Special relativity · Electrodynamics §7
Frequency and direction transform together
from the invariance of the phase.
How do the frequency and propagation direction of light transform between frames?
SR-09 · Doppler and aberration
Doppler principle and aberration
The frequency and direction of light transform together because a plane wave's phase is invariant between frames. Observers in relative motion see the same light wave shifted in frequency and arriving from a different apparent angle.
Under a boost along x, a light ray at angle θ in K has frequency ratio ν'/ν = γ(1 - β cos θ) and apparent angle cos θ' = (cos θ - β)/(1 - β cos θ). Along the line of motion, the Doppler factor is √(1-β)/√(1+β) for receding and √(1+β)/√(1-β) for approaching rays.
At right angles in the stationary system (θ = 90°), the frequency increases by γ (transverse Doppler effect) and the ray swings forward to cos θ' = -β (aberration). Ives and Stilwell confirmed the second-order shift in 1938.
Section 7 derives both principles strictly from phase invariance φ' = φ. Wavefront crossings counted by a moving detector match the transformed frequency ν' exactly. Frequency and wave vector transform as a 4-covector.
Type a speed and an angle, or choose a named ray. Frequency and direction come from one wave-vector transform. The transverse case (θ = 90°) is the purely relativistic shift: the medium formulae give no change.
Accepted snapshot
| Relativistic ν′/ν | 0.5 |
|---|---|
| Lorentz factor γ | 1.25 |
| Medium, moving observer | 0.4 |
| Medium, moving source | 0.625 |
| Receding line of sight | 0.5 |
| Approaching line of sight | 2 |
| Angle in k | 0 ° |
At θ = 90° the two medium formulae both give 1. The paper's factor is γ. That is the transverse Doppler shift, which has no classical counterpart. The medium formulae are not declared refuted: they are right for sound and agree with the paper to first order in v/c.
Not modeled: media and dispersion; sound in a medium; gravitational redshift; finite packets (SR-10); telescope optics and atmospheric refraction; photon picture; canal-ray apparatus beyond published values.
Worked case (readable without JavaScript)
Consider a light wave with frequency ν = 500 THz propagating at angle θ = 0° along the x-axis in the stationary system K, viewed by an observer moving along the x-axis at speed v = 0.6c (β = 0.6, γ = 1.25).
For a ray at right angles in the stationary system (θ = 90°):
The phase φ = k·x − ωt is an invariant scalar that takes the exact same numerical value in both reference frames at every spacetime event.