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formalization in first-order logic of the standard account of Minkowski spacetime in Maudlin (2012) and Malament (unpublished).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 (The Bulletin of Symbolic Logic, 5(2), 175–214 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 ${\mathscr{M}}^{1}$ that satisfy second order continuity to the mathematical structure $\langle \mathbb {R}^{4}, \eta _{ab}\rangle$, called ‘Minkowski spacetime’ in physics textbooks.