Definable coordinate geometries over fields, part 2: applications

TL;DR

This paper applies a previous result on the relationship between geometries and their automorphism groups to analyze and compare various classical and relativistic geometries and spacetimes using a formal logical framework.

Contribution

It demonstrates how to use automorphism groups to quickly determine relationships between different geometries and spacetimes, extending the previous theoretical results to practical applications.

Findings

Relationships between geometries are determined by their automorphism groups.

Applied the method to ordered affine, Euclidean, Galilean, Newtonian, and relativistic spacetimes.

Identified open problems regarding intermediate geometries.

Abstract

In Part 1 of this study we showed, for a wide range of geometries, that the relationships between their concept-sets are fully determined by those between their (affine) automorphism groups. In this (self-contained) part, we show how this result can be applied to quickly determine relationships and differences between various geometries and spacetimes, including ordered affine, Euclidean, Galilean, Newtonian, Late Classical, Relativistic and Minkowski spacetimes (we first define these spacetimes and geometries using a Tarskian first-order language centred on the ternary relation $\mathsf{Bw}$ of betweenness). We conclude with a selection of open problems related to the existence of certain intermediate geometries.