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Mathematics for Physics

166 Four-Momentum

A four-vector formed from energy and momentum that is used to package the relativistic energy and three-momentum of a particle into one Lorentz-covariant object.

Unified energy-momentum. Energy and three-momentum together form one four-vector. A four-vector is a four-component object that transforms as spacetime coordinates do under a Lorentz transformation. This principle is used to treat energy and momentum as one geometric quantity.

The four-momentum is

\[ P^{\mu} = m U^{\mu} = \left(\dfrac{E}{c},\, \mathbf{p}\right) \]

where

  • \(P^{\mu}\) is the four-momentum.
  • \(m\) is the rest mass.
  • \(U^{\mu}\) is the four-velocity.
  • \(E\) is the total energy.
  • \(\mathbf{p}\) is the relativistic three-momentum.
  • \(c\) is the speed of light.

Definition by four-velocity. Four-momentum is rest mass times four-velocity. Four-velocity is the derivative of spacetime position with respect to proper time. This principle is used to give a momentum that transforms correctly under boosts.

The mass-shell condition. The Minkowski square of the four-momentum is the invariant \(-m^{2}c^{2}\). The mass-shell condition is that constraint on the four-momentum. This principle is used to recover \(E^{2}=(pc)^{2}+(mc^{2})^{2}\) without computing the velocity.

Conservation of four-momentum. Conservation of energy and conservation of three-momentum are the time and space parts of one conservation law for four-momentum. Four-momentum conservation is the requirement that the total incoming four-momentum equals the total outgoing four-momentum. This principle is used to analyze collisions, decays, and pair production.

Note: Also called the energy-momentum four-vector. Also called the momentum four-vector.

166.1 References

  1. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(P^{\mu}=mU^{\mu}\) and \(P^{\mu}=(E/c,\mathbf{p})\); mass-shell; four-momentum conservation.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — energy-momentum \(4\)-vector \(p^{\mu}=m\eta^{\mu}\); conservation.
  3. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — \(p^{\mu}=mU^{\mu}\); invariant square.
  4. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(P=(E/c,\mathbf{p})\); unified conservation.