Physics Using Geometric Algebra/Relativistic Classical Mechanics/Spacetime position

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

$x=x^0+𝐱,$

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 ${\tau }_{}$

$cu=\frac{dx}{d\tau }$

The proper velocity can be written in terms of the velocity

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

where

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

and of course

$𝐯=\frac{d𝐱}{dt}.$

The proper velocity is unimodular

$u\overline{u}=1$

Spacetime momentum

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

$p_{}=mcu$

The spacetime momentum contains the energy as the scalar part

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

where the energy $E$ is defined as

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

The shell condition of the spacetime momentum is

$p\overline{p}=(mc{)}^2$