Energy–momentum

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 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

The quantity mv 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 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 x⁰ = ct. Some authors use the convention x⁰ = t, which yields a modified definition with p⁰ = E/c². It is also possible to define covariant four-momentum p_μ where the sign of the energy is reversed.

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:

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