The spacetime position
can be encoded in a paravector
![{\displaystyle x=x^{0}+\mathbf {x} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8851b7ac266d8596270398431dcbafeb48b39af4)
with the scalar part of the spacetime position in terms of the time
![{\displaystyle x^{0}=ct_{{}_{}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb87a4aa94bde1c4579cbefa8d2e41d529bdfe4b)
The proper velocity
is defined as the derivative of the spacetime position with respect to the proper time
![{\displaystyle c\,u={\frac {dx}{d\tau }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca21ff9604886e188968fa1ef57d3f0f11e27dd4)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64f4bad14b7bc42edbe883b8f719215914d0f8a6)
where
![{\displaystyle \gamma ={\frac {dx^{0}}{d\tau }}={\frac {1}{\sqrt {1-{\frac {\mathbf {v} ^{2}}{c^{2}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b96e9b06840cd3869bb32e1813bac6c87ff95b)
and of course
![{\displaystyle \mathbf {v} ={\frac {d\mathbf {x} }{dt}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/640757f778275e3b51c788d3ad3e0786d968670f)
The proper velocity is unimodular
![{\displaystyle u{\bar {u}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95a1d22264af03aef5f15b2342da0e47acd6cf72)
The spacetime momentum is a paravector defined in terms of the proper velocity
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The spacetime momentum contains the energy as the scalar part
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where the energy is defined as
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The shell condition of the spacetime momentum is
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