Light changes pitch with motion — and it does so exactly, not by the sound-wave rule. A source rushing toward you is blueshifted by √((1+β)/(1−β)); rushing away, redshifted by √((1−β)/(1+β)); and — with no classical analog at all — a source crossing sideways is redshifted by exactly 1/γ, pure time dilation. Down the center the frequency goes in, the boost acts, the shifted frequency comes out. The blue team builds it; the red team attacks it.
source A. Einstein, Zur Elektrodynamik bewegter Körper (1905), Ann. Phys. 17, 891–921, §7 — doi:10.1002/andp.19053221004; transverse term confirmed by Ives & Stilwell (1938), JOSA 28, 215. Rendered, not quoted.
Set β = v/c and γ = 1/√(1−β²). The full relativistic shift factor for light emitted at angle θ (in the observer's frame) is D = 1 / [ γ (1 − β cos θ) ]. Three cases fall out:
| geometry | fobs / f | shift |
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Approaching (θ=0) and receding (θ=π) are reciprocal: their factors multiply to 1. The transverse case (θ=π/2) leaves only the 1/γ time-dilation term — a shift Newtonian acoustics cannot produce.
Take the classical the-doppler-effect — pitch rising and falling with approach and recession — and correct the clock doing the counting. The moving emitter's own time runs slow by 1/γ, from the-time-dilation.
Multiply the classical geometric factor by that 1/γ and the whole thing collapses to Einstein's √((1−β)/(1+β)) — and to a shift that survives even when the classical factor is 1 (sideways motion). Each sphere is the next one's premise. Neighbour slugs AMBER — verify at publish.
The blue team's live check: recompute the receding, approaching, reciprocal and transverse-1/γ relations from the pure functions and confirm them against the known truth. If red tampers with the γ term, this badge is where it shows.
Three inputs feed the engine: a rest-frame source frequency f, a relative speed β = v/c (0 ≤ β < 1), and a geometry — is the source approaching, receding, or crossing transverse to the line of sight?
Only the relative velocity enters — there is no medium and no preferred rest frame, unlike sound. That symmetry is the whole difference, and it is what you feed the panel below.
Move any control — fobs is computed live from √((1∓β)/(1±β)) and 1/γ, never looked up.
What the machine produces, proven: a receding source is redshifted (fobs < f), an approaching source blueshifted (fobs > f), the two factors reciprocal to 1e−12, and a transverse source redshifted by exactly 1/γ with no classical analog. The current fobs is above; the invariants are the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
And "transverse" hides an angle. The 1/γ redshift holds when the photon arrives at 90° in the observer's frame. If instead emission is perpendicular in the source frame, the same word "transverse" gives a blueshift by γ. One number, two conventions — the panel uses the observer-frame definition.
"Doppler is source-vs-observer motion, like sound." Cut. For light there is no medium and no rest frame; the shift depends only on the relative velocity — symmetric between the two.
"A source moving sideways causes no shift." Cut. The transverse Doppler redshifts by 1/γ — pure time dilation — measured by Ives & Stilwell in 1938. Acoustics predicts zero.
"Shift ≈ 1∓β, linear in speed." Kept, corrected. That is only the first-order term; the exact factor is √((1∓β)/(1±β)), and the two directions must multiply to 1 — which linear factors do not.
The red team's move: strike the γ out of the transverse term — set the transverse shift to 1 (the classical "no sideways shift"). Slip that past and the transverse-redshift-by-1/γ claim is false. The blue team's witness (window 7) is watching.
Delete the time-dilation factor and a transverse source shows no shift — the witness recomputes fobs against f/γ, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.