG =

c = 1).

Several authors have studied cosmological models within

the frame work of Lyra geometry with constant gauge vector

in the time direction. The study of interacting fields, one of

them being zero-mass scalar field, is basically an attempt to

look into the yet unsolved problem of the unification of the

gravitational and quantum theories. Interacting scalar fieldsin Lyra geometry have been studied by Casana et al. (2004,

2005, 2006, 2007). Pradhan and Pandey (2003) have dis-

cussed bulk viscous cosmological models in Lyra geometry.

In this paper, we have investigated bulk viscous cosmolog-

ical models in the presence of interacting zero mass scalar

field in Lyra geometry with the aid of FRW line element. We

have obtained power law solutions and also solutions using

Hubble’s law. We have also discussed the physical proper-

ties of the models.

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88 Astrophys Space Sci (2009) 323: 87–90

2 Metric and field equations

We consider the spatially homogeneous and isotropic FRW

line element in the form

ds 2 = dt 2 − R2(t )

dr2

1 − Kr 2 + r2(dθ 2 + sin2 θ dϕ2)

, (2)

where K is the curvature index which can take the values

(−1, 0, +1), R(t) represents the radius of the universe and

the signature of the metric is (+, −, −, −).

The energy-momentum tensor due to the bulk-viscous

fluid and zero-mass scalar fields is written in the form

T ij = (p + ρ)U i U j − pgij +

ψ,i ψ,j −

1

2gij ψ,mψ ,m

,

(3)

together with

U i U i = 1 (4)

and

p = p − ηU i;i , (5)

where U i is the four velocity vector of the distribution, ρ

is the energy density, p is the pressure, η is the coefficient

of bulk-viscosity, ψ is the zero mass scalar field and semi-

colon (;) denotes covariant differentiation.

The scalar field ψ satisfies the equation

ψ i;i = 0. (6)

Using co-moving co-ordinates, the field equations (1) with

the help of (2) and (3) can be written as

2R44

R+

R4

R

2

+K

R2 +

3

4β2 = −X

p +

1

2ψ 2

4

, (7)

3

R4

R

2

+ 3K

R2 −

3

4β2 = X

ρ +

1

2ψ 2

4

, (8)

ψ44 − 3ψ4

R4

R = 0, (9)

p = p − 3ηH, (10)

where

H =R4

R, (11)

is the Hubble’s parameter and suffix (4) indicates the differ-

entiation with respect to t .

3 Solution of the field equations

We solve the field equations (7)–(9) by using the special law

of variation for Hubble’s parameter, proposed by Bermann

(1983) as

H = DR−m (12)

where D and m ( = 0) are constants.From (11) and (12), we obtained

R(t) =

m(Dt + C) 1

m , (13)

where C is the constant of integration.

Using (13), (9) yields

ψ(t) =a0

4

m(Dt + C)

4m + a1, (14)

where a0 and a1 are constants of integration.

Using (13) and (14) in the field equations (7)–(9), we get

ρ = 3D2

{m(Dt + C)}2 + 3A(t) − B(t) − 3

4β2, (15)

p = −

D2(3 − 2m)

{m(Dt + C)}2 + A(t) + B(t) +

3

4β2

, (16)

where

A(t) =K

{m(Dt + C)}2m

and B(t) =a2

0

2{m(Dt + C)}

6m .

Now using the borotropic equation of state

p = (γ − 1)ρ, 0 ≤ γ ≤ 2. (17)

In (12), we obtain the physical quantities p and η as

p = (γ − 1)

3D2

{m(Dt + C)}2 + 3A(t) − B(t) −

3

4β2

(18)

and

η =1

3H

D2(3γ − 2m)

{m(Dt + C)}2 + (3γ − 2)A(t)

+ (2 − γ)B(t) + (2 − γ )3

4β2

. (19)

Thus using (13), the FRW model for interacting bulk-

viscous fluid and zero-mass scalar-fields in Lyra geometry

can be written as

ds 2 = dT 2 − (mT )2m

dr 2

1 − Kr 2 + r 2dθ 2 + r2 sin2 θ dϕ2

.

(20)

This model has no initial singularity and represents expand-

ing universe.



Astrophys Space Sci (2009) 323: 87–90 89

Physical models

Here we discuss three physical models corresponding to γ =

0, 2, 43

of the equation of state given by (17).

Case (I) False vacuum model (i.e. γ = 0)

For γ = 0 , we have p + ρ = 0 which represents false vac-

uum or degenerate vacuum or ‘ρ vacuum’ (Cho 1992). The

physical significance of this fluid in non-viscous case has

been studied by Mohanty et al. (1989).

In this case the physical quantities take the explicit form:

ρ = −p =3D2

{m(Dt + C)}2 + 3A(t) − B(t) −

3

4β2 (21)

and

η =1

3H

D2(6 − 2m)

{m(Dt + C)}2 + 4A(t)

. (22)

Case (II) Zel’dovich-fluid model (i.e. γ = 2)

For γ = 2, we have ρ = p which represents Zel’dovich fluid

distribution and we get

ρ = p =3D2

{m(Dt + C)}2 + 3A(t) − B(t) −

3

4β2, (23)

η =1

3H

D2(6 − 2m)

{m(Dt + C)}2 + 4A(t)

. (24)

Case (III) Radiating model (i.e. γ = 43

)

For γ = 43

, we have ρ = 3p, which represents disordered ra-

diation and the physical quantities in this case take the form:

ρ =3D2

{m(Dt + C)}2 + 3A(t) − B(t) −

3

4β2, (25)

p =1

3

3D2

{m(Dt + C)}2 + 3A(t) − B(t) −

3

4β2

(26)

and

η =1

3H

D2(4 − 2m)

{m(Dt + C)}

2 + 2A(t) +

2

3

B(t) +1

2

β2

. (27)

4 Models with power law relation

If we assume that the scale factor R(t) is taken to be simple

power law function of time as

R(t) = n0t n, (28)

where n0 and n are constants.

