Ketland, Jeffrey
(2023)
Axiomatization of Galilean Spacetime.
[Preprint]
Abstract
In this article, we give a second-order synthetic axiomatization $\Gal(1,3)$ for Galilean spacetime, the background spacetime of Newtonian classical mechanics. The primitive notions of this theory are the 3-place predicate of betweenness $\bet$, the 2-place predicate of simultaneity $\sim$ and a 4-place congruence predicate, written $\equiv^{\sim}$, restricted to simultaneity hypersurfaces. We define a standard coordinate structure $\G^{(1,3)}$, whose carrier set is $\R^4$, and which carries relations (on $\R^4$) corresponding to $\bet$, $\sim$ and $\equiv^{\sim}$. This is the standard model of $\Gal(1,3)$. We prove that the symmetry group of $\G^{(1,3)}$ is the (extended) Galilean group (an extension of the usual 10-parameter Galilean group, with two additional parameters for length and time scalings). We prove that each full model of $\Gal(1,3)$ is isomorphic to $\G^{(1,3)}$.
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