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

$x^{0}=ct_{{}_{}}.$

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

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

where

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

and of course

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

The proper velocity is unimodular

$u{\bar {u}}=1$

Spacetime momentum

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

$p_{{}_{}}=mcu$

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

$E_{{}_{}}=\gamma mc^{2}$

The shell condition of the spacetime momentum is

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