The spacetime position ${\displaystyle x}$ can be encoded in a paravector

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

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

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

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

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

The proper velocity can be written in terms of the velocity

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

where

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

and of course

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

The proper velocity is unimodular

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

Spacetime momentum

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

${\displaystyle p_{{}_{}}=mcu}$

The spacetime momentum contains the energy as the scalar part

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

where the energy ${\displaystyle E}$ is defined as

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

The shell condition of the spacetime momentum is

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