Natural vs. Artificial Topologies for a Relativistic Spacetime.

Consider a set equipped with a structure . We call a natural topology , on , the topology induced by . For example, a natural topology for a metric space is a topology induced by the metric and for a linearly ordered set a natural topology should be the topology that is induced by the order . This fundamental property, for a topology to be called “natural”, has been largely ignored while studying topological properties of spacetime manifolds where is the Lorentz “metric”, and the manifold topology has been used as a natural topology, ignoring the spacetime “metric” .

In this survey we review critically candidate topologies for a relativistic spacetime manifold, we pose open questions and conjectures with the aim to establish a complete guide on the latest results in the field, and give the foundations for future discussions.

We discuss the criticism against the manifold topology, a criticism that was initiated by people like Zeeman, Goebel, Hawking-King-McCarthy and others, and we examine what should be meant by the term “natural topology” for a spacetime. Since the common criticism against spacetime topologies, other than the manifold topology, claims that there has not been established yet a physical theory to justify such topologies, we give examples of seemingly physical phenomena, under the manifold topology, which are actually purely effects depending on the choice of the topology; the Limit Curve Theorem, which is linked to singularity theorems in general relativity, and the Theorem of Gao-Wald type of “time dilation” are such examples.