Abstract

Presented at the Fifth Workshop Applied Category Theory, Graph-Operad-Logic, Merida, May 2006. 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. Such transformations-links are said to be pure transformations, or the boost transformations. In the first part of this essay we provide the complete solution for the link problem for arbitrary isometry. We prove that a solution of the link problem is not given uniquely by the initial and final vectors. Each solution needs the third vector called the privileged or preferred vector. The triple of vectors determine the unique pure isometry-link. We apply these considerations for the Lorentz-boost, parameterized by a relative velocity, and we show that the Lorentz boost needs a choice of the preferred time-like observer, an {ae}ther. Non-uniqueness of the isometric relative velocity, apparently was not the Einstein's intention.