4-momentum

Including a time element of energy as

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

This four element vector transforms as a tensor

p ′ μ = ∂ x ′ μ ∂ x ν p ν {\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

m 2 c 2 = g μ ν p μ p ν {\displaystyle 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 {\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

p μ = m U μ {\displaystyle p^{\mu }=mU^{\mu }}