ON SOME FLUID AND ELECTROGMAGNETIC DISTRIBUTIONS IN GENERAL RELATIVITY

Abstract

Einstein's field equations are second order non-linear partial differential equations. There are basically three approaches to study them: exact solutions, approximation schemes and numerical computation. The exact solutions are necessarily very special and constitute a measure of zero in the space of all solutions. However, exact solutions play an important role in generating tests for numerical codes (Centrella et al. 1986) and also for providing checks on the validity of the approximation schemes. They can also provide models for important, though oversimplified, physical situations e.g. the Schwarzschild, and Kerr black holes and the Friedman cosmological model on which almost whole of the relativistic astrophysics and astrophysical cosmology is based. Besides, they can also provide counter-examples to conjectures (Misner 1963); for a discussion of these aspects one may refer to (MacCallum 1984, 1985 a, b). Due to the nonlinearity of the Einstein's equations it is impossible to solve these equations in their full generality. The usual practice is to assume, at the outset, some symmetry e.g. Spherical, Planar, Hyperbolic or the ones described in terms of Killing vectors. In addition to this, one has to resort to some specific energy momentum tensor. The most commonly used energy momentum tensors are that characterising dust, perfect fluid, electromagnetic field, imperfect fluid and radiation field. At this point it is to be noted that the solutions where energy momentum tensor is ii generalized by way of introducing, say null radiation, electromagnetic field, heat conducting fluid etc., and which do not leave any of its trace in the metric coefficients are not important unless a physical interpretation of decomposing the energy momentum tensor and an explanation of how ambiguities in the decomposition are to be removed is provided (MacCallum 1987, Krasinski 1993). Kinnersely (1974) remarked "the study of exact solutions has acquired a rather low reputation in the past, for which there are several explanations. Most of known exact solutions describe situations which are frankly unphysical and these do have a tendency to distract attention from useful ones. But the situation is also partially the fault of us who work in the field. We toss in null currents, macroscopic neutrino fields and tachyons for the sake of greater "generality"; We seem to take delight at the invention of confusing anti-intuitive notion; and when all is done we leave our newborn wobbling on its vierbien without any visible means of interpretation". More or less similar views have been expressed by later workers. There is another aspect of the problem. We have too many solutions rather than too few solutions. Simple minded attempts to derive a new solution from natural assumptions are likely to result in yet another discovery. A physically motivated approach in the process of finding exact solutions in general relativity would start with supplying a realistic equation of state and a given set of initial conditions; then solving the set of Einstein equations would determine the dynamical evolution of the configuration. Proceeding with this physical strategy, even assuming some symmetry and that matter describes by iii an energy momentum tensor, one usually finds intractable equations requiring numerical integration. Therefore in order to find analytical solutions which (one hopes) would be physically relevant, many authors turn out to a strategy of "mathematical simplicity" which inverts the priorities of the physically motivated strategy. Additionally there are attempts to link the group of motions of flat embedding space to the internal symmetries of the elementary particle physics. 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. Thus, the theory of isometric embedding class of space is also found useful in deriving physically meaningful solutions. Thus, the present thesis entitled "On some fluid and electromagnetic distributions in general relativity" deals with the spherical, plane and hyperbolic symmetric exact interior solutions of the Einstein's field equations with physically motivated approach, taking perfect fluid, charged fluid and radiation as interior material. The work has been compiled in form of seven chapters containing the following matter: Chapter I First chapter is of introductory type and starts with a brief account of the general relativity and describing four-fold structure. The remaining part of the chapter contains brief ideas about various concepts likely to be used in forthcoming chapters. Chapter II Second chapter contains two types of solutions describing non-comoving non-conformal spherical, plane and hyperbolic symmetric perfect fluid distributions with non- iv vanishing acceleration and shear. It is found that the first type of solutions can be represented in terms of elliptic functions while solutions of second type exist only for perfect fluid of embedding class one (Gupta et al. (1984)) barring for plane symmetric metric for which some new solutions have been found. Chapter III In this chapter quite a wide range of perfect fluid solutions of Einstein field equations has been derived by considering a metric in the conformally flat form admitting a 3-parametric group of isometries with 2-dimensional space-like trajectories i.e. G3 (2, S). The solutions obtained in the chapter are in conformally flat form and can be regarded as a generalization of the earlier work by others. Chapter IV A class of conformally flat radiating fluid sphere has been derived which joins smoothly to the Vaidya's radiating metric at the pressure free interface. Chapter V In this chapter the point, non-point transformation and other analytical methods are used to find solutions of Einstein's field equations for neutral and charged non-static shear-free spherically symmetric perfect fluid distribution. Chapter VI The chapter starts with the spherically symmetric metric with hypersurfaces t = constant as spheroids (Vaidya & Tikekar (1982)) and hyperboloids. twelve solutions are found and analysed numerically subject to energy conditions by taking the surface density of the order of 2x1014 gm.cm-3 and consequently the maximum mass of any such model was found to be 10.4 times the Solar Mass. The problem of solving field equations for real charge density is also considered. Chapter VII In the present chapter the problem of finding nonsingular charged analogue of Schwarzschild's interior solutions has been reduced to that of finding a monotonically decreasing function f .A generalization procedure has been utilized to generalize solutions by Guilofyle (1999). Recently found solutions by Gupta, et al. (2005) are generalized by taking particular form of f and the resulting solutions are seen to have higher mass and stability. The maximum mass is found to be 1.6 times the Solar Mass.