Madarász, J.X., Stannett, M. orcid.org/0000-0002-2794-8614 and Székely, G. (2022) Groups of worldview transformations implied by Einstein’s special principle of relativity over arbitrary ordered fields. The Review of Symbolic Logic, 15 (2). pp. 334-361. ISSN 1755-0203
Abstract
In 1978, Yu. F. Borisov presented an axiom system using a few basic assumptions and four explicit axioms, the fourth being a formulation of the relativity principle; and he demonstrated that this axiom system had (up to choice of units) only two models: a relativistic one in which worldview transformations are Poincare transformations and a classical one in which they are Galilean. In this paper, we reformulate Borisov’s original four axioms within an intuitively simple, but strictly formal, first-order logic framework, and convert his basic background assumptions into explicit axioms. Instead of assuming that the structure of physical quantities is the field of real numbers, we assume only that they form an ordered field. This allows us to investigate how Borisov’s theorem depends on the structure of quantities.
We demonstrate (as our main contribution) how to construct Euclidean, Galilean, and Poincare models of Borisov’s axiom system over every non-Archimedean field. We also demonstrate the existence of an infinite descending chain of models and transformation groups in each of these three cases, something that is not possible over Archimedean fields.
As an application, we note that there is a model of Borisov’s axioms that satisfies the relativity principle, and in which the worldview transformations are Euclidean isometries. Over the field of reals it is easy to eliminate this model using natural axioms concerning time’s arrow and the absence of instantaneous motion. In the case of non-Archimedean fields, however, the Euclidean isometries appear intrinsically as worldview transformations in models of Borisov’s axioms and neither the assumption of time’s arrow, nor the rejection of instantaneous motion, can eliminate them.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 Association for Symbolic Logic. This is an author produced version of a paper subsequently published in Review of Symbolic Logic. Article available under the terms of the CC-BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). |
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 The Royal Society IE110369 London Mathematical Society 41508 |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 07 Apr 2021 13:07 |
Last Modified: | 26 May 2022 12:10 |
Status: | Published |
Publisher: | Cambridge University Press (CUP) |
Refereed: | Yes |
Identification Number: | 10.1017/s1755020321000149 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:172887 |