Self-Gravitating Fluid Sphere with Specified Equation of State

1. INTRODUCTION

In 916, Schwarzschild [10] considered perfect fluid spheres with homogenous density and isotopic pressure in general relativity and obtained the solutions of relativistic field equations. Tolman [16] developed a mathematical method for solving Einstein‘s field equations applied to a static fluid sphere in such a manner as to provide explicit solutions in terms of known functions. A number of new solutions were thus obtained and the properties of three of them were examined in detail. No stationary in homogeneous solutions to Einstein‘s equation for an irrotational perfect fluid have featured equations of state (Letelier [14], Letelier and Tabensky [15] and Singh and Yadav [23]). Solutions to Einsteins equation with a simple equation of state have been found in various cases, e.g. for constant (Whittaker [7]) for = 3p (Klein [12], Singh and Abdussattar [11], Feinstein and Senovilla [1], Kramer [2]), for constant (Buchdahl and Land [6], Alluntt [9]) and for (Buchdahl [44]). But if one takes, e.g. olytropic fluid sphere (Klein [12], Tooper [18], Buchdahl [5]) or a mixture of ideal gas and radiation (Suhonen [3]), one soon has to use numerical methods. Yadav and Saini [20] have also studied the static fluid sphere with equation of state (i.e. stiff matter). Davidson [25] has presented a solution that provides a non stationary analog to the static case when In the present paper, we have obtained some exact, static spherically symmetric solutions of Einstein‘s field equations for the perfect fluid with equation of state where We have also taken in one case while in second case. For different values of a and n we get many previously known solutions. To overcome the difficulty of infinite density at the centre, it is assumed that distribution has a core of radius r0 and constant density which is surrounded by the fluid with the specified equation of state.

We take the line element in the form where A and B are functions of r only. The field equations for (2.1) are [1] where a prime denotes differentiation with respect to r. The energy momentum tensor for perfect fluid is given by We choose the equation of state as where a is positive constant a[0,1] in this case we find that We use commoving co-ordinate so that The non-vanishing components of the energy momentum tensor are We can then write Using equations (2.7), (2.8) and (2.10) we get Case I : We choose A1ek(a constant) which reduces (2.11) to the form Integrating w.r.t.r, we get where k2 is a constant, Now (2.8) and (2.9) lead to k1 = 2, so that Hence the metric (2.1) can be cast into the form Absorbing the constant k2 is the co-ordinate differential dt the metric (2.15) is reduced to the form. The non-zero components of Reimann-Christoffel curvature tensor Rhijk for the metric (2.16) are Choosing the orthonormal tetrad ijA as The physical components R(abcd) of the curvature tensor defined by hijk(abcd)(a)(b)(c)(d)hijkRAAAAR Are Since a is finite +ve constant, we see that Hence it follows that the space time is asymptotically homaloidal. For the metric (2.16) the fluid velocity ui is given by The scalar of expansion ijuis identically zero. The non vanishing components of the tensor of rotation ijis defined by Are

The components of the shear tensor ijdefined by with the projection tensor Are while other components are zero. For the particular values of constant a, several previously known solutions are contained here in. When a = 1, results of this case reduce to that of Singh and Yadav [23]. Also in this case the relative mass m of a particle in the gravitational field of (2.16) is related to its proper mass m0 (Narlikar [24]) through k being a constant. As the particle moves towards the origin, m increases and r,m0 i.e. the relative mass goes on decreasing continuously. The case when a = 3 gives the distribution of disordered radiation already obtained by Singh and Abdussattar [11]. Case II : From (2.11) we see that if B is known A can be obtained. So we choose where is constant Use of (2.27) reduces the equation (2.11) to the form We put Ayeso that equation (2.28) is transformed to where E is integration constant. Therefore we get Consequently the metric (2.1) can be put into that form Absorbing the constant in co-ordinate differential dt, the metric (2.32) goes to the form The non vanishing components of Reimann-Christoffel curvature tensor Rhijk for the metric (2.33) are Choosing the orthonormal tetrad ijAas The physical components R(abcd) of the curvature tensor are We see that (abcd)0Rasr. It follows that the space-time is asymptotically homaloidal. Also the metric (2.33) the fluid velocity ui is given by The scalar of expansion i;iuis identically zero. The non vanishing components of the tensor of rotation The non-zero components of the shear tensor ijare

Pressure and density for the metric (2.33) 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. The field inside the core is given by Schwarzschild internal solution. Where AandB are constants and . And the density of the core which complete the solution for the perfect fluid core of radius r0 surrounded by considered fluid. The energy condition Tij Ui Uj > 0 and the Hawking and Penrose condition (Hawking and Penrose, 1970). Both reduces to u > 0 which is obviously satisfied. For different values of a and n, solutions obtained above in case II provide many previously known solutions. For a = 1, n = 1, we get the results due to Yadav and Saini [20]. For n = 2 and by suitable adjustment of constant we get the solution due to Singh and Yadav [23]. Also for a = 3and n = 2 we get the solution due to Yadav and Purushottom [21] and Yadav et al. [22] by suitable adjustement of constants.

In this paper the equation of state for the fluid has been taken as p=a which (for a=1) describes several important cases, e.g. relativistic degenerate Fermi gas and probably very dense baryon matter (Zeldovich and Novikon [26]; Walecka [8]. The casual limit for ideal gas has also form =p (Zeldovich and Novkove [26]). Furthermore, if the fluid satisfies the equation of state p= and if in addition its motion is irrotational, then such a source has the same stress energy tensor as that of a m assless scalar field (Tabensky and Taub [19]. Also the solution in this case can be transformed to the solution of Brans-Dicke Theory in vacuum. (Dicke [17]).

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