View source for Physics Using Geometric Algebra/Relativistic Classical Mechanics/Spacetime position

The spacetime position <math>x</math> can be encoded in a paravector

:<math>
 x = x^0 + \mathbf{x},
</math>

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

:<math>x^0 = c t_{{}_{}}.  </math>

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

:<math>
c\, u  = \frac{dx}{d \tau } 
</math>

The proper velocity can be written in terms of the velocity

:<math>
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),
</math>

where

:<math>
\gamma =  \frac{dx^0}{d \tau} = \frac{1}{\sqrt{1-\frac{\mathbf{v}^2}{c^2}}}
</math>

and of course

:<math>
\mathbf{v} = \frac{d \mathbf{x}}{d t }.
</math>

The proper velocity is unimodular

:<math>
 u \bar{u} = 1
</math>

==Spacetime momentum==

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

:<math>
 p_{{}_{}} = m c u
</math>

The spacetime momentum contains the energy as the scalar part

:<math>
 p = mc ( \gamma  + \gamma \frac{\mathbf{v}}{c} ) = 
 \frac{E}{c} + \mathbf{p},
</math>

where the energy <math>E</math> is defined as

:<math>
E_{{}_{}} = \gamma m c^2 
</math>

The shell condition of the spacetime momentum is

:<math> p \bar{p} = (mc)^2 </math>

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