Relativistic Combination of Non-Collinear 3-Velocities Using Quaternions

Universe

Thomas Berry¹  Matt Visser¹ 

[1] School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand;

关键词: special relativity;    combination of velocities;    wigner angle;    quaternions;   

DOI  :  10.3390/universe6120237

来源: DOAJ

【 摘 要 】

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 new and relatively nice result for the relativistic combination of non-collinear 3-velocities. We work with the relativistic half-velocities w defined by $v=\frac{2w}{1+w^2}$, so that $w=\frac{v}{1+\sqrt{1-v^2}}=\frac{v}{2}+\mathcal{O}\left(v^3\right)$, and promote them to quaternions using $\mathbf{w}=w\widehat{\mathbf{n}}$, where $\widehat{\mathbf{n}}$ is a unit quaternion. We shall first show that the composition of relativistic half-velocities is given by ${\mathbf{w}}_{1\oplus 2}\equiv {\mathbf{w}}_1\oplus {\mathbf{w}}_2\equiv {(1-{\mathbf{w}}_1{\mathbf{w}}_2)}^{-1}({\mathbf{w}}_1+{\mathbf{w}}_2)$, and then show that this is also equivalent to ${\mathbf{w}}_{1\oplus 2}=({\mathbf{w}}_1+{\mathbf{w}}_2){(1-{\mathbf{w}}_2{\mathbf{w}}_1)}^{-1}$. Here as usual we adopt units where the speed of light is set to unity. Note that all of the complicated angular dependence for relativistic combination of non-collinear 3-velocities is now encoded in the quaternion multiplication of ${\mathbf{w}}_1$ with ${\mathbf{w}}_2$. This result can furthermore be extended to obtain novel elegant and compact formulae for both the associated Wigner angle $\Omega$ and the direction of the combined velocities: ${\mathrm{e}}^{\mathbf{\Omega }}={\mathrm{e}}^{\Omega \widehat{\mathbf{\Omega }}}={(1-{\mathbf{w}}_1{\mathbf{w}}_2)}^{-1}(1-{\mathbf{w}}_2{\mathbf{w}}_1)$, and ${\widehat{\mathbf{w}}}_{1\oplus 2}={\mathrm{e}}^{\mathbf{\Omega }/2}\frac{{\mathbf{w}}_1+{\mathbf{w}}_2}{|{\mathbf{w}}_1+{\mathbf{w}}_2|}$. Finally, we use this formalism to investigate the conditions under which the relativistic composition of 3-velocities is associative. 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 non-commutative algebra of quaternions.

【 授权许可】

Unknown