ON SOME PERFECT FLUID SPHERES, PLATES AND CYLINDERS IN GENERAL RELATIVITY

Abstract

Eddington (1924) has pointed out that 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-dimensinal space-time can be locally and isometrically embedded in a flat space of ten dimensions. The postulates of general relativity do not provide any physical meaning to higher dimensional embedding space. Therefore the purpose of representation of 4-fold as hypersurface 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. In connection of Sakharov's work (1967), the concept of bending of space-time in gravitational theories is dealt by considering space-time embedded in a higher- dimensional space (Regge and Teitelboim, 1975; Maia, 1989 and Tapia, 1989). Recently, due to a proposal by Randall and Sundrum (1999) and discussions by L. Anchordoqui and S. P'erez Berglia (2000), this idea has again attracted much attention. 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, Friedman—Robertson-Walker—Lemaitre spaces and Schwarzschild's interior model were seen to be embeddable in 5-d flat space, which has increased the thrust to investigate the space-time, which are embeddable in 5-flat space-time. iii Stephani (1967), Barnes (1974) and Gupta (1984) have shown that there are two types of perfect fluid distributions, one which are conformally flat and the other one are non-conformally flat. All the fluid distributions of former type are known (Stephani, 1967). Some of the workers have found these solutions. Some of these start with the 5-flat metric, (Pandey and Gupta, 1970). Second type of solutions with non-vanishing conformal curvature tensors is rarely known. One of these is the Kohler and Chao's static solution (1965) and the others are Zeldovich (1972) fluids and non-static analogue of Kohler and Chao's solution (Gupta et al. (1984); Gupta and Sharma (1996); Kumar and Gupta (2010) and Gupta et al. (2010)). In the present thesis, the latter problems have been investigated further by considering the 5-flat metric in plane and spherical symmetry. Cylindrically symmetric space-time plays an important role in the study of the universe on a scale in which anisotropy and inhomogeneity are not ignored. Inhomogeneous cylindrically symmetric cosmological models have significant contribution in understanding some essential features of the universe such as the formation of galaxies during the early stages of their evolution. Bali and Tyagi (1989), Pradhan et al. (2001, 2006) have investigated cylindrically symmetric inhomogeneous cosmological models in presence of electromagnetic field. Barrow and Kunze (1997, 1998) found a wide class of exact cylindrically symmetric flat and open inhomogeneous string universes. In their solutions all physical quantities depend on at most one space coordinate and the time. In this thesis we have investigated the simple models of nonlinear cylindrically symmetric cosmological model when the source of gravitation is a perfect fluid. Besides the above topics we have considered spherically symmetric space-time in isotropic coordinate system iii which is capable of describing various material distributions some of which are completely new. The present thesis is composed of five chapters containing the following matter: Chapter 1 First chapter is an introductory one and starts with a compact account of the general theory of relativity and then in continuation, .it contains basic definitions of the material distributions such as perfect fluid, charge, current etc. Also, it contains basic data pertaining to space-time metric with spherical and allied symmetries (including plane symmetry). The theory of isometric embedding class of a space is also mentioned along with the related important theorems and results obtained by various workers. Furthermore continuous single parameter Lie-group of transformations technique (Similarity transformations method) of solving ordinary and partial differential equations has also been discussed in adequate detail. The chapter ends with the summary of the whole work embodied in the thesis. Chapter 2 Second chapter contains the derivation of several similarity solutions of Einstein field equations for non-static non-conformally spherically symmetric perfect fluid distributions described by the space-time metric in 5-flat form. Few new and a couple of already derived solutions are obtained and analyzed subject to the reality conditions. This is first ever attempts to derive such solutions by Similarity method. Chapter 3 In this chapter, number of plane symmetric non-conformally flat perfect fluid distributions of embedding class one are derived using Lie group of transformations, (i.e. Similarity transformations method). The solutions so obtained include many new similarity solutions. These solutions thus obtained are analyzed physically. iv Chapter 4 Perfect fluid cylinders have been obtained by solving the Einstein's field equations for the material distributions by considering the Einstein-Rosen metric. The derivation of the solutions is made on the basis of the kinematical properties of the fluid distributions. The metric potential of the metric are taken in separable form. A set of solutions with expansion proportional to shear scalar is also obtained. Most of the solutions are new. Chapter 5 Last chapter includes the static spherically perfect fluid solutions represented by a static metric in isotropic coordinate system. In all, nine solutions are obtained out of which two of them are old (Schwarzschild (1916) and Nariai (1950) interior solutions), other solutions are tested for reality conditions along with the causality conditions.