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Mathematics for Physics

198 Relativistic Momentum

The momentum of a particle at high speed, equal to mass times velocity times the Lorentz factor, so momentum conservation holds in all inertial frames, that is used to generalize classical momentum to high speeds.

Frame-invariant momentum conservation. Relativistic momentum is defined so that if the total momentum of an isolated system is conserved in one inertial frame, it is conserved in every inertial frame. Relativistic momentum is rest mass times ordinary velocity times the Lorentz factor. This principle is used to keep momentum conservation valid under Lorentz transformations.

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.

Non-linear momentum growth. For a massive particle, \(p\) grows without bound as the speed approaches the speed of light. This principle is used to show that a constant force can raise the momentum indefinitely while the speed stays below \(c\).

Newtonian correspondence. When the speed is much smaller than the speed of light, \(p\) reduces to \(mv\). This principle is used to recover Newtonian momentum at everyday speeds.

Force as the rate of change of relativistic momentum. Force at high speed is the rate of change of relativistic momentum with coordinate time. Relativistic force is \(\dfrac{dp}{dt}\). This principle is used to compute the paths of high-energy charges in magnetic fields.

The relativistic force is

\[ \mathbf{F} = \dfrac{d\mathbf{p}}{dt} \]

where

  • \(\mathbf{F}\) is the force.
  • \(\mathbf{p}\) is the relativistic momentum.
  • \(t\) is coordinate time.

198.1 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. — \(p=\gamma mv\); conservation in all frames; Newtonian limit.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(p=\gamma mv\); \(\mathbf{F}=\dfrac{d\mathbf{p}}{dt}\).
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — unbounded \(p\) as \(v\to c\); force as \(\dfrac{dp}{dt}\).
  5. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — conservation of relativistic momentum; Newtonian correspondence.