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Mathematics for Physics

199 Relativistic Momentum and Energy

The relativistic relations among momentum, energy, and mass for particles moving at high speed.

Relativistic momentum and the speed limit. Relativistic momentum grows faster than linearly with speed and becomes unbounded as the speed approaches the speed of light. The Lorentz factor is the velocity-dependent scaling \(\gamma\). This principle is used to compute accelerator momenta and to show that a massive body cannot reach the speed of light.

The relativistic momentum is

\[ p = \gamma mv \]

where

  • \(p\) is the relativistic momentum.
  • \(\gamma\) is the Lorentz factor.
  • \(m\) is the rest mass.
  • \(v\) is the speed.

Rest energy. A particle has rest energy even when it is at rest, so mass and energy are interchangeable. Rest mass is the mass measured in the particle’s rest frame. This principle is used to compute the energy released when rest mass converts to kinetic energy.

The total energy is

\[ E = \gamma mc^{2} \]

where

  • \(E\) is the total energy.
  • \(c\) is the speed of light.

The energy-momentum invariant. The squared total energy equals the squared momentum term plus the squared rest energy, and every inertial observer agrees on that combination. An invariant is a quantity whose value is the same in every inertial frame. This principle is used to relate energy and momentum in collisions without finding the velocity.

The energy-momentum relation is

\[ E^{2} = (pc)^{2} + (mc^{2})^{2} \]

Massless kinematics. A massless particle has zero rest mass and travels at the speed of light. A massless particle is a quantum of a field with vanishing rest mass. This principle is used to treat photons and other lightlike carriers.

The massless energy-momentum relation is

\[ E = pc \]

where

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

Related definitions:

199.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(p=\gamma mv\); \(E=\gamma mc^{2}\); rest energy.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(p=\gamma mv\); \(E^{2}=(pc)^{2}+(mc^{2})^{2}\).
  4. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — energy-momentum invariant; \(E=pc\) for \(m=0\).
  5. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — rest energy; massless kinematics.
  6. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — relativistic momentum; \(E^{2}=(pc)^{2}+(mc^{2})^{2}\); photons.