Relativity Groupoid Instead of Relativity Group

Scientific Paper
Title	Relativity Groupoid Instead of Relativity Group
Read in full	Link to paper
Author(s)	Zbigniew Oziewicz
Keywords	associative addition of binary relative velocities, groupoid category
Published	2007
Volume	4
Number	5
No. of pages	11
Pages	739-749

Read the full paper here

Abstract

International Journal of Geometric Methods in Modern Physics, 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.