# 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 momentumEdit

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

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