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 relativistic energy E and three-momentum p = (px, py, pz) = γmv, where v is the particle's three-velocity and γ the Lorentz factor, isConsider initially a system of one degree of freedom q. In the derivation of the equations of motion from the action using Hamilton's principle, one finds (generally) in an intermediate stage for the variation of the action,The variation of the action is E = m c 2 1 − v 2 c 2 = m r c 2 , {displaystyle E={frac {mc^{2}}{sqrt {1-{frac {v^{2}}{c^{2}}}}}}=m_{r}c^{2},} p = E v c 2 , {displaystyle mathbf {p} =E{frac {mathbf {v} }{c^{2}}},} E 2 c 2 = p ⋅ p + m 2 c 2 . {displaystyle {frac {E^{2}}{c^{2}}}=mathbf {p} cdot mathbf {p} +m^{2}c^{2}.} η μ ν ∂ S ∂ x μ ∂ S ∂ x ν = − m 2 c 2 . {displaystyle eta ^{mu u }{frac {partial S}{partial x^{mu }}}{frac {partial S}{partial x^{ u }}}=-m^{2}c^{2}.} 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 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 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. 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. 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: