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formalization in first-order logic of the standard account of Minkowski spacetime in [Maudlin 2012] and [Malament, unpublished].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 M of our system M 1 that satisfy second order continuity to the mathematical structure xR 4 , η ab y, called 'Minkowski spacetime' in physics textbooks. * We are grateful to John Burgess, Dino Calosi, Harold Hodes and Chris Wüthrich for discussion and comments on parts of this paper.

Towards Formalising Schutz' Axioms for Minkowski Spacetime in Isabelle/HOL

“…More recently, an extension of Tarski's Euclidean ideas using Goldblatt's approach to Minkowski spacetime was given by Cocco and Babic [6]. Their system is mostly formulated in first-order logic, but with a second-order continuity axiom in order to show the usual four-dimensional Minkowski spacetime is a model.…”

“…In what follows, rather than giving formal proofs for all of these results, we sketch the proofs given by Schutz and highlight interesting features of their formalisation. We refer to the Isabelle proof document 6 for the complete proof script, and the original monograph [44] for sometimes more extensive prose, when we do not reproduce it. We endeavour to present proof procedures at a comfortable level of detail.…”

Formalising Geometric Axioms for Minkowski Spacetime and Without-Loss-of-Generality Theorems

“…More recently, an extension of Tarski's Euclidean ideas to Goldblatt's approach to Minkowski spacetime was given by Cocco and Babic [3]. Their system is mostly formulated in first-order logic, but with a second-order continuity axiom in order to show the usual four-dimensional Minkowski spacetime is a model.…”

“…Since we are interested in intervals on paths, the predicate is a bit more complicated than P above, in that it takes an event-set and two events, and indicates that these two events define an interval equal to the set. 3 The interval between a and b is defined as |ab| = {x : [a x b]} ∪ {a, b}. The first assumption states that the predicate P is symmetric in its two arguments (both of which are intervals), and can be compared to the third assumption of linorder_less_wlog.…”

Complexity in the interdefinability of timelike, lightlike and spacelike relatedness of Minkowski spacetime

“…Pambuccian [13] explicitly defines collinearity and equidistance from lightlike relatedness in metric-affine Fano spaces in order to present Alexandrov-Zeeman type theorems as definability results. For a contemporary axiomatization of Minkowski spacetime pursued in the style of Tarski, see [3].…”