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Mathematics for Physics

180 Massless Particles

A particle with zero rest mass that always travels at the speed of light in vacuum, with energy and momentum proportional to each other.

Perpetual motion at \(c\). A massless particle is never at rest. It travels at the speed of light in vacuum for every inertial observer. This principle is used to treat photons as the velocity standard of spacetime.

The linear energy-momentum coupling. For zero rest mass, total energy is proportional to momentum. Rest mass is the mass in a rest frame, and a massless particle has none. This principle is used to compute the momentum of radiation from its energy.

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.

Absence of a rest frame. No inertial frame exists in which a massless particle is at rest. A rest frame is a coordinate system in which the particle’s spatial coordinates stay fixed. This principle is used to explain why proper time is not defined along a photon’s path.

Null worldlines. The worldline of a massless particle is a null path. A lightlike worldline is a curve on which the spacetime interval vanishes. This principle is used to draw the light cone that bounds causal influence.

The indeterminate \(\gamma m\) limit. Energy and momentum remain finite because \(m=0\) and \(v=c\) together make the usual \(\gamma m\) expressions an indeterminate limit. An indeterminate limit is that singular form of those expressions. This principle is used to allow massless energy carriers that Newtonian mechanics forbids.

180.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. — source for the heading explanation.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(E=pc\); photons always at \(c\); indeterminate \(\gamma m\) limit.
  3. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — \(v=c\); \(E=pc\); no rest frame.
  4. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — null paths; no proper time for photons.
  5. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — lightlike worldlines; no rest frame.
  6. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — photons travel at \(c\); \(E=pc\).