Some Topological Aspects of Spacetime Metrics

Abstract

Minkowski space and Lorentz manifold are the well-known mathematical models of spacetimes propounded in the special theory of relativity and the general theory of relativity, respectively. Some non-manifold physically relevant topologies on these spacetimes have been introduced by Zeeman using Lorentz metric. In the past several decades, such topologies have attracted a great deal of research in mathematical physics. newlineThe present thesis is focused on the study of the fundamental group, leading to the first singular homology group of these spacetimes in the context of some non-manifold physically relevant topologies, namely time topology, fine topology, geodesic topology and zeeman topology. These topological notions of spacetimes are relevant in various areas including particle physics, cosmology, sensor networks, robotics, molecular biology and solid-state physics. newlineIn this thesis, it has been obtained that the fundamental group of Minkowski space with the time and the fine topologies and Lorentz manifold with their respective generalizations, contains uncountably many subgroups isomorphic to the additive group of integers. Study of its commutator subgroup and the corresponding quotient group leads to a similar result for the first singular homology group. Furthermore, both the groups have been obtained to be non-isomorphic as one is found to be abelian and the other to be non-abelian. newlineIt has been observed in the present work that changing the topology is not merely a matter of transplanting familiar ideas into a new setting. When the topology undergoes transformation, techniques are drastically altered, necessitating meticulous analysis and profound insight to yield meaningful results. Consequently, each shift in topology introduces a distinct set of challenges, continually elevating the complexity of our explorations. newline newline