Abstract

This work is concerned with the geometrical structure of flat spacetimes, especially Minkowski spacetime. In Part 1 we develop synthetic or axiomatic representations of these spacetime geometries, in analogy to classical synthetic spatial geometry. Following Hilbert, we take as primitive the relations of affine betweenness and congruence. We modify the Hilbert primitives slightly, to accommodate the distinction between space and time. Using these primitives, we give categorical axiomatizations for three classical spacetime geometries associated with classical mechanics and Maxwellian electrodynamics, and for the Minkowski geometry of special relativity. Quite simple proofs of categoricity are obtained by appealing to previous work in foundations of geometry in the Hilbert tradition. We also give a direct synthetic development of the elements of Minkowski geometry from our axioms, in the manner of classical synthetic geometry. On the basis of our axiomatic representation of Minkowski geometry, we formulate a physical interpretation which is intended to give explicitly the physical content of the geometry when considered as a physical theory. ;In Part 2, we compare our formulation of Minkowski geometry with others known in the literature, in both physical and formal respects. The principal alternative formulations are the COORDINATE formulation, used in practical physics, and the various CAUSAL formulations of Robb, Reichenbach, and others. We also consider a representation due to Weyl, related to ours, which has received insufficient attention in the literature. The primary advantages which we claim for our approach are: As against the coordinate formulations, our formulation avoids the arbitrariness of coordinates, and the misleading concept of an "observe" as part of the theoretical basis for spacetime geometry. Our formulation shows explicitly that the entire physical content of Minkowski geometry may be expressed purely in terms of the physical properties and relations of physical objects. As against the causal formulations, our axiomatization has a significantly simpler formal structure; and our physical interpretation corresponds much more closely to the actual empirical basis for a theory of physical spacetime geometry, than does that of the causal formulations