Space-time structure may be topological and not geometrical

TL;DR

This paper explores conditions under which topological structures in physics can give rise to geometrical space-time, suggesting that at Planck scale, space-time may remain topological without geometric properties.

Contribution

It provides necessary and sufficient conditions for topological structures to produce geometrical manifolds, and discusses their potential failure at quantum gravity scales.

Findings

Conditions for topological to geometric transition identified

At Planck scale, space-time may lack geometric structure

Topological space-time may persist without geometry at quantum scales

Abstract

In a previous effort [arXiv:1708.05492] we have created a framework that explains why topological structures naturally arise within a scientific theory; namely, they capture the requirements of experimental verification. This is particularly interesting because topological structures are at the foundation of geometrical structures, which play a fundamental role within modern mathematical physics. In this paper we will show a set of necessary and sufficient conditions under which those topological structures lead to real quantities and manifolds, which are a typical requirement for geometry. These conditions will provide a physically meaningful procedure that is the physical counter-part of the use of Dedekind cuts in mathematics. We then show that those conditions are unlikely to be met at Planck scale, leading to a breakdown of the concept of ordering. This would indicate that the mathematical structures required to describe space-time at that scale, while still topological, may not be geometrical.