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Mathematics for Physics

208 Velocity Addition

A transformation rule for combining relative velocities that is used to convert a particle’s speed between inertial frames moving at constant relative speed.

Non-linear velocity combination. Relative velocities combine non-linearly, so two speeds each below the speed of light still yield a combined speed below the speed of light. Non-linear combination is a rule whose result does not grow as a simple sum. This principle is used to compute a particle’s speed as measured by a moving observer.

The Lorentz velocity transformation is

\[ u' = \dfrac{u - v}{1 - \dfrac{uv}{c^{2}}} \]

\[ u = \dfrac{u' + v}{1 + \dfrac{u'v}{c^{2}}} \]

where

  • \(u\) is the particle’s velocity along \(x\) in frame \(S\).
  • \(u'\) is the particle’s velocity along \(x'\) in frame \(S'\).
  • \(v\) is the velocity of \(S'\) relative to \(S\) along the shared \(x\)-axis.
  • \(c\) is the speed of light in vacuum.

Invariance of the speed of light. A signal that travels at the speed of light in one inertial frame travels at the speed of light in every inertial frame. This principle is used to keep \(c\) an unchanged upper speed under a change of frame.

Low-speed Galilean correspondence. When the speeds are much smaller than the speed of light, the relativistic rule reduces to ordinary Galilean addition. Galilean velocity addition is the classical rule that adds relative speeds. This principle is used to recover Newtonian relative motion at everyday speeds.

Velocity as a ratio of coordinate differentials. Velocity is the ratio of a spatial displacement to a time displacement. A coordinate differential is a small change in a space or time coordinate. This principle is used to obtain the velocity rule by dividing the Lorentz space transformation by the time transformation.

Note: Also called the Lorentz velocity transformation. Also called the Einstein velocity addition law.

208.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — Lorentz velocity transformations; invariance of \(c\); Galilean limit.
  2. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — inverse relation \(u=\dfrac{u'+v}{1+\dfrac{u'v}{c^{2}}}\); Galilean limit.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — non-linear combination; Galilean correspondence.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — Einstein velocity addition; velocity as \(\dfrac{dx}{dt}\).
  5. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — velocity as a ratio of Lorentz coordinate differentials.