DOPPLER (SR)
At 0.6c the light ahead exactly doubles — and abeam, time dilation shows its bare face.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A lamp of one pure color sweeps past three receivers: one dead ahead, one dead behind, one exactly abeam. Sound has a medium that referees such questions; light has none — so only β can matter, and the postulate does the rest.
Ahead and behind
Ahead, the wavefronts crowd; behind, they stretch — by the factor √((1+β)/(1−β)) and its reciprocal. At β = 0.6 the light ahead exactly doubles in frequency and the light behind exactly halves, and the product of the two returns f₀² to the bit. The recession stretch is the z astronomers hang on galaxies.
The tell
The abeam receiver is the purely relativistic tell: no approach, no recession, so classical physics predicts no shift at all. Relativity disagrees — f₀/γ, time dilation with no classical alibi. And the kinship runs deeper: the full relativistic formula IS the classical moving-source result times 1/γ, an exact identity this page verifies.
The sweep
Each ring is centered where the source was at emission, and successive lab-frame emissions are separated by the time-dilated interval γ/f₀. The resulting forward and rear spacings are proportional to γ(1−β) and γ(1+β), exactly matching the relativistic Doppler factors.
Audit
Audited: the approach–recession product returns f₀² exactly across the whole β range; the transverse shift lands on f₀/γ bitwise at the defaults; and the kinship identity — relativistic equals classical-source times 1/γ — holds at machine epsilon everywhere. The classical formula was one slow clock short of the truth.