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 three-momentum p = (px, py, pz) and energy E is
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 Pμ where the sign of the energy is reversed.
where we use the convention that
is the metric tensor of special relativity. The magnitude ||P||2 is Lorentz invariant, meaning its value is not changed by Lorentz transformations/boosting into different frames of reference.
Relation to four-velocity
where the four-velocity is
Conservation of four-momentum
The conservation of the four-momentum yields two conservation laws for "classical" quantities:
Note that the invariant mass of a system of particles may be more than the sum of the particles' rest masses, since kinetic energy in the system center-of-mass frame and potential energy from forces between the particles contribute to the invariant mass. As an example, two particles with four-momenta (5 GeV/c, 4 GeV/c, 0, 0) and (5 GeV/c, −4 GeV/c, 0, 0) each have (rest) mass 3 GeV/c2 separately, but their total mass (the system mass) is 10 GeV/c2. If these particles were to collide and stick, the mass of the composite object would be 10 GeV/c2.
One practical application from particle physics of the conservation of the invariant mass involves combining the four-momenta PA and PB of two daughter particles produced in the decay of a heavier particle with four-momentum PC to find the mass of the heavier particle. Conservation of four-momentum gives PCμ = PAμ + PBμ, while the mass M of the heavier particle is given by −||PC||2 = M2c2. By measuring the energies and three-momenta of the daughter particles, one can reconstruct the invariant mass of the two-particle system, which must be equal to M. This technique is used, e.g., in experimental searches for Z′ bosons at high-energy particle colliders, where the Z′ boson would show up as a bump in the invariant mass spectrum of electron–positron or muon–antimuon pairs.
If the mass of an object does not change, the Minkowski inner product of its four-momentum and corresponding four-acceleration Aμ is simply zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so
Canonical momentum in the presence of an electromagnetic potential
This, in turn, allows the potential energy from the charged particle in an electrostatic potential and the Lorentz force on the charged particle moving in a magnetic field to be incorporated in a compact way, in relativistic quantum mechanics.