Role of Relativistic Charged Perfect Fluid in Bianchi type-III Space-time in Brans-Dicke theory of gravitation

In this paper


Introduction
Brans -Dicke theory of gravitation is a well-known modified version of Einstein's theory.Brans and Dicke [1] formulated a theory of gravitation in which, besides a gravitational part, a dynamical scalar field is introduced to account for variable gravitational constant and to incorporate Mach's Principle in Einstein's theory.In this theory, the scalar field has the dimension of the universe of a gravitational constant and its role is confined to its effect on gravitational field equations.Brans-Dicke scalar-tensor theory of gravitation is quite important in view of the fact that scalar fields play a vital role in inflationary cosmology.There has been a renewed interest in gravitational constants in recent years."The new inflationary models [2], the potential problem of "graceful exit" [3] and extended chaotic inflation [4] are based on the gravity theory of Brans-Dicke.
The scalar-tensor theories have been the subject of considerable interest in the study of various cosmological models due to their relevance for the inflationary expansion of the universe and to solve many outstanding problems in cosmology.Several aspects of Brans-Dicke theory have been widely examined by many authors.Bardeen et al. [5] explored the inflationary universe models which provide a mechanism for galaxy formation by generating small scale density fluctuation in the universe, Bianchi type-I string cosmological models with and without a source-free magnetic field have been examined by Banerjee et al. [6].Johari and Desikan [7] have investigated cosmological models with constant deceleration parameter in Nordtvedt's theory.In Brans-Dicke theory of gravity, Bianchi type-III cosmological model with a negative constant deceleration parameter in presence of perfect fluid have been studied by Adhav et al. [8], Katore et al. [9] explored a plane symmetric space-time filled with dark energy models in Brans-Dicke theory, Bhoyar et al. [10] studied Bianchi type-III and Kantowski Sachs cosmological model containing a magnetic field with variable cosmological constant.Lorenz-Petzold [11], Kumar et al. [12], Pawar et al. [13], Rao et al. [14], Naidu et al. [15], Kandalkar et al. [16],Mete et al. [17],Sireesha et al. [18],Hegazy et al. [19],Trivedi et al. [20] are some of the authors who have investigated several aspects of Brans-Dicke theory of gravitation.In the presence of the magnetic field, Tikekar and Patel [21] have acquired some exact Bianchi type-III cosmological solutions of massive string.Singh and Shri Ram [22] have presented a technique to generate new exact Bianchi type-III cosmological solutions of massive strings in the presence and absence of the magnetic field.Tripathy et al. [23] and Pradhan [24] studied string cosmological models in the presence of the electromagnetic field.A detailed discussion of Brans-Dicke cosmology is given by Singh et al. [25,26] have investigated some Bianchi type-II cosmological models in Brans-Dicke theory, Shamir et al. [27] explored anisotropic dark energy Bianchi type-III cosmological models in Brans-Dicke theory of gravity, Solanke and Karade [28,29] have investigated Bianchi type-I and III universe field with perfect fluid and scalar field coupled with electromagnetic fields in theory of gravity.Katore et al. [30] have investigated Bianchi type-I dark energy cosmological model with power-law relation in Brans-Dicke theory of gravitation.Recently Jumi Bharali et al. [31] studied bulk viscous magnetized locally rotationally symmetric Bianchi type-I cosmological model in general relativity.In the context of the Brans-Dicke theory of gravitation, Nimkar et al. [32] investigated the Bianchi type-VI Cosmological model andstudiedsome observational parameters such as jerk parameter, redshift, Look-back time, Luminosity distance redshift, and angular diameter distance.
Inspired by the aforementioned study, we examined the Bianchi type-III perfect fluid cosmological model in the Brans-Dicke theory of gravitation with an electromagnetic field.We also discussed some physical and kinematical properties of the model.

The metric and field equation
We consider a spatially homogeneous Bianchi Type-III space time in the form where A , B and C are functions of t and m is a constant.
Brans-Dicke field equations for the combined scalar and tensor fields are where i j G is Einstein's tensor,  is a dimensionless coupling constant and i j T is energy momentum tensor for perfect fluid with conservation equation

Energy Momentum Sources
The energy momentum tensor for matter under discussion given by where i j p T is energy momentum tensor for perfect fluid and i j E is energy momentum tensor for electromagnetic field is given by Here the electromagnetic field tensor j i F has the expression where i A is a four potential vector.
For this, we consider the expression as )

Solution of field equations
Considering the non-vanishing component of Einstein's tensor from equation ( 3), we derive Integrating (39) with respect to t , we get where k is a constant of integration.Without loss of generality let us assume 4 k as unity so that equation (40) written as B A  .

……………… (41)
Using equations ( 41) and ( 35) to (38), we yield Since the field equations are highly nonlinear with three equations and five unknown.Therefore two extra conditions can be considered to solve the field equations.
Let us choose power law form of metric potential [30] given by The power law relation between scalar field  and the scale factor a has already been used by Johri and Desikan [7] in the context of Robertson Walker Brans-Dicke models.Thus the power law relation between  and the scale factor a is , where a is constant of proportionality.The average scale factor is given by   n t k ABC a Using equations ( 24) and ( 25), we have where k is a constant of integration.
The metric in (1), with the help of (45), can be redefined in the form

Physical and Kinematical Properties of the Model:
The physical and kinematical properties of the model (52) are obtained as follows.
For the investigated model, the pressure p and the density  are given by where 5 k is a constant of integration.
The physical quantities of observational interest in cosmology are, The spatial volume is obtained as The expansion scalar becomes The shear scalar is given by 0 The mean anisotropic parameter m A as 0 3 The mean Hubble parameter t n H  .

……………… (59)
The deceleration parameter is given by The cosmic Jerk parameter is given as, H q q q J    

Conclusion
In this paper, we have presented Bianchi Type-III charged fluid universe in Brans-Dicke theory of gravitation in presence of perfect fluid with electromagnetic field.The spatial volume V of the model increases with time showing the spatial expansion of the universe.It is observed that Hubble's parameter H vanishes with extremely large value and continues to decrease with time.The scalar expansion and the physical parameters energy density and pressure diverge at 0  t and they all vanish as t approaches to infinity.The scalar field increases with time and at 0  t , it vanishes.The recent observations of type Ia supernovae reveal that the present universe is accelerating and the value of deceleration parameter lies somewhere in the range 0 1    q . It follows that one can choose the value of n in the range 1 0   n to ensure that the derived model is consistent with observations.It is also seen that the value of the cosmic jerk parameter is positive throughout the entire history of this model.This shows that our model strongly agrees with present day observations.

where 1 k , 2 k and 3 k
are constants.