# 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 = (px, py, pz) and energy E is

$\mathbf {P} ={\begin{pmatrix}P^{0}\\P^{1}\\P^{2}\\P^{3}\end{pmatrix}}={\begin{pmatrix}E/c\\p_{\text{x}}\\p_{\text{y}}\\p_{\text{z}}\end{pmatrix}}$ 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 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.

## 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:

$-\|\mathbf {P} \|^{2}=-P^{\mu }P_{\mu }=-\eta _{\mu \nu }P^{\mu }P^{\nu }={E^{2} \over c^{2}}-|\mathbf {p} |^{2}=m^{2}c^{2}$ 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 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:

$P^{\mu }=mU^{\mu }\!$ where the four-velocity is

${\begin{pmatrix}U^{0}\\U^{1}\\U^{2}\\U^{3}\end{pmatrix}}={\begin{pmatrix}\gamma c\\\gamma v_{\text{x}}\\\gamma v_{\text{y}}\\\gamma v_{\text{z}}\end{pmatrix}}$ and

$\gamma ={\frac {1}{\sqrt {1-\left(v/c\right)^{2}}}}$ 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:

1. The total energy E = P0c is conserved.
2. The classical three-momentum 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 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 electronpositron 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

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

${\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}={\begin{pmatrix}\phi /c\\A_{\text{x}}\\A_{\text{y}}\\A_{\text{z}}\end{pmatrix}}$ where φ is the scalar potential and A = (Ax, Ay, Az) the vector potential, the "canonical" momentum four-vector is

$Q^{\mu }=P^{\mu }+qA^{\mu }.\!$ 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.