dorsal/arxiv
View SchemaFrom Reference Frame Relativity to Relativity of Mathematical Models : Relativity Formulas in a Variety of Non-Archimedean Setups
| Authors | Elemer E Rosinger |
|---|---|
| Categories | |
| ArXiv ID | physics/0701117 |
| URL | https://arxiv.org/abs/physics/0701117 |
Abstract
Galilean Relativity and Einstein's Special and General Relativity showed that the Laws of Physics go deeper than their representations in any given reference frame. Thus covariance, or independence of Laws of Physics with respect to changes of reference frames became a fundamental principle. So far, all of that has only been expressed within one single mathematical model, namely, the traditional one built upon the usual continuum of the field $\mathbb{R}$ of real numbers, since complex numbers, finite dimensional Euclidean spaces, or infinite dimensional Hilbert spaces, etc., are built upon the real numbers. Here, following [55], we give one example of how one can go beyond that situation and study what stays the same and what changes in the Laws of Physics, when one models them within an infinitely large variety of algebras of scalars constructed rather naturally. Specifically, it is shown that the Special Relativistic addition of velocities can naturally be considered in any of infinitely many reduced power algebras, each of them containing the usual field of real numbers and which, unlike the latter, are non-Archimedean. The nonstandard reals are but one case of such reduced power algebras, and are as well non-Archimedean. Two surprising and strange effects of such a study of the Special Relativistic addition of velocities are that one can easily go beyond the velocity of light, and rather dually, one can as easily end up frozen in immobility, with zero velocity. Both of these situations, together with many other ones, are as naturally available, as the usual one within real numbers.
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"abstract": "Galilean Relativity and Einstein\u0027s Special and General Relativity showed that\nthe Laws of Physics go deeper than their representations in any given reference\nframe. Thus covariance, or independence of Laws of Physics with respect to\nchanges of reference frames became a fundamental principle. So far, all of that\nhas only been expressed within one single mathematical model, namely, the\ntraditional one built upon the usual continuum of the field $\\mathbb{R}$ of\nreal numbers, since complex numbers, finite dimensional Euclidean spaces, or\ninfinite dimensional Hilbert spaces, etc., are built upon the real numbers.\nHere, following [55], we give one example of how one can go beyond that\nsituation and study what stays the same and what changes in the Laws of\nPhysics, when one models them within an infinitely large variety of algebras of\nscalars constructed rather naturally. Specifically, it is shown that the\nSpecial Relativistic addition of velocities can naturally be considered in any\nof infinitely many reduced power algebras, each of them containing the usual\nfield of real numbers and which, unlike the latter, are non-Archimedean. The\nnonstandard reals are but one case of such reduced power algebras, and are as\nwell non-Archimedean. Two surprising and strange effects of such a study of the\nSpecial Relativistic addition of velocities are that one can easily go beyond\nthe velocity of light, and rather dually, one can as easily end up frozen in\nimmobility, with zero velocity. Both of these situations, together with many\nother ones, are as naturally available, as the usual one within real numbers.",
"arxiv_id": "physics/0701117",
"authors": [
"Elemer E Rosinger"
],
"categories": [
"physics.gen-ph"
],
"title": "From Reference Frame Relativity to Relativity of Mathematical Models : Relativity Formulas in a Variety of Non-Archimedean Setups",
"url": "https://arxiv.org/abs/physics/0701117"
},
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