| Groups of Worldview Transformations Implied by Isotropy of Spac
Judit X. Madarasz, Mike Stanett and Gergely Szekely
Given any Euclidean ordered field, $Q$, and any `reasonable' group, $G$, of (1+3)-dimensional spacetime symmetries, we show how to construct a model $M_{G}$ 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, Poincar'e and Euclidean transformations, respectively, where $c in Q$ is a model-specific parameter corresponding to the speed of light in the case of Poincar'e 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 $W subseteq gal$, $W subseteq cpoi$, or $W subseteq ceucl$ 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 Poincar'e 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 $mathbb{R}$ 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 $taxis$, 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 $g in G$: $g[taxis]$ is a line, and if $g[taxis] = taxis$ then $g$ is a trivial transformation (ie $g$ is a linear transformation that preserves Euclidean length and fixes the time-axis setwise).
December 2020
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