Physics Using Geometric Algebra/Relativistic Classical Mechanics/Spacetime position

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The spacetime position can be encoded in a paravector

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

The proper velocity is defined as the derivative of the spacetime position with respect to the proper time

The proper velocity can be written in terms of the velocity

where

and of course

The proper velocity is unimodular

Spacetime momentum[edit]

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

The spacetime momentum contains the energy as the scalar part

where the energy is defined as

The shell condition of the spacetime momentum is