page_title: Velocity-addition formula
section_title: Relativistic Doppler shift
hierarchical_section_title: Velocity-addition formula / Applications / Relativistic Doppler shift
caption_attribution_description: English: Photographic portrait of Christian Doppler (1803-1853)
context_page_description: In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as Thomas precession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.
  Standard applications of velocity-addition formulas include the Doppler shift, Doppler navigation, the aberration of light, and the dragging of light in moving water observed in the 1851 Fizeau experiment.
  The notation employs u as velocity of a body within a Lorentz frame S, and v as velocity of a second frame S′, as measured in S, and u′ as the transformed velocity of the body within the second frame.
context_section_description: Here velocity components will be used as opposed to speed for greater generality, and in order to avoid perhaps seemingly ad hoc introductions of minus signs. Minus signs occurring here will instead serve to illuminate features when speeds less than that of light are considered.
  For light waves in vacuum, time dilation together with a simple geometrical observation alone suffices to calculate the Doppler shift in standard configuration (collinear relative velocity of emitter and observer as well of observed light wave).
  All velocities in what follows are parallel to the common positive x-direction, so subscripts on velocity components are dropped. In the observers frame, introduce the geometrical observation
  as the spatial distance, or wavelength, between two pulses (wave crests), where T is the time elapsed between the emission of two pulses. The time elapsed between the passage of two pulses at the same point in space is the time period τ, and its inverse ν = ​¹⁄τ is the observed (temporal) frequency. The corresponding quantities in the emitters frame are endowed with primes.
  For light waves
  and the observed frequency is
  where T = γVT′ is standard time dilation formula.
  Suppose instead that the wave is not composed of light waves with speed c, but instead, for easy visualization, bullets fired from a relativistic machine gun, with velocity s′ in the frame of the emitter. Then, in general, the geometrical observation is precisely the same. But now, s′ ≠ s, and s is given by velocity addition,
  The calculation is then essentially the same, except that here it is easier carried out upside down with τ = ​¹⁄ν instead of ν. One finds
  Details in derivation
  Observe that in the typical case, the s′ that enters is negative. The formula has general validity though. When s′ = −c, the formula reduces to the formula calculated directly for light waves above,
  If the emitter is not firing bullets in empty space, but emitting waves in a medium, then the formula still applies, but now, it may be necessary to first calculate s′ from the velocity of the emitter relative to the medium.
  Returning to the case of a light emitter, in the case the observer and emitter are not collinear, the result has little modification,
  where θ is the angle between the light emitter and the observer. This reduces to the previous result for collinear motion when θ = 0, but for transverse motion corresponding to θ = π/2, the frequency is shifted by the Lorentz factor. This does not happen in the classical optical Doppler effect.
caption: Christian Doppler (1803–1853) was an Austrian mathematician and physicist who discovered that the observed frequency of a wave depends on the relative speed of the source and the observer.
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