Physics Using Geometric Algebra/Relativistic Classical Mechanics/Spacetime position

The spacetime position $$x$$ can be encoded in a paravector



x = x^0 + \mathbf{x}, $$

with the scalar part of the spacetime position in terms of the time


 * $$x^0 = c t_{{}_{}}. $$

The proper velocity $$u$$ is defined as the derivative of the spacetime position with respect to the proper time $$\tau_{{}_{}} $$



c\, u = \frac{dx}{d \tau } $$

The proper velocity can be written in terms of the velocity



c\, u =  \frac{dx^0}{d \tau} + \frac{d\mathbf{x}}{d \tau} = \gamma\left( 1 + \frac{d\mathbf{x}}{d x^0}  \right) = \gamma\left( 1 + \frac{\mathbf{v}}{c}  \right), $$

where



\gamma = \frac{dx^0}{d \tau} = \frac{1}{\sqrt{1-\frac{\mathbf{v}^2}{c^2}}} $$

and of course



\mathbf{v} = \frac{d \mathbf{x}}{d t }. $$

The proper velocity is unimodular



u \bar{u} = 1 $$

Spacetime momentum
The spacetime momentum is a paravector defined in terms of the proper velocity



p_{{}_{}} = m c u $$

The spacetime momentum contains the energy as the scalar part



p = mc ( \gamma + \gamma \frac{\mathbf{v}}{c} ) = \frac{E}{c} + \mathbf{p}, $$

where the energy $$E$$ is defined as



E_{{}_{}} = \gamma m c^2 $$

The shell condition of the spacetime momentum is


 * $$ p \bar{p} = (mc)^2 $$