In special relativity, four-momentum (also called momentum–energy or momenergy[1]) 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 relativistic energy E and three-momentum p = (px, py, pz) = γmv, where v is the particle's three-velocity and γ the Lorentz factor, is
The quantity mv of above is the ordinary non-relativistic momentum of the particle and m its rest mass.
The four-momentum is useful in relativistic calculations because it is a Lorentz covariant vector.
Calculating the Minkowski norm squared 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:
The negativity of the norm reflects that the momentum is a timelike four-vector for massive particles.
The other choice of signature would flip signs in certain formulas (like for the norm here).
Another, more satisfactory, approach is to begin with the principle of least action and use the Lagrangian framework to derive the four-momentum, including the expression for the energy.
it is immediate (recalling x0 = ct, x1 = x, x2 = y, x3 = z and x0 = −x0, x1 = x1, x2 = x2, x3 = x3 in the present metric convention) that
is a covariant four-vector with the three-vector part being the canonical momentum.
When the equations of motion are known (or simply assumed to be satisfied), one may let go of the requirement δq(t2) = 0.
In this case the path is assumed to satisfy the equations of motion, and the action is a function of the upper integration limit δq(t2), but t2 is still fixed.
This time, the system is allowed to move through configuration space at "arbitrary speed" or with "more or less energy", the field equations still assumed to hold and variation can be carried out on the integral, but instead observe
which constitute the standard formulae for canonical momentum and energy of a closed (time-independent Lagrangian) system.
With this approach it is less clear that the energy and momentum are parts of a four-vector.
The energy and the three-momentum are separately conserved quantities for isolated systems in the Lagrangian framework.
More pedestrian approaches include expected behavior in electrodynamics.
The transformation properties of the electromagnetic field tensor, including invariance of electric charge, are then used to transform to the lab frame, and the resulting expression (again Lorentz force law) is interpreted in the spirit of Newton's second law, leading to the correct expression for the relativistic three- momentum.
It is also possible to avoid electromagnetism and use well tuned experiments of thought involving well-trained physicists throwing billiard balls, utilizing knowledge of the velocity addition formula and assuming conservation of momentum.
As shown above, there are three conservation laws (not independent, the last two imply the first and vice versa): 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.
If these particles were to collide and stick, the mass of the composite object would be 10 GeV/c2.
Conservation of four-momentum gives pCμ = pAμ + pBμ, while the mass M of the heavier particle is given by −PC ⋅ PC = 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 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
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.
In the case when there is a moving physical system with a continuous distribution of matter in curved spacetime, the primary expression for four-momentum is a four-vector with covariant index: [9] Four-momentum
is the generalized four-momentum associated with the action of fields on particles; four-vector
The following formulas are obtained for the energy and momentum of the system: Here