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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 $$\vec p = (p_x, p_y, p_z)$$ and energy E is

$$\mathbf{P} = \begin{pmatrix} P^0 \\ P^1 \\ P^2 \\ P^3 \end{pmatrix} = \begin{pmatrix} E/c \\ p_x \\ p_y \\ p_z \end{pmatrix}$$

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 $$P_{\mu}$$ where the sign of the 3 momentum 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:

$$-||\mathbf{P}||^2 = - P^\mu P_\mu = - \eta_{\mu\nu} P^\mu P^\nu = {E^2 \over c^2} - |\vec p|^2 = m^2c^2$$

where we use the convention that

$$\eta^{\mu\nu} = \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$

is the reciprocal of the metric tensor of special relativity. $$||\mathbf{P}||^2\!$$ 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:

$$P^\mu = m \, U^\mu\!$$

where the four-velocity is

$$\begin{pmatrix} U^0 \\ U^1 \\ U^2 \\ U^3 \end{pmatrix} = \begin{pmatrix} \gamma c \\ \gamma v_x \\ \gamma v_y \\ \gamma v_z \end{pmatrix}$$

and $$\gamma = \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$$is the Lorentz factor and c is the speed of light.
Conservation of four-momentum

The conservation of the four-momentum yields two conservation laws for "classical" quantities:

1. The total energy E = P0c is conserved.
2. The classical three-momentum $$\vec p$$ is conserved.

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 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 = 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 an object's mass does not change, the Minkowski inner product of its four-momentum and corresponding four-acceleration Aμ is zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so

$$P^{\mu} A_{\mu} = \eta_{\mu\nu} P^{\mu} A^{\nu} = \eta_{\mu\nu} P^{\mu} \frac{d}{d\tau} \frac{P^{\nu}}{m} = \frac{1}{2m} \frac{d}{d\tau} ||\mathbf{P}||^2 = \frac{1}{2m} \frac{d}{d\tau} (-m^2c^2) = 0 .$$

Canonical momentum in the presence of an electromagnetic potential

For applications in relativistic quantum mechanics, it is useful to define a "canonical" momentum four-vector, Q^\mu , which is the sum of the four-momentum and the product of the electric charge with the electromagnetic four-potential:

$$Q^{\mu} = P^{\mu} + q A^{\mu} \!$$

where the four-vector potential is a result of combining the scalar potential and the vector potential:

$$\begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix} = \begin{pmatrix} \phi / c \\ A_x \\ A_y \\ A_z \end{pmatrix}$$

This 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 into the Schrödinger equation.

Momentum
Four-force
Four-vector
Special relativity

References

Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-19-853952-5.

Physics Encyclopedia