paper

Relativity grupoid insted of relativity group

arXiv:math/0608770 · doi:10.1142/S0219887807002260

Abstract

Minkowski in 1908 used space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector. We propose 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 Frölicher-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.

10 p., invitated lecture presented at the 42nd Karpacz Winter School of Theoretical Physics, Ladek, Poland, 6-11 February 2006, "Current mathematical topics in gravitation and cosmology". Added references, title changed. Version to be published in IJGMMP

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