2020/07/28 by Judit X. Madarász, Madarász, Judit X., Mike Stannett +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #History and Theory of Mathematics #Mathematics and Applications #math-ph #math.LO #math.MG #math.MP
paper · pdf · doi:10.48550/arxiv.2007.14261
published as Journal of Applied Logics - IfCoLog Journal 8(3) pp. 809-876 (2021) · 51 pages, 20 figures
arxiv created 2020/07/28 · openalex publication_date 2020/07/28 · openalex created_date 2025/10/10 · arxiv updated 2026/07/30 · openalex updated_date 2026/07/31
Given any Euclidean ordered field, Q, and any 'reasonable' group, G, of (1+3)-dimensional spacetime symmetries, we show how to construct a model MG of kinematics for which the set W of worldview transformations between inertial observers satisfies W=G. This holds in particular for all relevant subgroups of Gal, cPoi, and cEucl (the groups of Galilean, Poincaré and Euclidean transformations, respectively, where c∈ Q is a model-specific parameter orresponding to the speed of light in the case of Poincaré transformations). In doing so, by an elementary geometrical proof, we demonstrate our main contribution: spatial isotropy is enough to entail that the set W of worldview transformations satisfies either W⊆ Gal, W⊆ cPoi, or W⊆ cEucl for some c>0. So assuming spatial isotropy is enough to prove that there are only 3 possible cases: either the world is classical (the worldview transformations between inertial observers are Galilean transformations); the world is relativistic (the worldview transformations are Poincaré transformations); or the world is Euclidean (which gives a nonstandard kinematical interpretation to Euclidean geometry). This result considerably extends previous results in this field, which assume a priori the (strictly stronger) special principle of relativity, while also restricting the choice of Q to the field of reals. As part of this work, we also prove the rather surprising result that, for any G containing translations and rotations fixing the time-axis t, the requirement that G be a subgroup of one of the groups Gal, cPoi or cEucl is logically equivalent to the somewhat simpler requirement that, for all g∈ G: g[t] is a line, and if g[t]=t then g is a trivial transformation (i.e. g is a linear transformation that preserves Euclidean length and fixes the time-axis setwise).