# 4-momentum

Including a time element of energy as

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

This four element vector transforms as a tensor

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

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

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

which for the metric of special relativity yeilds

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

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

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