Abstract
In the context of the relativistic Doppler effect [DE] and the Lorentz–Fitzgerald contraction [LF], we investigate the consequences of two abstract axioms [R] and [M] expressed in terms of an operation \({\oplus}\) generalizing the addition of velocities and a function \({L:(\lambda, v)\mapsto L(\lambda, v)}\) . The latter can represent either the Doppler effect or the Lorentz– Fitzgerald Contraction. In words, these axioms state the following: [R] iterating the function L has the same effect as adding velocities; [M] adding a velocity via the operation \({\oplus}\) preserves the order of the function L. We prove that these axioms are equivalent to each other, and also to generalized forms of the Doppler effect and the standard expression [AV] for the relativistic addition of velocities, taken jointly. We also show that if [AV] holds, then the axioms [R] and [M] are inconsistent with [LF].
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This paper is fondly dedicated to Janos Aczél, the first author’s friend and mentor for more that four decades
We are grateful to Janos Aczél, Chris Doble, Michael Kiessling, Alexey Krioukov, David Malament, Pat Suppes, and a referee for their reactions to earlier versions of the results presented here. The second author’s work is partially supported by an ARC grant from the Communauté Française de Belgique.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Falmagne, JC., Doignon, JP. Axiomatic derivation of the Doppler factor and related relativistic laws. Aequat. Math. 80, 85–99 (2010). https://doi.org/10.1007/s00010-010-0035-0
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DOI: https://doi.org/10.1007/s00010-010-0035-0