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.