Static Fluid Sphere with Pressure Equal to Energy Density

INTRODUCTION

Solution with a simple equation of state have been found in various cases, e.g. for = P (Letelier[6], Letelier and Tabenaky [7], + 3p = constant = 3p (Klein [5], Singh and Abdussattar [8], (Buchdahl and Land [3], and for (1a)pap(Buchdahl [2]). But if one takes e.g. polytropic flud sphere 11npa(Klein [4]. Topper [11]. Buchdahl [1]) or a mixture of ideal gas and radiation (Suhonen [10]), one soon has to use numerical method. Singh and Yadav [9] have also studied the static fluid sphere with the equation of state p = (i.e. Zeldowich fluid). Further study in this line has been done by Yadav and saini [12] which is more general than one due to Singh and Yadav [9]. In this paper, we have obtained an exact, static spherically symmetric solution of Einstein‘s field equation for the perfect fluid with p = (i.e. stiff matter). To overcome the difficulty of infinite density at the centre, it is assumed that the distribution has a copy of redius r0 and constant density 0 which is surrounded by the fluid with pressure equal to energy density.

We take the static spherically symmetric metric in the form where and are functions of r only. The field equations. For the element (2.1) are Where a prime denotes differentiation with respect to r. Throughout the investigation we set velocity of light C and gravitational constant K to be unity. A zeldovich fluid can be regarded as a perfect fluid having the energy momentum tensor. Characterized by the equation of state We use commoving co-ordinates so that The non – vanishing components of the energy momentum tensor are

Using equation (2.7), (2.8) and (2.10), we have From (2.11) we see that if is known can be obtained. So we choose where A is constant. Using (2.12) equation (2.11) takes the form Putting Zethe equation (2.13) is reduced to which is a linear differential equation whose solution is Therefore we get where c is an integration constant. Consequently the metric (2.1) can be put into the form If we set the integration constant c = 0 then absorbing the constants A and

2

3in co-ordinate differentials dt and dr respectively, we get. Also for the metric (2.18) the fluid velocity ui is given by The scalar of expansion 1u;iis identically zero. The non-vanishing components of the tensor of rotation wij defined by are The components of the shear tensor ijdefined by With the projection tensor are The other components being zero.

Pressure and density for metric (2.18) are It follows from (3.1) that the density of the distribution tends to infinity as r tends to zero. In order to get rid of the singularity at r = 0 in the density we visualize that the distribution has a core of radius r0 and constant density 0. The field inside the core is given by the Schwarzschild interial solution. where A and B are constants and The constants appearing in the solution can be evaluated by the continuity conditions for the metric (2.18) and (3.2) at the boundary r = r0

In this paper the equation of state for the fluid has been taken as p = which describes several important cases, e.g. radiation, relativistic degenerated Fermal gas and probably very dense baryon matter Further if the fluid satisfies the equation of stae p = and if in addition its motion is irrotational, then such a source has the same stress energy tensor as that of a massless field.

1. Buchdahl, H.a. (1964); Astrophys. J. 140, pp. 1512 2. Buchdahi, H.A. (1967) ;Astrophys. J. 147, pp. 310. 3. Buchhahl, H.A. and Land, W.J. (1968) ; J. austar. Math. Sco., 8, pp. 6. 5. Klein, O. (1955); Ark. Fys. 7, pp. 487. 6. Letalier, P.S. (1975); J. Math. Phys., 16, pp. 1488. 7. Letalier, P.S. and Tabensky, R.R. (1975); Nuovo Cimento, 328, pp. 407. 8. Singh, K.P. and abdussattar (1973); Indian Jour. Pure and appl. Math. 4, pp. 468. 9. Singh, T. and Yadav, R.B.S. (1981); Jour. Math. Phys. Sci., 15(3), pp. 283. 10. Suhonen, E. (1968); Kgl. Danske Vidensksels., Math. Fys. Medd., 36, p. 1. 11. Tooper, R.F. (1964) ;Astrophys. J., 140, pp. 434. 12. Yadav, R.B.S. and Saini, SL (1991); Astrophys. And Space sci. 186. Pp. 331-336.

Research Scholar, P.G. Department of Mathematics, Magadh University, Bodh-Gaya