SOME FLUID DISTRIBUTIONS IN SPHERICAL AND ALLIED SYMMETRIES IN GENERAL THEORY OE RELATIVITY

Abstract

Eddington (1924) has pointed out the 4-dimensional space-time manifold of general relativity can be represented as a surface of 4-dimensions drawn in-a pseudo Euclidean flat space of sufficient number of dimensions. It is well known (Eisenhart, 1966) that a 4-dimensional space-time can be locally and isometrically embedded in a flat space of ten dimensions. The postulates of general relativity doesn't provide any physical meaning to higher dimensional embedding space. Therefore the purpose of the representation of 4-fold as hypersurf ace is to picture more vividly the metrical properties of the space-time. There are attempts to link the group of motions of flat embedding space to the internal symmetries of elementary particle physics. Some have utilized the higher dimensions to study the singularity of the space-time. Eddington has utilized the concept of embedding for explaining the geometry of de Sitter and Einstein Universe and called these spherical and cylindrical respectively. Besides this most of the popular cosmological models like Friedman, Robertson-Walker metric, Schwarzschild's interior solutions were found to be embeddable in 5-d flat space which has increased the thrust to investigate the space-time which are embeddable in 5-d flat space-time. Stephani, Gupta, Barnes (1967, 1971, 1974) have shown that there are two types of perfect fluid distributions, one which are conformally flat and the other which are non conformally flat. The former type of solutions are all known (Stephani 1967). Pandey, and Mishra, S., have found these solutions. Some of these starting with the 5-D flat metric, In the present thesis the same approach has been used to derive some more solutions considering the spherical and allied symmetries. Second type of solutions with non vanishing conformal curvature tensors are rarely known. One of these is the Kohler and Chao's static solution (1965) and the other are Zeldowich fluids and non static analogue of Kohler and Chao's solution (Gupta et. al. 1984). In the present thesis, the latter problems have been investigated further and some more results and solutions in spherical and related symmetries have been obtained . No real astrophysical object is composed of a perfect fluid. In most astrophysical applications it seems that perfect fluid models are adequate. Some theoretical work on realistic stellar models has suggested that superdense stellar matter may be locally anisotropic (principal stresses unequal). In the present thesis .Tolman's (1939) perfect fluid models have been generalized to produce anisotropic fluid models joining smoothly with the Schwarzschild's exterior solution and some of these have been analysed numerically. During the last decade much interest has been focussed on finding interior solutions of the Einstein's field equations corresponding to static charged spheres. Interior solutions for charged fluid spheres have been presented by Efinger (1965), Kyle and 'Martin (1967), Wilson (1967), Kramer and Neugebauer (1971) and others. Spheres of charged dust have been investigated by Bonner, W.B., and Wickramasuriya (1975) and Roy Chaudhuri, A.K. (1975). Here, in this thesis two charged models are derived, one which reduces to Whitaker's (1968) interior solution and the other which presents the charged generalization of Knutsen, Henning (1990) interior solutions. CAPTER 1 In this chapter the basic aspects regarding the physical situations in terms of geometrical properties of space-time are summarised and restrictions imposed on the coordinate system utilized to describe the physical field are mentioned. The space-time metric used to describe the various physical models have been analysed and classified in classes. The definitions -of embedding class one, Weyl tensor, perfect fluid distributions, charged fluid distribution are given. Also the important results related to above have been stated. In the end of the chapter the main results of the thesis are summarized. CHAPTER 2 In the second chapter the solutions with vanishing Weyl's conformal tensor are discussed by considering the metric in 5-flat form i.e. ds2 A dr2 (mr no2[de_ 2 2 (0) d-L21 i+Cdt2+DdU2 provided ((Cm2 - An2) / AC) - K for K 1, -1, 0 and 40) is Sine, Sinhe and 0 respectively and A, C D, m and n are real constants. Consequently, solutions in spherical, planer and hyperbolic symmetry are obtained subject to certain restrictions. CHAPTER 3 This chapter again deals with the perfect fluid distributions of class one with vanishing conformal tensor. This time the metric potentials of the metric in (2) have been expressed in terms of pressure b (--8Thp) and energy density a (iEircp). Consequently, the corresponding field equations (now in terms of a and b) are solved completely. The solution consists of an expression of 'b' in terms of 'a' 'along with a first order differential equation in 'a' . The solutions with spherical iii symmetry (K 1, n s 0) and hyperbolic symmetry (K -1) are obtained at the very first time. The plane symmetric case has already been discussed by S. Mishra (1981). CHAPTER 4 The fourth chapter includes the problems of obtaining the perfect fluid distribution of class one with non-vanishing conformal curvature tensor. The solutions with the equation of state a a b have been investigated fully and concluded that the solutions are possible only for a -1 (the result is quite general). In addition to the above, a class of solutions have been obtained in each of the spherical, plane and hyperbolic symmetry. CHAPTER 5 In the fifth chapter some static spherically symmetric anisotropic fluid distributions are considered as the generalization of the Tolman's solutions and some of these are analysed numerically. CHAPTER 6 In the last and final chapter, (i) a very general charged fluid model has been analysed numerically which can be understood as a charged generalization of Mishra's (1981) and Knutsen, Henning (1990) interior solution. The charged fluid model obtained reduces to the ordinary perfect fluid distribution due to said authors. (ii) A charged analogue of Wittaker's interior solutions has also been constructed. Both the solutions have been analysed numerically.