A Study Of Spacetime Topologies

Abstract

Ideas of defining non-manifold topologies on flat and curved spacetime taking into account the causal structure of spacetime, paved the way for the use of topology in general relativity. The A, f, s, fine, t and space topologies are some of the topologies defined on Minkowski space and the path topology, space topology, separating topology, Zeeman topology and geodesic topology are some of the non-manifold topologies defined on Lorentz manifolds with a reasonable homeomorphism group making these topologies physically significant and interesting for their mathematical study. The work carried out in the present thesis revolves around a topological study of various physically relevant non-manifold topologies on Lorentz manifolds. newlineIn the present thesis, it is obtained that the Minkowski space with the A-topology is Hausdorff, path connected, separable, non-first countable, non-regular, non-compact and non-simply connected. It is also found that a set is compact in Minkowski space with the A-topology if and only if it is compact in the Euclidean space and does not contain the image of a Zeno sequence. The A-topology on the Minkowski space has been generalized to Lorentz manifolds and studied its compact sets and topological properties. newlineFurther, it is obtained that the homeomorphism group of Minkowski space with the f-topology is the group generated by Lorentz transformations together with translations and dilatations. It is obtained that the nonempty open sets of different dimensional Minkowski spaces with each of the fine topology, A-topology, s-topology, space topology and f-topology are not homeomorphic. These results are obtained for the Zeeman topology. This leads to the introduction of locally Minkowskian manifolds in the context of the fine and t-topologies. newlineExponential map has been obtained to be non-continuous for path, geodesic, space, separating and Zeeman topologies. Topological properties of a Lorentz manifold with each of the geodesic and Zeeman topologies have also been studied and obtained. newline