Electric and Scalar Charged Fluid Sphere in General Relativity

1. INTRODUCTION

An interior solution of electroscalarly charged static ‗dust‘ sphere has been found by Teixeira et al. [4] and it has been shown that the geometric mass of an electroscalarly charged sphere is 1/222bmq where q is the electric charge, b the scalar charge and 1. Florides [1] has shown that the geometric mss of a charged sphere is m(a)(a), where, (a) is the contribution from the mass density and (a) is that from electric charge. So et al. [3] have found Reissner Nordstrom solution from the Schwarzschild solution by a co-ordinate transformation and have shown that the geometric mass of a charged sphere is 1/222smmq where, ms is the Schwarzschild mass and q is the charge of the sphere. Paul [2] has obtained an interior solution of electroscalarly charged fluid sphere and has shown that the geometric mass of the sphere has contribution from its mass density and scalar electric charges. In the present paper, we have investigated the interior solution of an electric and scalar charged fluid sphere in general relativity under certain assumptions.

We consider the spherically symmetric metric given by where r, and t are numbered 1, 2, 3 and 4 Einstein – Maxwell scalar field equations are where P, and q are the mass density, pressure, charge density and scalar charge density respectively, s is the scalar field. The matter is at rest in the co-ordinate system of (1) so that along radial direction where is the electric potential. The suffix 1 indicates differentiation with respect to r. equation (2.2) with the help of equation (2.1) gives

22111111132

22r



Now since there are five equations and eight variables, let us assume where A, B and C are constant. The first two assumptions of equation (2.13) ensure flatness at the centre and the third makes the scalar field zero at r = 0 Now with the help of equation (2.13) equations (2.8) to (.12) give Now we see that e, p, i.e. mass density, pressure, charge density and scalar charge density all become infinite at the centre r = 0. So there is singularity in these quantities at centre. However if we choose Then at the centre r = 0, we get Here if we assume that constant A and B are The geometric mass of the sphere is where dv (the proper elementary volume) Thus, the geometric mass of the sphere [for the choice equation (19)] is given by

Thus the mass density, electric charge and scalar charge densities contribute to the geometrical mass of the charged fluid sphere as is evident from equation (2.27).

1. Florides, P.S. (1977). The complete field of a general static spherically symmetric distribution of charge Il Nuovo Cim., 42A., pp. 343. 2. Paul, B.B. (1980). Electro-scalarly charged fluid spheres in general relativity. Indian Jour. Pure. Appl. Math. 11(8), pp. 1055. 3. Som, M.M.; Santos, N.O. and Teixeira, A.F. daF (1977). 0 Geometrical mass in the Reissner-Nordstrom solution. Phy. Rev., D16 , pp. 2417. 4. Teixeira, A.F. daF.; Wolk Idel and Som, M.M. (1976). Generalised charged static dust spheres in general relativity. J. Phys. A9, pp. 1267.

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