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arXiv · 2006.13830

Space-time structure may be topological and not geometrical

Gabriele Carcassi·Christine A. Aidala
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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.

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Gabriele Carcassi, Christine A. Aidala. 2020-06-18. Space-time structure may be topological and not geometrical. https://doi.org/10.1088/1402-4896/ab9bdb

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