Lorentz Manifolds A Topological Study

Abstract

Lorentz manifold is the most suitable model for General Theory of Relativity. Lorentz manifold is associated with a geometric structure i.e., Lorentz metric. This metric is a solution of the Einstein Field Equations. The work presented in the thesis revolves around the Lorentz manifolds and its non-manifold topologies. Hawking defined the path topology on the Lorentz manifold. Hawking explored the topological properties and the homeomorphism group of the Lorentz manifold with the path topology. newline The topological properties of the n-dimensional Minkowski space with the non-Euclidean order topology are explored. Further, it is shown by example that there exists a compact set of the Minkowski space with the order topology which is different from that of the Euclidean case and from that of the t-topology. Unlike the t-topology, a compact set of the n-dimensional Minkowski space with the order topology may contain nonempty open set of Rn and it may not be closed and bounded unlike the Euclidean topology. newline The sufficient conditions for two loops in the n-dimensional Minkowski space with the t-topology to be non-path homotopic have been obtained. A similar result is obtained for the generalization of the t-topology to Lorentz manifolds. Moreover, it is obtained that the fundamental group of the n-dimensional Minkowski space with the t-topology contains uncountably many copies of Z. This result is extended for the generalization of the t-topology to Lorentz manifolds. newline The first singular homology group of the n-dimensional Minkowski space with the t-topology is found to contain uncountably many copies of Z. The result obtained is different from that of the Euclidean case and the technique used to obtain the result is also different from that of the Euclidean case. Moreover, it is proved that the first singular homology group of the n-dimensional Lorentz manifold with the path topology contains uncountably many copies of Z. newline It is obtained that both the first singular homology group and the fundamental group of the n-dimensional Minkowski space with the t-topology are not isomorphic. The result obtained is different from that of the Euclidean case. Similar result is proved for the generalization of the t-topology to Lorentz manifold. newline newline