Bianchi Type - III Charged Fluid Universe in Brans-Dicke Theory of Gravitation

– We investigate the spatially homogeneous Bianchi Type-III space time with electromagnetic field tensor and relativistic charged perfect fluid in Brans-Dicke (B-D) theory of gravity. Solutions have been obtained by using a general approach of solving the partial differential equations. It is observed that the convergent and isotropic solution of the metric function can be derived with the components of the vector potentials


I. INTRODUCTION
In recent years there has been a lot of interest in several alternative theories of gravitation; out of which the most important among them is scalar-tensor theory of gravitation formulated by Brans-Dicke [1].This theory of gravity is one of the most competent theory due to its vast cosmological implications [2].In this theory, the scalar field has the dimensions of universe of the gravitational constant and its role is confined to its effect on gravitational field equations.This theory of gravity is mediated by a scalar field  in addition to the usual metric tensor field ij g present in Einstein's theory.Among the various modifications of general relativity, the B-D theory of gravity is well known example of a scalar tensor theory in which the gravitational interaction involves a scalar field and the metric tensor.In recent years, the study of Bianchi type models in the context of B-D theory has attracted many authors Pawar et.al [3], Sharif et.al [4], Kandalkar et.al [5], Raut et.al [6], Katore et.al [7].A detailed discussion of B-D cosmology is given by Singh et al. [8].Lorenz-Petzold [9] studied exact Bianchi type-III solutions in the presence of electromagnetic field.Bianchi type-I space-time in scalartensor theory have been investigated by Kumar et al. [10].Adhav et al. [11] studied LRS Bianchi type-II cosmological model with anisotropic dark energy, Katore et al. [12,13] explored Bianchi type-V and plane symmetric space-time filled with dark energy models in B-D theory.Bianchi type -III dark energy model in scalar tensor theory of gravitation explained by Naidu et al. [14].Adhav et al. [15] explored Bianchi type-III cosmological model with negative constant deceleration parameter in B-D theory of gravity in presence of perfect fluid.Shamir et al. [16] have studied anisotropic dark energy Bianchi type-III cosmological models in B-D theory of gravity.The Brans-Dicke field equations are given by where  is a dimensionless coupling constant.The function  is known as B-D scalar field.Karade and Solanke [17]   investigated Bianchi type-III universe field with the perfect fluid and scalar field coupled with electromagnetic fields in ) , ( T R f theory of gravity.Recently Bhoyar et al. [18] discussed the Bianchi type-III and Kantowski Sachs cosmological model containing magnetic field with variable cosmological constant.This motivates us to investigate Bianchi type-III charged fluid universe in B-D Theory of gravitation.The paper is organized as follows: Section II, deals with the derivation and solutions of the field equations.A brief summary is given is section III.

II.THE METRIC AND FIELD EQUATIONS
Here, we consider a spatially homogeneous Bianchi Type-III space time in the form , where

Electromagnetic field
The energy momentum tensor for electromagnetic field is given by , 4 Here the electromagnetic field tensor where i V is a four potential vector.
To achieve the compatibility with space time (1), we assume electromagnetic vector potential as Noting ( 4) and ( 5) we can deduce easily the following , , , , From equations ( 4), ( 5) and ( 6), we can deduce , 2 Using (3) we can deduce the components of energy momentum tensors , 2 , The stress energy tensor of a perfect fluid with density  , pressure p and four velocity i u is given by   where 1 This equation with different combination of i and j , gives following equations , 0 , 0 From the vanishing components of Einstein tensor, using equations ( 2) and (4), we deduce , 0 where D is an unknown function of t Integrating this with respect to t , we get where With the aid of equation ( 12), we can write the equation ( 10) as, , 0 From equations (15a), (15b)and (15c) ,we have Integrating with respect to t , we get , , , Now, considering the non-vanishing component of Einstein tensor, from equation ( 2), we derive Integrating (19e) with respect to t , we get where 7 k is constant.From equations (19a) and (19b), we get From equations (19b) and (19c), we get , 0 Using equations (19c) and (19a), we obtain , 0 Upon integration of (20a) and (20e), yields We can express the values of Using equations (15)

III. CONCLUSION
In this present paper, we have presented Bianchi Type-III space time with electromagnetic field tensor and relativistic charged perfect fluid in the context of Brans-Dicke theory of gravity.We have derived and solved the gravitational field equations corresponding to B-D theory.It is observed that the convergent, non-singular, isotropic solutions can be obtained along with the components of vector potential.It is also interesting to note that the investigated models are free from singularity.

jT
is energy momentum tensor for perfect fluid with conservation equation.and notations have their conventional meanings.