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Quaternions have an (over a century-old) extensive and quite complicatedinteraction with special relativity. Since quaternions are intrinsically4-dimensional, and do such a good job of handling 3-dimensional rotations, thehope has always been that the use of quaternions would simplify some of thealgebra of the Lorentz transformations. Herein we report a relatively niceresult for the relativistic combination of non-collinear 3-velocities. If wework with the relativistic half-velocities $w$ defined by $v={2w\over1+w^2}$,and promote them to quaternions using $\mathbf{w} = w \; \mathbf{\hat n}$,where $\mathbf{\hat n}$ is a unit quaternion, then we shall show \[\mathbf{w}_{1\oplus2} = \mathbf{w}_1 \oplus \mathbf{w}_2=(1-\mathbf{w}_1\mathbf{w}_2)^{-1} (\mathbf{w}_1 +\mathbf{w}_2) = (\mathbf{w}_1+\mathbf{w}_2)(1-\mathbf{w}_2\mathbf{w}_1)^{-1}. \] All of the complicatedangular dependence for relativistic combination of non-collinear 3-velocitiesis now encoded in the quaternion multiplication of $\mathbf{w}_1$ with$\mathbf{w}_2$. This result can furthermore be extended to obtain an elegantand compact formula for the associated Wigner angle: \[\mathrm{e}^{\mathbf{\Omega}} = \mathrm{e}^{\Omega \; \mathbf{\hat\Omega} } =(1-\mathbf{w}_1\mathbf{w}_2)^{-1} (1-\mathbf{w}_2\mathbf{w}_1), \] in terms ofwhich \[ {\mathbf{\hat{n}}}_{1\oplus2} = \mathrm{e}^{\mathbf{\Omega}/2} \;\;{\mathbf{w}_1+\mathbf{w}_2\over |\mathbf{w}_1+\mathbf{w}_2|}; \qquad\qquad{\mathbf{\hat{n}}}_{2\oplus1} = \mathrm{e}^{-\mathbf{\Omega}/2} \;\;{\mathbf{w}_1+\mathbf{w}_2\over |\mathbf{w}_1+\mathbf{w}_2|}. \] Thus, we wouldargue, many key results that are ultimately due to the non-commutativity ofnon-collinear boosts can be easily rephrased in terms of the algebra ofquaternions.

Relativistic combination of non-collinear 3-velocities using quaternions