What is "four-momentum"

n. (context physics English) A generalization of momentum in four-dimensional spacetime

In special relativity, four-momentum 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 and three-momentum , where is the particle's and the Lorentz factor, is

$$p = (p^0 , p^1 , p^2 , p^3 ) = \left({E \over c} , p_x , p_y , p_z\right).$$

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

The above definition applies under the coordinate convention that . Some authors use the convention , which yields a modified definition with . It is also possible to define covariant four-momentum where the sign of the energy is reversed.