Relativistic screw
A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by
- $$\mathbf Q(\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,
where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The half rate of rotation about the line is specified by $$\dot\phi$$, and the half rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to
- $$\mathbf Q(\tau) = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi) - [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \gamma c\tau\,\mathbf e_{321}$$.
The operator $$\mathbf Q$$ transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product
- $$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.
The bulk norm of a unitized relativistic screw operator is given by
- $$\left\Vert\mathbf Q\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - Q_{mx}^2 - Q_{my}^2 - Q_{mz}^2 - Q_{mw}^2}$$ ,
and it corresponds to the distance that the origin is moved. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).