A System of Axioms for 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formalization in first-order logic of the standard account of Minkowski spacetime in [Maudlin 2012] and [Malament, unpublished].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.