It is proved that the Lorentz boost entails the relative velocity to be ternary: the ternary relative velocity is a velocity of a body with respect to an interior observer, as seen by a preferred exterior-observer. The Lorentz-boost imply non-associative addition of ternary relative velocities. Within Einstein’s special relativity theory, each preferred observer (fixed stars, aether, etc), determine the unique relative velocity among each pair of massive bodies. Therefore, the special relativity founded on axiom, that each pair of reference systems must be related by Lorentz isometry, needs a preferred reference system in order to have the unique Einstein’s relative velocity among each pair of massive bodies. This choice-dependence of relative velocity violate the Relativity Principle that all reference systems must be equivalent. This astonishing conflict of the Lorentz relativity group, with the Relativity Principle, can be resolved in two alternative ways. Either, abandon the Relativity Principle in favor of a preferred reference system. Or, within the Relativity Principle, replace the Lorentz relativity group by the relativity groupoid, with the choice-free binary relative velocities (not parametrizing isometry). The axiomatic definition of the kinematical unique binary relative velocity as the choice-free Minkowski space-like vector, leads to the groupoid structure of the set of all deduced relativity transformations (instead of the Lorentz relativity group), with the associative addition of binary relative velocities. Observer-independence and the Lorentz-group-invariance are distinct concepts. This suggest the possibility of formulating many-body relativistic dynamics without Lorentz/Poincare invariance.

Keywords: Special relativity, groupoid category, isometry, Minkowski geometry, Non-Euclidean geometry, Lorentz metric, ternary relative velocity, binary relative velocity

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