Spacetime Singularities vs. Topologies of Zeeman-G\"obel Class

TL;DR

This paper explores various Zeeman-G"obel topologies in curved spacetimes, examining their properties, the failure of the Limit Curve Theorem, and implications for the topology's role in understanding spacetime singularities and causality.

Contribution

It reviews and analyzes multiple Zeeman-G"obel topologies, discusses the significance of the Limit Curve Theorem, and questions the necessity of certain topological conditions in spacetime physics.

Findings

Several Zeeman-G"obel topologies admit a countable basis and include causal structures.

The Limit Curve Theorem fails in these topologies unless light-cones are added to basic open sets.

The choice of topology influences the physical significance of spacetime symmetries.

Abstract

In this article we first observe that the Path topology of Hawking, King and MacCarthy is an analogue, in curved spacetimes, of a topology that was suggested by Zeeman as an alternative topology to his so-called Fine topology in Minkowski spacetime. We then review a result of a recent paper on spaces of paths and the Path topology, and see that there are at least five more topologies in the class $\mathfrak{Z}-\mathfrak{G}$ of Zeeman-G\"obel topologies which admit a countable basis, incorporate the causal and conformal structures, but the Limit Curve Theorem fails to hold. The "problem" that L.C.T. does not hold can be resolved by "adding back" the light-cones in the basic-open sets of these topologies, and create new basic open sets for new topologies. But, the main question is: do we really need the L.C.T. to hold, and why? Why is the manifold topology, under which the group of homeomorphisms of a spacetime is vast and of no physical significance (Zeeman), more preferable from an appropriate topology in the class $\mathfrak{Z}-\mathfrak{G}$ under which a homeomorphism is an isometry (G\"obel)? Since topological conditions that come as a result of a causality requirement are key in the existence of singularities in general relativity, the global topological conditions that one will supply the spacetime manifold might play an important role in describing the transition from the quantum non-local theory to a classical local theory.