Static Charged Fluid Spheres in General Relativity

INTRODUCTION

A number of authors have already studied the charged fluid distribution in equilibrium Bonnor [1], Effinger [3] and kyle and Martin [7] have considered the interior solution for a static charged sphere. As the fluid equations do not completely determine the system different solutions were obtained the Effinger [3], kyle and Martin [7] by using different conditions. Some exact static solutions of Einstein- Maxwell equations representing a charged fluid sphere were obtained by Singh and Yadav [11], Shi-Chang [10] found some conformal flat interior solutions of the Einstein- Maxwell equations for a charged stable static sphere. These solution an exact solution satisfy physical conditions inside the sphere. Xingxiang [13] obtained by satisfying matter distribution and charge distribution. The metric is regular and can be matched to the Reissner-Nordstrom metric and pressure is finite. In the limit of vanishing charge, the soluteion can reduce to the interior solution of an uncharged sphere. Buchdahal [2] has also considered some regular general relativistic charged fluid spheres. Some other cases of the interior solutions for charged fluid sphere have been presented by Whitman and Burch [12], Krori and Barua [6], Junevicous [5], Nduka [8], Sah and Chandra [9], Fulare and Sah [4]. In this paper spherically symmetric metric, we have solved Einstein-Maxwell field equations by using different assumptions on metric potential and (ie. and These solutions satisfy physical conditions. The central and boundary conditions have been also discussed. The pressure and density have been found for the distribution.

We use here the line element in the form where and are function of r only. where is the energy momentum tensor, is the charged current four vector, is the Ricci tensor and R the scalar of curvature tensor. For the system under study the energy momentum tensor splits up into two parts viz and for matter and charge respectively where with

Here p is internal pressure, and are densities of matter and charges respectively and is the velocity vector of matter. The static condition is given by The electromagnetic energy momentum tensor is given by We assume the field to be purely electrostatic, i.e. = 0 and = , where is the electrostatic potential. Thus the Einstein- Maxwell field equations are cast into the form where and By the use of equations (2.11) – (2.13), we get the expressions for p, and E as We take and of the form Where A,B,C, and K are constants and m is a positive integer (m Then equations (2.15)- (2.18) yield Now matching the solution with Reissner-Nordstrain metric at the boundary R = we have In particular if we take A= At r = 0, these results give (taking c= 0)

For physically realistic solution pressure and density should by greater than or equal to zero so, we have from equation (3.4) and (3.15). Further for From conditions (3.20) and (3.21) we have Thus the constant B lies between

1. Bonnor, W.B. (1960). Z. Physik, 160, pp. 59. 2. Buchdahl, H.A. (1979). Acta. Phys., B10, pp. 673. 3. Effinger, H.J. (1965). Z. Phys., 188, pp. 31. 4. Fulare, P. C. and Sah, A. (2018). International Journal of Astronomy and Astrophys., 8(01), 46. 5. Junevicous, G.J.G. (1976). J. Phys. , A 9, pp. 2069. 6. Krori, K. D. and Barua, J. (1975). J.Phys A Math,8, pp. 508. 7. Kyle, C. F. and Martin, A.W. (1967). II Nuovo Cim, 50, pp. 583. 8. Nduka, A. (1977). Acta. Phys., B8, 75. 9. Sah, A. and Chandra, P. (2016). International Journal of Astronomy and Astrophys. , 6, pp. 494. Phys., B9, pp. 475. 12. Whitman, P.G. and Burch, R. C. (1981). Phys. Rest, D24, 2049. 13. Xingxiang, W. (1987). G.R.G. 19, pp. 729.

Department of Mathematics, Govt. Women‘s Polytechnic, Patna, Bihar, India