In special relativity, four-momentum is a four-vector that replaces classical momentum; the four-momentum of a particle is defined as the particle's mass times the particle's four-velocity.

where

is the energy of the moving body.

Calculating the norm of the momentum-energy quad-vector we obtain:

and since c is a constant we may say that the norm of the four-momentum vector is equal to the body's mass; although, when computing values, it is really only equal to the mass if we choose to work in units of measurement in which the speed of light is simply c = 1.

The conservation of the four-momentum yields 3 laws of "classical" conservation:

1. The energy (p⁰) is conserved.
2. The classical momentum is conserved.
3. The norm of the four-momentum is conserved.

In reactions between an isolated handful of particles, four-momentum is conserved. The mass of a system of particles may be more than the sum of the particle's masses, since kinetic energy counts as mass. As an example, two particles with the four-momentums {5, 4, 0, 0} and {5, -4, 0, 0} both have the mass 3, but their total mass is 10. Note that the length of the four-vector {t, x, y, z} is

The scalar product of a four-momentum and the corresponding four-acceleration is always 0.

See also

• four-vector
• four-velocity
• four-acceleration
• four-force