4-momentum

Including a time element of energy as

$\mathbf {p} =\left({\frac {E}{c}},p^{x},p^{y},p^{z}\right)$

This four element vector transforms as a tensor

$p'^{\mu }={\frac {\partial x'^{\mu }}{\partial x^{\nu }}}p^{\nu }$

And as such constitutes a four-vector, a rank 1 tensor, called the 4-momentum. The mass of a particle is an invariant given by the spacetime length of the 4-momentum according to

$m^{2}c^{2}=g_{\mu \nu }p^{\mu }p^{\nu }$

which for the metric of special relativity yeilds

$E^{2}=p^{2}c^{2}+m^{2}c^{4}$

For a particle with mass, the 4-momentum can be related to the 4-velocity by

$p^{\mu }=mU^{\mu }$

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