In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy E and three-momentum p = (p_x, p_y, p_z) = γmv, where v is the particle's three-velocity and γ the Lorentz factor, is $${\displaystyle p=\left(p^{0},p^{1},p^{2},p^{3}\right)=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right).}$$

The quantity mv of above is the ordinary non-relativistic momentum of the particle and m its rest mass. The four-momentum is useful in relativistic calculations because it is a Lorentz covariant vector. This means that it is easy to keep track of how it transforms under Lorentz transformations.

Calculating the Minkowski norm squared of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass:

$${\displaystyle p\cdot p=\eta _{\mu \nu }p^{\mu }p^{\nu }=p_{\nu }p^{\nu }=-{E^{2} \over c^{2}}+|\mathbf {p} |^{2}=-m^{2}c^{2}}$$ where the following denote:

${\textstyle p}$, the four-momentum vector of a particle,

${\textstyle p\cdot p}$, the Minkowski inner product of the four-momentum with itself,

${\textstyle p^{\mu }}$and ${\textstyle p^{\nu }}$, the contravariant components of the four-momentum vector,

${\textstyle p_{\nu }}$, the covariant form,