On the metric structure of space-time

TL;DR

This paper explores the foundational assumptions necessary to derive the metric structure of space-time, proposing a shift from fixed Lorentzian manifolds to a more flexible 'event manifold' framework influenced by general relativity.

Contribution

It introduces the concept of 'event manifolds' as a weaker alternative to Lorentzian manifolds and emphasizes the importance of variable physical structures in space-time.

Findings

Distinguishing Lorentzian from other manifolds requires the idea of variable physical structure.

Combines axiomatic and Weyl's 'Raumproblem' ideas to analyze space-time structure.

Proposes 'event manifold' as a foundational concept in space-time geometry.

Abstract

I present an analysis of the physical assumptions needed to obtain the metric structure of space-time. For this purpose I combine the axiomatic approach pioneered by Robb with ideas drawn from works on Weyl's "Raumproblem". The concept of a Lorentzian manifold is replaced by the weaker concept of an "event manifold", defined in terms of volume element, causal structure and affine connection(s). Exploiting properties of its structure group, I show that distinguishing Lorentzian manifolds from other classes of event manifolds requires the key idea of general relativity: namely that the manifold's physical structure, rather than being fixed, is itself a variable.