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Mathematics for Physics

162 Energy-Momentum Relation

An identity relating total energy, momentum, and rest mass that is used to compute energy from momentum without finding the velocity.

Velocity-independent kinematics. The energy-momentum relation gives total energy from momentum, and momentum from total energy, without measuring the velocity. Total energy is rest energy plus kinetic energy. Relativistic momentum is rest mass times velocity times the Lorentz factor. This principle is used to solve high-speed kinematics from calorimeter or magnet data.

The energy-momentum relation is

\[ E^{2} = m^{2}c^{4} + p^{2}c^{2} \]

where

  • \(E\) is the total energy.
  • \(m\) is the rest mass.
  • \(p\) is the magnitude of the relativistic momentum.
  • \(c\) is the speed of light.

Invariance of the four-momentum square. That identity is the invariant square of the four-momentum, so every inertial observer agrees on the rest mass. Four-momentum is the four-vector \(\left(\dfrac{E}{c},\,\mathbf{p}\right)\). This principle is used to move calculations into the center-of-mass frame while keeping the relation unchanged.

Massless kinematics. For zero rest mass the relation reduces to \(E=pc\), and the particle travels at the speed of light. A massless particle is a field quantum with no rest frame. This principle is used to assign momentum to light and to treat Compton scattering.

The massless energy-momentum relation is

\[ E = pc \]

Newtonian correspondence. At speeds much smaller than the speed of light, the relation reduces to rest energy plus Newtonian kinetic energy. This principle is used to recover \(E\approx mc^{2}+\dfrac{p^{2}}{2m}\) at everyday speeds.

Note: Also called the energy-momentum invariant relation. Also called the mass-shell relation.

162.1 References

  1. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(E^{2}=m^{2}c^{4}+p^{2}c^{2}\); mass-shell; \(E=pc\) for \(m=0\).
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(E^{2}-p^{2}c^{2}=m^{2}c^{4}\); energy from momentum.
  3. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — energy from momentum; low-speed limit; photons.
  4. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — invariant four-momentum square; massless kinematics.
  5. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — four-momentum norm; \(E=pc\); Newtonian correspondence.