Four-momentum

In special relativity, four-momentum (also called momentum–energy or momenergy^[1]) 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 $p = (p x, p y, p z) = γm v$ , where $v$ is the particle's three-velocity and $γ$ the Lorentz factor, is

p=\left(p^{0},p^{1},p^{2},p^{3}\right)=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right).

The quantity $m v$ of above is 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.

^ Taylor, Edwin; Wheeler, John (1992). Spacetime physics introduction to special relativity. New York: W. H. Freeman and Company. p. 191. ISBN 978-0-7167-2327-1.