SINGULAR LORENTZ LIMIT OF SOME GRAVITATIONAL METRICS AND SOME PSEUDO-SPHERICALLY SYMMETRIC FLUIDS

Abstract

The Chapter 1 is introductory . The conventions and notations are explained . Some preliminary remarks are made on negative energy density, use of singular Lorentz limit and mass tensor. A brief account of the main results of investigations made is also given. In Chapter 2, we have raised the question : what happens to the gravitational field of a centrally symmetric body if the gravitating body moves with a velocity comparable to that of light ? In order to answer this question we have first expressed the usual Schwarzschild metric into the well-known Eddington form. The quasi-cartesian coor-dinates are then introduced. The resulting metric contains a Minkowskian flat from together with a term which manifests the deviation from flatness. We have then applied the usual Lorentz transformation to this metric. Here we have used the longitudinal mass variation law for the variation of the Schwarzschild mass. We then take the limit of the metric so obtained as the velocity v I. The metric which has been obtained by this limiting process has been studied in details and it has been expressed in a variety of forms. From the study , it seems plausible to assume that the Schwarzschild space-time goes into a new manifold with two sheets separated by a singular boundary. Ona component of the manifold describes Taub type plane symmetric space-time and the other describes the non-static gravitational field of Kasner. These two components are analogous to the two components of Schwarzschild manifold separated by r = 2m. We have also demonstrated the salient features of the singular Lorentz limit of Schwarzschild space-time on a Kruskal type diagram. In Chapter 3, we have first obtained and studied the time-like and null.geodesics in terms of quasi -cartesian coordinates. The Lorentz transformation together with the longitudinal mass variation law has been applied to various types of geodesics. The corresponding limits have been obtained and studied. A careful investigation of these limits leads to the conclusion that the limit of the first integrals of the geodesics in the Eddington coordinates goes into the corresponding geodesic equat-ions of the plane symmetric space-time obtained as a singular Lorentz limit of the spherically symmetric space-time. However, the singular Lorentz limit of the integral curves of spherically symmetric field do not represent the geodesic curves of plane symmetric Taub space-time. In Chapter 4, we investigate the singular Lorentz limit of Kerr- metric and its geodesics. The methodology similar to that used in Chapter 2 has been adopted and it is shown that the singular Lorentz limit of the Kerr metric exists and it is a generalization of plane symme-tric Taub metric in the same sense in which Kerr metric is a generali-zation of Schwarzschild metric. It has also been demonstrated that the singular Lorentz limit of equations of geodesics of Kerr metric leads to the geodesic equations of the limiting space-time. The geodesics of the singular Lorentz limit of the Kerr space-time have been studied in Boyer-Lindquist coordinates. These geodesics reveal the fact that the field of this spinning ,mass is repulsive in character. In Chapter 5, we have studied the geometrical and gravitational implications of the spherically symmetric space-time from the point of view of the interaction between the Gaussian curvatures of the two 2-spaces that appear in the metric. The empty space-time conditions R.. = 0 do not allow the simultaneous vanishing of the two Gaussian curvatures except in a trivial case. We have first taken the static case. In all eight space-time metrics have been derived. Similarly the non-static case leads to the eight distinct metrics. Taking into consideration the time-like and space-like character of the coordinates in these sixteen space-time metrics together, we come to the conclusion that in all ten distinct types of manifold are possible and these satisfy the empty space-time conditions R . = 0. We have studied the physical features of these space-times by means of motion of a test mass. The type of fields obtained in this way include spherically symmetric, pseudo-spherically symmetric and plane symmetric fields. Some of these fields exhibit repulsion instead of attraction to a normal test particle. In Chapter b, we have studied the geometrical and physical features of pseudo-spherically symmetric space-time of Class 1. Kasner, Allendoerfer, Karmarkar, Eddington, Fronsdal, Kruskal, Singh and Pandey, Fujitani et. al. and many others have carried out investigations which display the restriction on embedding class of a Lorentzian manifold by a gravitational situation. Considering a pseudo-spherically symmetric space-time a relation has been obtained that expresses the Gaussian curvature of a 2-fold in terms of quantities which have very clear physical and geometrical meaning in the 4-dimensional context. With the help of this relation we have investigated the possibility of fluid distributions of Class 1 and Class 2. The fluid models so obtained differ from the conventional fluids in some cases. The salient features of some models have been discussed. (iv) Based on some investigations in the thesis the following papers have been prepared : 1. Gravitational significance of speed of light and longitudinal mass , Abstracts of contributed papers , 11th International Conference on General Relativity and Gravitation , Stockholm , Sweden , July 1986 , Vol. 1, p. 267. 2 . Schwarzschild 1 s source running into a naked singularity , Abstracts (contributed papers) , XIV Conference of the Indian Association for General Relativity and Gravitation , And hra University , Visakha-patnam , Jan . 1987 , pp . 19-20 .