Relativistic combination of non-collinear 3-velocities using quaternions
Abstract
Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that the use of quaternions would simplify some of the algebra of the Lorentz transformations. Herein we report a relatively nice result for the relativistic combination of non-collinear 3-velocities. If we work with the relativistic half-velocities w defined by v=2w1+w2, and promote them to quaternions using w = w \; n, where n is a unit quaternion, then we shall show \[ w12 = w1 w2 =(1-w1w2)-1 (w1 +w2) = (w1 +w2)(1-w2w1)-1. \] All of the complicated angular dependence for relativistic combination of non-collinear 3-velocities is now encoded in the quaternion multiplication of w1 with w2. This result can furthermore be extended to obtain an elegant and compact formula for the associated Wigner angle: \[ e = e \; = (1-w1w2)-1 (1-w2w1), \] in terms of which \[ n12 = e/2 \;\; w1+w2 |w1+w2|; n21 = e-/2 \;\; w1+w2 |w1+w2|. \] Thus, we would argue, many key results that are ultimately due to the non-commutativity of non-collinear boosts can be easily rephrased in terms of the algebra of quaternions.