In special relativity, fourmomentum is the generalization of the classical threedimensional momentum to fourdimensional spacetime. Momentum is a vector in three dimensions; similarly fourmomentum is a fourvector in spacetime. The contravariant fourmomentum of a particle with threemomentum and energy E is
The fourmomentum 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. It is also possible to define covariant fourmomentum P_{μ} where the sign of P_{0} is reversed.)
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Calculating the Minkowski norm of the fourmomentum 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 reciprocal of the metric tensor of special relativity. Because is Lorentz invariant, its value is not changed by Lorentz transformations, i.e. boosts into different frames of reference.
For a massive particle, the fourmomentum is given by the particle's invariant mass m multiplied by the particle's fourvelocity:
where the fourvelocity is
and is the Lorentz factor and c is the speed of light.
The conservation of the fourmomentum 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 centerofmass frame and potential energy from forces between the particles contribute to the invariant mass. As an example, two particles with fourmomenta (−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 fourmomenta P(A) and P(B) of two daughter particles produced in the decay of a heavier particle with fourmomentum P(C) to find the mass of the heavier particle. Conservation of fourmomentum 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 threemomenta of the daughter particles, one can reconstruct the invariant mass of the twoparticle system, which must be equal to M. This technique is used, e.g., in experimental searches for Z' bosons at highenergy particle colliders, where the Z' boson would show up as a bump in the invariant mass spectrum of electronpositron or muonantimuon pairs.
If an object's mass does not change, the Minkowski inner product of its fourmomentum and corresponding fouracceleration A^{μ} is zero. The fouracceleration is proportional to the proper time derivative of the fourmomentum divided by the particle's mass, so
For applications in relativistic quantum mechanics, it is useful to define a "canonical" momentum fourvector, Q^{μ}, which is the sum of the fourmomentum and the product of the electric charge with the electromagnetic fourpotential:
where the fourvector potential is a result of combining the scalar potential and the vector potential:
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 Schroedinger equation.
