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Forthcoming papers


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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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