Bi-quaternion square roots, rotational relativity, and dual space-time intervals

Summary: "Diagonal quadratic forms of any dimension, built on a sign-indefinite metric, are represented as squares of six-dimensional (6D) bi-quaternion (BQ) vectors having a definable norm. In particular, the line element of 4D Minkowski space-time is written as the square of a BQ-vector whose spatial and temporal parts are mutually orthogonal. Lorentz transformations of BQ-vector components with simultaneous {rm SO}(3,{bf C}) transformations of the quaternion frame yield a correlation between matrix representations of these groups, distinguishing the {rm SO}(1,2) subgroup of mixed space-time rotations. The admitted variability of the subgroup parameters leads to a BQ-vector formulation of relativity theory, comprising all features and effects of special relativity with an additional ability to describe the motion of arbitrary non-inertial frames. Abandoning the requirement of the existence of a BQ-vector norm leads to an unconventional 6D model of relativity, such that the imaginary part of a space-time interval is observed on the light cone."