GRAVITATIONAL SIGNIFICANCE OF THE PLANE SYMMETRIC SPACE-TIME

Abstract

In the first chapter which is introductory, the basic concepts^ mathematical structure of general relativity (3)82. explained. The problem of imbedding is briefly stated. The possibility of negative masses and negative density is dis cussed. A brief summary of the main results obtained in the thesis is given. In the second chapter entitled 'Imbedding class of the Plane Symmetric Space-time' the problem of local isommetric imbedding of the Riemannian four fold, (1.1) ds2 =-A2dx2-S2(du2+u2d(J)2)+C2dt2+2Ddxdt, where A,S,C, and D are functions of x and t, is examined thoroughly. A set of necessary and sufficient condition for the space-time (1.1) to be of class-1 has been obtained. The summary of the results has been published in GRG Vol.8, No.2, page 147-15& (1977). It is shown that the plane symmetric space-time under investigation is at the most of class-3 and a set of sufficient conditions for this to be of class-2 has been obtained. We have obtained a para bolic imbedding of the plane symmetric space-time analogous to the one obtained by Fujitani et al.(196l) in connection with spherical symmetry. It is worthwhile mentioning here that elliptic and hyperbolic imbeddings have already been obtained by Goyal (1974). The summary of the results obtained in this investigation were presented as a paper at the -11- Einstein's Centenary Symposium held at Physical Research Laboratory Ahmedabad (1979). In the third chapter with the title 'The Plane Symmetric Gravitational Fields' a space-time metric involving two arbitrary functions has been derived by solving the field equations Ri . =0. The metric has been expressed in six different forms and the connecting transformation equa tions are given with a view to comparing the plane symmetric metrics with analogous spherically symmetric cases. There is one form out of these six which describes the metric of the space-time against a flat background in the same manner in which the Eddington form of the Schwarzschild exterior field describes the gravitational field of a mass m, located at the centre. By means of a Lorentz-type transforma tion introduced in the local Minkowskian background, it has been possible to demonstrate an interesting connection of the plane symmetric gravitational field with the spheri cally symmetric one through a limiting process. In this limiting process a catastrophic change takes place both in the metrical as well as topological structure of the spacetime manifold, the catastrophy being reflected in the breakdown of the transformation equations. The gravitational situation so obtained is interpreted as arising due to the gravitating source attaining the speed of light. The physical feature of the field are studied by means of geodesic equations. The elliptic, hyperbolic and parabolic imbeddings of this space-time are also given explicitly. These -111- imbeddings also manifest the complexity of the curvature of the space-time when compared with analogous situation in spherical symmetry. Geodesic equations have been used for examining the motion of test particles in the field. In the fourth chapter with the title 'Parabolic Analogue of Some Well-known Algebraically Special SoJuiicms' ,a method has been evolved to obtain a plane symmetric analogue of the Kerr m&tric This corresponds to the gravitational field of a rotating object moving with the fundamental velocity against the local Minkowskian background. The time-like geodesies have been studied with a view to investigating the motion of a test particle in these fields. A parabolic analogue of the well-known Taub-NUT space has also been derived. The gravitational metric describing the field due to a uniformly accelerated mass has also been considered and its parabolic analogue has been obtained. The fifth chapter entitled 'Plane Symmetric Perfect Fluid Distributions' contains investigations and results obtained by solving Einstein's field equations with perfect energy momentum tensor as source term. It has been shown that the plane symmetric perfect fluid distributions of imbedding class-1 can be of two types. In one type the space time is conformal to the flat space while the other type is characterised by non-vanishing Wcyl conformal curvature tensor. When the density p and pressure p satisfy the relat ion p = 3p and the space-time is of class-1, then both the cases become identical. Many space-time metrics have been -ivobtained which describe incoherent matter, tachyon dust or null fluids. A geometrical and physical distinction has been brought out between various perfect fluid distributions by studying their imbeddings, equations of state and behaviour of pressure and density. The sixth chapter entitled 'Plane Symmetric Electro magnetic Distributions' contains solutions of Einstein's field equations with the electromagnetic energy tensor as the source-term. It is demonstrated that such a field cannot be of class-1. Metrics describing charged null fluids have been obtained. A plane symmetric analogue of Reissner Nordstrom metric has been derived. It is shown that the case R2^23 =° loads to a null electromagnetic field. A pure gravitational field or a perfect fluid distribution is not possible when R^,0.. =0. 2323