Translation
A spacetime translation operator $$\mathbf T$$ is given by
- $$\mathbf T(\tau) = \dfrac{1}{2}\gamma\tau\,(-c\,\mathbf e_{321} + \dot x\,\mathbf e_{230} + \dot y\,\mathbf e_{310} + \dot z\,\mathbf e_{120}) + {\large\unicode{x1D7D9}}$$ .
The trivector part of this operator is $$\frac{1}{2}\tau (\mathbf e_4 \wedge \mathbf u)^\unicode{x2606}$$ for a velocity $$\mathbf u$$, where the dualization applies the metric and causes the temporal component to be negated.
The operator $$\mathbf T(\tau)$$ 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
- $$\left\Vert\mathbf T\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \dfrac{1}{2}\gamma\tau \sqrt{c^2 - \dot x^2 - \dot y^2 - \dot z^2}$$ ,
and this makes it clear that the bulk norm is real precisely when the velocity of the translation must be less than the speed of light. Setting $$dt = \gamma\tau$$, $$dx = \dot x\,dt$$, $$dy = \dot y\,dt$$, and $$dz = \dot z\,dt$$, we can rewrite this as
- $$\left\Vert\mathbf T\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \dfrac{1}{2}\sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2}$$ .
This spacetime interval is a Lorentz invariant having the same value for all inertial observers.