SOME SPHERICALLY, PLANAR AND CYLINDRICALLY SYMMETRIC FLUID DISTRIBUTIONS IN GENERAL THEORY OF RELATIVITY

Abstract

Since the inception of Einstein field equations of general relativity the efforts are on to solve these for various physical situations like fluid distributions, electromagnetic distributions, vacuum fields etc. Still, not very large number of solutions could be derived to fulfil the requirements due to highly nonlinear character of the field equations. The numerical and perturbation techniques of finding the solutions may turn out to be unreliable. However, exact solutions have played a most important role in the study of various physical phenomena and may help in checking the validity of approximation techniques by comparing the exact solutions with the approximate results. The present thesis deals mainly with the exact solutions of the Einstein field equations describing isotropic (perfect) and anisotropic fluid distributions with spherical, hyperbolic, planar, and cylindrical symmetries and contains six chapters of which the first chapter is introductory. The main contents of each chapter are furnished in the following text. In chapter 1, the basic concepts 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 metrics possessing spherical, hyperbolic, plane and cylindrical symmetries are analysed and classified. A brief account of embedding class along with the material distributions is mentioned. In addition to this the similarity transformation method for solving the ordinary and partial differential equations is discussed briefly. Lastly, the chapter ends with the summary of the main results embodied in the thesis. Chapter 2 deals with the fluid distributions of embedding class one referred to spherical and allied symmetries. A space-time metric expressible in 5-flat form is subjected to perfect fluid distributions. In this process around sixteen new solutions have been derived and analysed subject to reality conditions. All of the said solutions are accelerating and possess non-vanishing Weyl tensor. Beside these, the well known results are the static Kohler-Chao solution, Zeldovich fluid ( p=p) and Tolman's radiation ( p=3p). Chapter 3 is based upon the similarity transformation method for solving differential equations, which has been used to solve the ordinary and partial differential equations pertaining to the problem of non-conformallyflat relativistic fluid distributions of embedding class one. All the similarity solutions with spherical and hyperbolic symmetries are seen to follow the equation of state energy density equal to pressure (Zeldovich fluid). However, the plane symmetric case yields different type of solutions. All the solutions so obtained are new with regard to their origin or/and orientation. In chapter 4, we have discussed the cylindrically symmetric perfect fluid distributions with reference to kinematical considerations. The space-time metric used is that of Einstein-Rosen's and coordinates are taken to be comoving. The cases dealt, are classified under the headings; (i) shear free, (ii) shear and expansion free, (iii) acceleration free, (iv) expansion free, and (v) with shear, acceleration and expansion. Some important subclasses of the solutions are presented with equation of state p = a p. As far as author is aware the solutions has never appeared earlier in this context (kinematical considerations). Chapter 5 is supplementary to the earlier chapter and deals with the plane symmetric perfect fluid distributions described by a space-time metric with separable metric coefficients. The case with equation of state p = f(p) reduces to the case p = a p 13. All the solutions with 13 = 0 have been obtained. The condition a = 1 and f3 # 0 corresponds to unrealistic situation p < p. However, the later can be considered as plane symmetric analogue of Wesson's spherical symmetric solutions. The chapter ends with the discussion of non-conformally flat class one solutions. As a result, two types offluid solutions, viz. Zeldovich fluid, and radiation (p=3p) is the major outcome. The latter describes the conformally flat perfect fluids. In chapter 6 , which is also the concluding chapter of the thesis, we have derived the exact solutions assuming spherically symmetric anisotropic and isotropic (perfect) fluid distributions considering the curvature and isotropic coordinates, respectively. The former deals with some of the important particular cases of Florides' general solution with radial stress zero. The cases considered, satisfy one of the properties; (i) embedding class one, (ii) conformally flatness, and (iii) gaseous state. The solutions are carried out with or without cosomological constant (A) and joined continuously to the Schwarzschild exterior solution. In the case of isotropic fluid spheres, around six new solutions are worked out in isotropic coordinate system. The solutions thus obtained are analysed subject to the reality conditions