Translation
A spacetime translation operator $$\mathbf T$$ is given by
- $$\mathbf T(\tau) = \dfrac{1}{2}\gamma\tau\,(\dot x\,\mathbf e_{230} + \dot y\,\mathbf e_{310} + \dot z\,\mathbf e_{120} - c\,\mathbf e_{321}) + {\large\unicode{x1D7D9}}$$ .
It transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product
- $$\mathbf r' = \mathbf T \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}}$$.
The bulk norm of a translation operator is given by
- $$\left\Vert\mathbf T\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - x^2 - y^2 - z^2}$$ ,
and it must be real for any motion that's physically possible (i.e., without exceeding the speed of light).