Static Spherically Symmetric Charged Fluid Sphere in Einstein-Cartan Maxwell Theory

1. INTRODUCTION

Various authors have made their attempts to investigate the problem of charged fluid spheres in E-C theory. Nduka [7], Singh and Yadav [10], Prasanna [8], Kopenzynski [3, 4] and Raychaudhari [9] have considered the generalization of Maxwell‘s equations in space having torsion but this idea leads to a breakdown in the gauge invariance and charge conservation principle. However, Raychaudhari [9] and Nduka [7] have taken the equation in a form so as to mepressure the charge conservation principle. With this formulation Raychaudhari [9] has investigated the possibility of bounce in the pressure of magnetic field for Bianchi type I universe with p = 0 and p = 0. Further Singh and Yadav [10] have discussed the static charged fluid sphere in E-C theory and have found that the pressure is discontinuous at the bounding of the fluid sphere. Some other workers in this line are Krori et al. [3], Mehra and Gokhroo [6]. Suh [12], Som and Bedron [11]. Yadav and Prasad [13]. Thomas, Maurya, Pant, Patel, Ratanpal, et al. [14-21]. In the paper, we have studied the interior field of a static spherically symmetric charged fluid distribution with spin. Assuming that the spins of the individual particles compositing the fluid area all aligned along radial direction, we have obtained solutions by choosing metric potential α(r) and β(r) in different suitable forms. Pressure and density have been also calculated for the distribution and physical constants appearing in the solution have been evaluated by matching the solutions to the Reissner-Nordstrom metric at the boundary.

The Einstein Cartan Maxwell equations are

(2.1)

iiijjj1RR8T2

(2.2)

kklklkijijljilijQQQ8S

(2.3)

1/2ij1/2i1/2igFi(g)J(g)u (2.4) ijF;k0 Where Rij is the Ricci Tensor of asymmetric connection and also the energy momentum tensor tij is not symmetric, Fij is the electromagnetic field tensor, Qij is torsion tensor, Sij is spin tensor, is charge density and j is current four vector (we have set C and gravitational constant also equal to unity) Now we have se the static spherically symmetric metric (2.5) 22222222dsedtedrrdrsind Where α and β are functions of r only.

electromagnetic field respectively as (2.6) iiijjjTTE where, iiijjjT(p)uup iiYilmjjYjlm1EFFFF4 we use comoving co-ordinates so that 1234/2uuu0ue The non-vanishing components of the energy momentum tensor are 12341234TTTpandt We can then write the field equations

(2.7) 22118pEerrr

 (2.8)

28pEe2442r



(2.9) 22118Eerrr



Here following Hehl [1, 2], we have defined effective density  and effective pressure pas

(2.10)

222kandpp2k (2.11) /2kHe Here H is constant and dashes denotes differentiation with respect to r.

(2.12) 4141EFF

and

(2.13)

414141/2dF2F4Fedrr2

 (3.1)

2

22e3118p22r2442rr2r

 (3.2)

2

22e11E22442r2rr2r

 (3.3)

2

223118e4r4884r2r2r Equation (3.1) and (3.3) using (2.10) gives pressure and density as

(3.4)

22222e3118p16k22r2442rr2r

 (3.5)

222225118e16k4r4884r2r2r With three equation (2.7)- (2.9) in five variable (p, E, p, α, β) the system determinate, we require two more equations or relations. For this we choose α and β as (3.6) 2arc (3.7) 21drk (3.8) 2221kHexp{(drk)}

(3.9)

2222(arc)1116p32Hexp{(drk)}e2



222114dadr(ad)r2r

 (3.10)

2222(arc)1116p32Hexp{(drk)}e2



222113ard(ad)r2r

 (3.11)

2(arc)2222111Eed(1rd)a(1rd)2r2r

(3.12)

21(drk)4141412dF2F4r(ad)Fedrr



Also using boundary condition at r = r0, we have

(3.13) 00 (3.14)

2012(drk)0200

2MQe1rr

 (3.15)

2012(drk)002300

MQ2dre2rr

 (3.16)

20120drk(arc)2020

e11H16e322r2



2020

13ard(ad)r



In this paper we have studied the interior field of a static spherically symmetric charged fluid distribution with spin. Assuming that the spins of the individual particles compositing the fluid area all aligned along radial direction, we have obtained solutions by choosing metric potentials α(r) and β(r) in different suitable forms. Pressure and density have been also calculated for the distribution and physical constants appearing in the solution have been evaluated by matching the solutions to the Reissner-Nordstrom metric at the boundary. Further for a realistic model p > 0, p > 0 which will impose further restrictions on our shlutions.

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Research Scholar, University Department of Mathematics, Magadh University, Bodh Gaya, Bihar