The Lorentz Boost-Link Is Not Unique. Relative velocity as a morphism in a connected groupoid category of null objects

Abstract

The isometry-link problem is to determine all isometry transformations among given pair of vectors with the condition that if these initial and final vectors coincide, the transformation-link must be identity on entire vector space. In the first part of this essay we provide the complete solution for the link problem for arbitrary isometry, for any dimension and arbitrary signature of the invertible metric tensor. We apply these considerations for pure Lorentz transformations, for the Lorentz boost, parameterized by relative velocity, and we are showing that the isometric pure Lorentz transformation-link is not given uniquely by the initial and final vectors. Lorentz's boost needs a choice of the preferred time-like observer. This leads to non-uniqueness of the relative velocity among two reference systems, that was apparently not intention of Einstein in 1905. Presently, as during the XX century, the Lorentz covariance is the cornerstone of physical theory. Our main conclusion is: observer-dependence (and -independence), and the Lorentz--covariance (and invariance), are different concepts. In order to have axiomatically the unique relative velocity among pair of massive bodies, we propose a connected groupoid category of massive bodies in mutual motions