A system of axioms for Minkowski spacetime

TL;DR

This paper introduces a simple, formal axiomatic system for Minkowski spacetime using first-order logic, aiming to facilitate the formalization of physical theories within this geometric framework.

Contribution

It provides a streamlined set of axioms for Minkowski spacetime, balancing simplicity and formal rigor, with a representation theorem linking models to standard Minkowski space.

Findings

Axiomatic system formalizes Minkowski spacetime in first-order logic.

Models satisfying second-order continuity correspond to standard Minkowski space.

The system enables future formalization of physical theories in Minkowski spacetime.

Abstract

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in [Maudlin 2012] and [Malament, unpublished]. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of [Tarski 1959]: a predicate of betwenness and a four place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional `vector space' that can be decomposed everywhere into a spacelike hyperplane - which obeys the Euclidean axioms in [Tarski and Givant, 1999] - and an orthogonal timelike line. The length of other `vectors' are calculated according to Pythagoras' theorem. We conclude with a Representation Theorem relating models $\mathfrak{M}$ of our system $\mathcal{M}^1$ that satisfy second order continuity to the mathematical structure $\langle \mathbb{R}^{4}, \eta_{ab}\rangle$, called `Minkowski spacetime' in physics textbooks.