Normal number

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

$\mathbf{P}=\left(\begin{array}{c}{P}^0\\ P^1\\ P^2\\ P^3\end{array}\right)=\left(\begin{array}{c}E/c\\ p_x\\ p_y\\ p_z\end{array}\right)$

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 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:

$-\Vert \mathbf{P}{\Vert }^2=-P^{\mu }{P}_{\mu }=-{\eta }_{\mu \nu }{P}^{\mu }{P}^{\nu }=\frac{E^2}{c^2}-|\mathbf{p}{|}^2=m^2{c}^2$

where we use the convention that

${\eta }_{\mu \nu }=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$

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:

$P^{\mu }=m{U}^{\mu }$

where the four-velocity is

$\left(\begin{array}{c}{U}^0\\ U^1\\ U^2\\ U^3\end{array}\right)=\left(\begin{array}{c}\gamma c\\ \gamma v_x\\ \gamma v_y\\ \gamma v_z\end{array}\right)$

and

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

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 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 A^μ is zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so

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

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:

$\left(\begin{array}{c}{A}^0\\ A^1\\ A^2\\ A^3\end{array}\right)=\left(\begin{array}{c}\varphi /c\\ A_x\\ A_y\\ A_z\end{array}\right)$

where φ is the scalar potential and A = (A_x, A_y, A_z) the vector potential, the "canonical" momentum four-vector is

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

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, in relativistic quantum mechanics.

See also

• Four-force
• Pauli–Lubanski pseudovector

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534