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Kantowski-Sachs plane symmetric models are investigated in bimetric theory of gravitation proposed by Rosen [1] in the context of bulk viscous fluid. Taking conservation law and the equation of state, two different models of the universe are obtained. It is observed that Kantowski-Sachs vacuum model obtained in first case and bulk viscous fluid model obtained in second case. It is also observed that the bulk viscous cosmological model always represents an accelerated universe and consistent with the recent observations of type-1a supernovae. Some physical and geometrical features of the viscous fluid model are studied.

General relativity established by Einstein serves as a basis for constructing mathematical models of the universe. This theory has some controversies and lapses for which various alternative and modified theories of it have been proposed by authors from time to time to unify gravitation and matter fields in various forms. Most of the cosmological models based on general relativity and its modified theories such as Barber’s second self creation theory, Einstein-Cartan, Gauge theory gravity, Brans-Dicke theory, Scalar-tensor theories, Scalar theories contain an initial singularity (the big-bang) from which the universe expands. Thus to get rid of the singularities that occur in general relativity and other theories, Rosen [

and

where

As in general relativity, the variation principle also leads to the conservation law

where (;) denotes covariant differentiation with respect to

The field equations of Rosen’s bimetric theory of gravitation are

where

and

Here the vertical bar (|) denotes the covariant differentiation with respect to

Usually the investigation of relativistic models has the energy momentum tensor of matter and generated by a perfect fluid. But to obtain more realistic models, one must consider the viscosity mechanism because the effect of bulk viscosity exhibits essential influence on the characteristic of the solution. The viscosity mechanism in cosmology has attracted the attention of many researchers as it can account for high entropy of the present universe (Weinberg [

In general relativity, the relativists are generally using various symmetries to get physically viable information from the complicated structure of the field equations. The field equations of general relativity are non-linear in nature with ten unknowns

Rosen [

Sahoo [

To the best of our knowledge no author has studied Kantowski-Sachs plane symmetric model in the context of bimetric theory of relativity, when source of the gravitational field is governed by bulk viscous fluid. Therefore, in this paper we are interested to study this problem for two different cases. The work reported in first case concludes that Kantowski-Sachs plane symmetric model does not accommodate bulk viscous fluid in bimetric theory of relativity. However, Kantowski-Sachs bulk viscous fluid model obtained in second case.

Consider the Kantowski-Sachs [

where the metric potentials A and B are functions of cosmic time “t” only.

The background flat space-time metric is

The energy momentum tensor for bulk viscous fluid distribution is given by

with

where p is the proper pressure, _{i} is the four velocity vector of the fluid and

Since the bulk viscous pressure represents only a small correction to the thermo dynamical pressure, it is reasonable assumption that the inclusion of viscous term in the energy momentum tensor does not change fundamentally the dynamics of the cosmic evolution. For the specification of

Here λ is called the adiabatic parameter.

Using comoving co-ordinate system, the field Equation (4) for the metrics (5) and (6) corresponding to the energy momentum tensor (7) can be written as

And

Equation (8) can be expressed as

Here and afterwards the suffix “4” after a field variable represents ordinary differentiation with respect to time “t” only.

Equations (10), (11) and (12) yield

Taking last two terms of Equation (14), we get

Equations (15) and (13) yield

Case-1: From the reality conditions, we have

So from Equation (15), we find

Use of (18) in Equation (9), we obtain

By help of Equation (18), equations (10) and (11) yield

On integration, (20) yields

where

Putting the values of A and B from (21), and use of Equations (18) and (19) in Equation (16), we have

Thus the metric (5) corresponding to Equations (21) & (22) takes the form

With proper choice of co-ordinates Equation (23) can be transformed to

As

It is observed from (18), (19) and (22) that

Thus the above result reduces to that of result already obtained by Sahoo [

Case-2:

With help of the conservation property (3), metric (5) takes the form

By the help of (15) Equation (26) yields

To avoid complexity in the problem substituting the relation

we have

where

Use of (28) and value of “k” from (4) in Equation (14), we get

where

Now Equation (29) can be expressed as

Integrating (30), one can obtain

where

As we have consider the relation

Thus (28) with the help of (32) yields

Now use of Equation (33) in Equation (15), we get

Putting the value of

and

Using (32), Equation (13) yields

By use of (35), (36), (37) separately in (38) and then using (34) in each case, we get

Therefore in view of (32), the line element (5) can be written in the form

The above model of the universe can be transformed through a proper choice of coordinates to the form

i. The Spatial Volume V of the Universe:

The spatial volume V of the universe is found to be

Now V → constant as t → 0 and V → 0 as t → ∞.

Thus we inferred from the results obtained above that the universe starts from a constant volume and collapse at infinite future.

ii. The Expansion Scalar θ:

The Expansion Scalar “θ” in the model is found to be

Hence as t → 0, θ → constant and as t → ∞, θ → 0.

This result shows that the model has the constant rate of expansion at initial time but as time increases the rate of expansion becomes slow and there will be no expansion at infinite future.

iii. Anisotropy of the Universe:

The shear scalar σ (Ray Choudhuri [

for the model yield

Therefore σ^{2} → a constant as t → 0 and σ^{2} → 0 as t → ∞. Thus it is inferred that the model is anisotropic at initial time but gradually approaches to isotropic as time increases. It is interesting that at infinite future the universe may turns to isotropic state. Since the universe in a smaller case is neither homogeneous, so the transition from anisotropic to isotropic state might have happened in the early universe which is not supported by any observed or experimental data. However there are theoretical arguments that sustain the existence of an anisotropic phase that approaches an isotropic case (Misner, [

received some attention (Hu & Parker, [

time and continues throughout the evolution.

iv. Hubble parameter:

The Hubble parameter H in the model is found to be

v. Scale factor:

The scale factor S^{3} in the model is found to be

vi. The deceleration parameter:

The deceleration parameter ‘q’ in the models defined by

model of the universe corresponds to an inflationary model. The model represents an accelerating universe in bimetric theory of gravitation and also consistent with the recent observations of type-Ia supernovae.

vii. Energy conditions for viscous fluid:

The strong, weak and dominant energy conditions i.e.

It is observed from above data that the strong energy conditions is satisfied in the model. The weak and dominant energy conditions are also satisfied when

viii. Bulk viscous coefficient:

The bulk viscous coefficient

and

In all the cases it is observed that as t → 0, h → ‒ve constant and as t → ∞, h → ‒∞. So it is evident from the above result that the solutions leads to unphysical situations and hence there is no singularity involved in the model.

In this paper, Kantowski-Sachs models are constructed in Rosen’s bimetric theory of gravitation when the energy momentum tensor is bulk viscous fluid. Applying the conservation equation and also the equation of state, two different models of the Kantowski-Sachs universe are obtained i.e. vacuum model and bulk viscous fluid model. It is observed that the bulk viscous cosmological model always represents an accelerated universe and also is consistent with the recent observations of type-1a supernovae. The model obtained is not of a steady state model and has no singularity. Also the model is anisotropic at initial time but approaches to isotropy at infinite future. As there is one way to avoid singularity is energy density ρ to vanish, so Rosen’s model in the context of bulk viscous fluid is only valid when the energy density ρ is not zero.

The authors thank the reverend referee for his constructive comments to bring the paper in improvement form.