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Madarasz, J.X., Stannett, M. orcid.org/0000-0002-2794-8614 and Szekely, G. (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
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 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 corresponding 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 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 ∈ 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).
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 The Author(s). |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Funding Information: | Funder Grant number LONDON MATHEMATICAL SOCIETY 41508 |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 02 Nov 2022 17:06 |
Last Modified: | 02 Nov 2022 17:06 |
Published Version: | https://www.collegepublications.co.uk/ifcolog/?000... |
Status: | Published |
Publisher: | College Publications |
Refereed: | Yes |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:192396 |
Available Versions of this Item
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Groups of worldview transformations implied by isotropy of space. (deposited 06 Aug 2020 08:26)
- Groups of worldview transformations implied by isotropy of space. (deposited 02 Nov 2022 17:06) [Currently Displayed]