Relativity Groupoid Instead of Relativity Group

TLDR: In this paper, the authors consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra, where a binary velocity is interpreted as a categorical morphism with the associative addition.

Abstract: In 1908, Minkowski [13] used space-like binary velocity-field of a medium, relative to an observer. In 1974, Hestenes introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector [7, 8]. We propose to consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra. Any concept of a binary axiomatic 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 isometric special relativity (loop structure). We consider an algebra of many time-plus-space splits, as an operator algebra generated by idempotents. The kinematics of relativity groupoid is ruled by associative Frobenius operator algebra, whereas the dynamics of categorical relativity needs the non-associative Frolicher–Richardson operator algebra. The Lorentz covariance is the cornerstone of physical theory. Observer-dependence within relativity groupoid, and the Lorentz-covariance within the Lorentz relativity group, are different concepts. Laws of Physics could be observer-free, rather than Lorentz-invariant.