Abstract
We investigate the question: what structures of numbers (as physical quantities) are suitable to be used in special relativity? The answer to this question depends strongly on the auxiliary assumptions we add to the basic assumptions of special relativity. We show that there is a natural axiom system of special relativity which can be modeled even over the field of rational numbers.
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Notes
That our theory is two-sorted means only that there are two types of basic objects (bodies and quantities) as opposed to, e.g., Zermelo–Fraenkel set theory where there is only one type of basic objects (sets).
That is, if m is an inertial observer, there is a is a positive quantity c m such that for all coordinate points \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) there is a light signal p coordinatized at \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) by observer m if and only if equation \(\mathsf{space}^{2}(\bar{\mathbf{x}},\bar{\mathbf{y}})= c_{m}^{2}\cdot\mathsf{time}(\bar{\mathbf{x}},\bar{\mathbf{y}})^{2}\) holds.
The supremum property (i.e., that every nonempty and bounded subset of the numbers has a least upper bound) implies the Archimedean property. So if we want to get ourselves free from the Archimedean property, we have to leave this one, too.
That is, for every vector \(\bar{\mathbf{v}}=\langle v_{1},\ldots,v_{d}\rangle\) determining a spatial direction and a slower than light speed, there is another vector \(\bar{\mathbf{w}}\) in the ε-neighborhood of \(\bar {\mathbf{v}}\), such that, if points \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) are on a line parallel to \(\bar{\mathbf{w}}\) (i.e., on a line corresponding to a uniform motion with the speed and in the direction determined by \(\bar{\mathbf{w}}\)), then there is an inertial observer k moving through \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\), see Fig. 1.
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This research is supported by the Hungarian Scientific Research Fund for basic research grants No. T81188 and No. PD84093, as well as by a Bolyai grant for J.X. Madarász.
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Madarász, J.X., Székely, G. Special Relativity over the Field of Rational Numbers. Int J Theor Phys 52, 1706–1718 (2013). https://doi.org/10.1007/s10773-013-1492-8
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DOI: https://doi.org/10.1007/s10773-013-1492-8