Ternary relative velocity; astonishing conflict of the Lorentz group   with relativity
  We are proving that the Lorentz boost entails the relative velocity to be ternary: ternary relative velocity is a velocity of a body with respect to an interior observer as seen by a preferred exterior-observer. The Lorentz boosts 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 Minkowski's 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 [de Abreu 2004; de Abreu and Guerra 2005, 2006]. Or, within the Relativity Principle, replace the Lorentz relativity group by the relativity groupoid, with the choice-free binary relative velocities introduced by Minkowski in 1908. Binary - means that the relative velocity is a function of a pair of reference systems. Following Minkowski [1908], we consider the binary relative velocity to be the Minkowski space-like vector (and not the Minkowski bivector as it is in the Hestenes theory [Hestenes 1974]). 
Zbigniew Oziewicz (UNAM)
physics.gen-ph