Madarász, Judit and Stannett, Mike and Székely, Gergely (2021) GROUPS OF WORLDVIEW TRANSFORMATIONS IMPLIED BY ISOTROPY OF SPACE. JOURNAL OF APPLIED LOGICS-IFCOLOG JOURNAL OF LOGICS AND THEIR APPLICATIONS, 8 (3). pp. 809-876. ISSN 2055-3706
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Abstract
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, Poincare and Euclidean transformations, respectively, where c is an element of Q is a model-specific parameter corresponding to the speed of light in the case of Poincare 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 subset of Gal, W subset of cPoi, or W subset of 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 Poincare 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 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 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 is an element of 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).
Item Type: | Article |
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Additional Information: | Export Date: 26 January 2022 |
Subjects: | Q Science / természettudomány > QA Mathematics / matematika |
SWORD Depositor: | MTMT SWORD |
Depositing User: | MTMT SWORD |
Date Deposited: | 07 Feb 2022 16:34 |
Last Modified: | 27 Apr 2023 07:38 |
URI: | http://real.mtak.hu/id/eprint/137551 |
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