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

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

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 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:

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

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:

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

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, 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

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

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:

${\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}={\begin{pmatrix}\phi /c\\A_{x}\\A_{y}\\A_{z}\end{pmatrix}}$

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

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

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

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• 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.
  
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• 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