Relativity Groupoid Instead of Relativity Group - Revision history

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	==Abstract==		==Abstract==

	<em>International Journal of Geometric Methods in Modern Physics</em>, V4, N5 (2007) 739-749. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents.[[Category:Scientific Paper]]		<em>International Journal of Geometric Methods in Modern Physics</em>, V4, N5 (2007) 739-749. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents.

			[[Category:Scientific Paper\|relativity groupoid instead relativity group]]

	[[Category:Relativity]]		[[Category:Relativity]]

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{{Infobox paper
| title = Relativity Groupoid Instead of Relativity Group
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_140.pdf Link to paper]
| author = [[Zbigniew Oziewicz]]
| keywords = [[associative addition of binary relative velocities]], [[groupoid category]]
| published = 2007
| journal = [[None]]
| volume = [[4]]
| number = [[5]]
| num_pages = 11
| pages = 739-749
}}

'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_140.pdf here]

==Abstract==

<em>International Journal of Geometric Methods in Modern Physics</em>, V4, N5 (2007) 739-749. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents. The Lorentz covariance and invariance are acepted to be the cornerstone of the physical theory. Observer-dependence within the relativity groupoid, and the Lorentz-covariance withinh the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than to be Lorentz-invariant. In 1908 Minkowski introduced space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced a relative velocity as a Minkowski bivector. Here we propose binary relative velocity as a traceless nilpotent endomorphism in a operator algebra. Each concept of a binary relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of ternary relative velocities in an isometric special relativity. We consider an algebra of many time-plus-space splits, as an operator algebra generated by observers-idempotents.[[Category:Scientific Paper]]

[[Category:Relativity]]