Physics Using Geometric Algebra/Relativistic Classical Mechanics/Spacetime position

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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 momentum[edit]

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