Equation (28) gives

R4

R=

n

t and

R44

R=

n(n − 1)

t 2 . (29)

With the help of (28), the field equation (12) gives

ψ(t) =b0t 3n+1

3n + 1

+ b, (30)

where b is a constant of integration.

With the help of (29), we obtain the equation for the ef-

fective pressure p and energy density ρ from the field equa-

tions (10) and (11) as

p = −

3n2

t 2 −

2n

t 2 + L(t) +

3

4β2 + M(t)

(31)

and

ρ =

3n2

t 2 −

3

4 β

2

+ 3L(t) − M(t), (32)

where

L(t) =K

n20

t −2n and M(t) =b2

0

2t 6n.

Using the borotropic equation of state (17), (5) and (32)

yields the expression for the pressure p and bulk-viscosity

coefficient η as

p = 3(γ − 1)n2

t 2

−3

4

β2 + 3L(t) − M(t) (33)

and

η =1

3n

n

t (3nγ − 2) +

3

4β2(2 − γ ) + tL(t)(3γ − 2)

+ tM(t)(2 − γ )

. (34)

Thus FRW model for bulk-viscous fluid with zero-mass

scalar field in Lyra geometry can be written as

ds 2 = dt 2 − t 2n dr2

1 − Kr2 + r2dθ 2 + r2 sin2 θ dϕ 2

. (35)

It is interesting to note that the model is free from singu-

larity.

Physical models

Here we discuss physical quantities for the model γ =

0, 2, 43

.



90 Astrophys Space Sci (2009) 323: 87–90

Case (I) False vacuum model (i.e. γ = 0)

For γ = 0, we have the false vacuum or the degenerate vac-

uum. In this case the physical quantities take the explicit

form:

ρ = −p =3n2

t 2 −

3

4β2 + 3L(t) − M(t) (36)

and

η =1

3n

3

2tβ2 −

2n

t + 2tM(t) − 2tL(t)

. (37)

Case (II) Zel’dovich fluid (i.e. γ = 2)

For γ = 2, we have ρ = p which represents stiff-fluid. In

this case the physical quantities take the form:

ρ = p =3n2

t 2 −

3

4β2 + 3L(t) − M(t), (38)

η = 13n

nt

(3n − 1) + 2tL(t)

. (39)

Case (III) Radiating model (i.e. γ = 43

)

For γ = 43

, the distribution reduces to the special case with

equation of state ρ = 3p and the physical quantities in this

case take the forms

ρ =3n2

t 2 −

3

4β2 + 3L(t) − M(t), (40)

p = n2

t 2 − 1

4β2 + L(t) − 1

3M(t), (41)

η =1

3n

n

t (4n − 2) −

t

2β2 + 2tL(t) −

2

3tM(t)

. (42)

5 Conclusions

Interacting bulk-viscous fluid and zero-mass scalar fields

play a vital role in understanding the early stages of evolu-

tion of universe. Here we have found spatially homogeneous

isotropic FRW models corresponding to bulk-viscous fluid

and zero-mass scalar fields. To obtain determinate solutions

of the field equations Hubble’s law and power law for metric

potentials have been used. We have discussed the physical

models corresponding to Zel’dovich fluid, false vacuum and

radiation respectively. It is observed that the models are free

from singularities and are expanding.

Acknowledgements Author’s are grateful to Dr. T.M. Karade and

Dr. D.R.K. Reddy for fruitful discussions. The constructive comments

of the referee are gratefully acknowledged. The authors also thank

IUCCA, Pune, IIT and TIFR, Mumbai for sending preprints.

References

Bermann, M.S.: Nuovo Cimento 74B, 182 (1983)

Casana, R., Pimentel, B.M., de Melo, C.A.M.: In: Proc. Sci., WC2004,

pp. 013/1–013/7 (2004)

Casana, R., Pimentel, B.M., de Melo, C.A.M.: Braz. J. Phys. 35(4B),

1151 (2005)

Casana, R., Pimentel, B.M., de Melo, C.A.M.: Astrophys. Space Sci.

305, 125 (2006)

Casana, R., Pimentel, B.M., de Melo, C.A.M.: Class. Quantum Gravity

24, 723 (2007)

Cho, Y.M.: Phys. Rev. Lett. 68, 3133 (1992)

Halford, W.D.: J. Math. Phys. 13, 1699 (1972)

Lyra, G.: Math. Z. 54, 52 (1951)

Mohanty, G., Pattanaik, R.R.: Astrophys. Space Sci. 161, 253 (1989)

Pradhan, A., Pandey, H.R.: arXiv:gr-qc/0307038v1 (2003)

Sen, D.K.: Z. Phys. 149, 311 (1957)

Sen, D.K., Dunn, K.A.: J. Math. Phys. 12, 578 (1971)

Weyl, H.: Sber. Preuss. Acad. Wiss. 465 (1918)

http://arxiv.org/abs/arXiv:gr-qc/0307038v1

http://arxiv.org/abs/arXiv:gr-qc/0307038v1

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