# Four-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 three-momentum **p** = (*p*_{x}, *p*_{y}, *p*_{z}) 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 *x*^{0} = *ct*. Some authors use the convention *x*^{0} = *t*, which yields a modified definition with *P*^{0} = *E*/*c*^{2}. It is also possible to define covariant four-momentum *P*_{μ} where the sign of the energy is reversed.

## Minkowski norm

Calculating the Minkowski norm 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:

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

For a massive particle, the four-momentum is given by the particle's invariant mass *m* multiplied by the particle's four-velocity:

where the four-velocity is

and

is the Lorentz factor, *c* is the speed of light.

## 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/*c*^{2} separately, but their total mass (the system mass) is 10 GeV/*c*^{2}. If these particles were to collide and stick, the mass of the composite object would be 10 GeV/*c*^{2}.

One practical application from particle physics of the conservation of the invariant mass involves combining the four-momenta **P**_{A} and **P**_{B} of two daughter particles produced in the decay of a heavier particle with four-momentum **P**_{C} to find the mass of the heavier particle. Conservation of four-momentum gives *P*_{C}^{μ} = *P*_{A}^{μ} + *P*_{B}^{μ}, while the mass *M* of the heavier particle is given by −||**P**_{C}||^{2} = *M*^{2}*c*^{2}. 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 *A*^{μ} 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

For a charged particle of charge *q*, moving in an electromagnetic field given by the electromagnetic four-potential:

where φ is the scalar potential and **A** = (*A*_{x}, *A*_{y}, *A*_{z}) the vector potential, the "canonical" momentum four-vector is

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.

## See also

## References

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