The Relativistic Doppler Effect

You already know the everyday Doppler effect: an ambulance siren drops in pitch the instant it passes you, a racing car's engine wails high on approach and groans low as it leaves. The pitch shifts because each wave crest is emitted from a slightly different place, bunching the crests up ahead of a source that's coming towards you and stretching them out behind. The same thing happens to light — a source rushing towards you looks bluer, one racing away looks redder. This is how we know the Universe is expanding: almost every distant galaxy's light is shifted towards the red, as if the whole cosmos were flying apart.

But light is special. Its "speed through the medium" is c for everyone, and a fast-moving source carries its own slowed-down clock. So the light Doppler formula is not quite the sound one — it carries an extra dose of relativity baked in. This page builds the relativistic Doppler shift, shows how it splits into a familiar wave-bunching piece and a pure time-dilation piece, and reveals a shift that has no classical counterpart at all — the transverse Doppler effect.

Two effects, multiplied together

Picture a source moving directly away from you at speed v, emitting light of frequency f_{\text{src}} in its own frame. Two things happen at once when you receive it:

Classical wave-stretching. Between emitting one crest and the next, the source has receded a little, so each successive crest has farther to travel and arrives later. This is the ordinary Doppler stretching — it lowers the frequency by a factor 1/(1+\beta) for a receding source (with \beta = v/c).
Time dilation. The source's own clock, which sets the emission rate, runs slow as seen by you — by the Lorentz factor \gamma. So even the rate of emission is reduced. This piece is purely relativistic; sound doesn't have it.

Multiply the two effects together and simplify (the algebra collapses the \gamma and the 1+\beta into a tidy square root), and you get the relativistic Doppler formula for a source and observer separating:

f_{\text{obs}} = f_{\text{src}}\sqrt{\frac{1 - \beta}{1 + \beta}} \qquad (\text{receding} \Rightarrow \text{redshift}).

If instead the source is approaching, flip the sign of \beta (or of the two terms):

f_{\text{obs}} = f_{\text{src}}\sqrt{\frac{1 + \beta}{1 - \beta}} \qquad (\text{approaching} \Rightarrow \text{blueshift}).

Receding (moving apart): f_{\text{obs}} = f_{\text{src}}\sqrt{\dfrac{1-\beta}{1+\beta}} < f_{\text{src}} — frequency drops, light reddens.
Approaching: f_{\text{obs}} = f_{\text{src}}\sqrt{\dfrac{1+\beta}{1-\beta}} > f_{\text{src}} — frequency rises, light blues.
The two factors are reciprocals: approach then recede at the same speed and the shifts exactly undo. Wavelength shifts the opposite way to frequency (\lambda_{\text{obs}} = \lambda_{\text{src}} / (\text{that factor})).

The shift factor, plotted

The graph shows the frequency ratio f_{\text{obs}}/f_{\text{src}} against \beta for three cases: a source approaching head-on (blueshift, above 1), receding head-on (redshift, below 1), and moving purely sideways (the transverse case, discussed next). Notice how the approach and recession curves are mirror images through the line f_{\text{obs}}/f_{\text{src}} = 1, and how both run away from 1 ever faster as \beta \to 1.

Two limits are worth carrying: at small \beta the shift is nearly linear, f_{\text{obs}} \approx f_{\text{src}}(1 \mp \beta) — the classical result, since time dilation is second order and negligible. As \beta \to 1 a receding source's light reddens all the way to zero frequency (infinite wavelength), while an approaching source's light blueshifts without bound.

The transverse Doppler effect: time dilation, laid bare

Here is the purely relativistic surprise. Suppose the source flies straight past you and you catch the light at the moment it is exactly abreast — moving sideways, neither approaching nor receding. Classically there'd be no Doppler shift at all: the distance isn't changing, so the crests shouldn't bunch or stretch. Yet relativity predicts a shift — a redshift:

f_{\text{obs}} = \frac{f_{\text{src}}}{\gamma} = f_{\text{src}}\sqrt{1 - \beta^2} \qquad (\text{transverse}).

With the classical wave-bunching piece switched off (no radial motion), the only thing left is the time dilation — the moving source's clock ticking slow — and so the transverse Doppler shift is direct, naked time dilation. It's a lovely payoff: an effect that cannot exist in the classical theory becomes a clean, independent confirmation that moving clocks really do run slow. This is exactly what the famous 1938 Ives–Stilwell experiment measured, giving one of the first direct laboratory tests of time dilation.

For light received from a source moving purely sideways, f_{\text{obs}} = f_{\text{src}}/\gamma = f_{\text{src}}\sqrt{1-\beta^2}.
It is always a redshift (frequency lowered), regardless of direction.
It is pure time dilation — a shift with no classical counterpart, and hence a direct test of relativity.

Worked examples

Example 1 — a receding star. A star recedes at \beta = 0.1 (about 30{,}000\ \text{km/s}). A spectral line it emits at 500\ \text{nm} arrives at what wavelength? Wavelength shifts inversely to frequency, so

\lambda_{\text{obs}} = \lambda_{\text{src}}\sqrt{\frac{1 + \beta}{1 - \beta}} = 500\sqrt{\frac{1.1}{0.9}} = 500 \times 1.106 \approx 553\ \text{nm}.

The green line arrives shifted towards orange-red — a redshift of about 10.6\%.

Example 2 — the redshift z of a quasar. Astronomers report shift as 1 + z = \lambda_{\text{obs}}/\lambda_{\text{src}}. A quasar shows z = 2, so its light is stretched threefold (\lambda_{\text{obs}} = 3\lambda_{\text{src}}). Reading it as a relativistic Doppler shift, 3 = \sqrt{(1+\beta)/(1-\beta)}. Squaring gives 9(1-\beta) = 1+\beta, so \beta = 8/10 = 0.8 — an apparent recession at 80\% of light speed.

Example 3 — transverse redshift. An ion in a storage ring moves at \beta = 0.6. Light seen from it edge-on (transverse) is shifted by 1/\gamma = \sqrt{1 - 0.36} = 0.8: a 500\ \text{nm} line is received at 500/0.8 = 625\ \text{nm}, a 25\% redshift from time dilation alone, with no approach or recession involved.

Not quite — and this is a beautiful subtlety. For nearby galaxies the redshift really is an ordinary relativistic Doppler shift from motion through space, and reading off a velocity is fine. But the redshift of very distant galaxies is better understood as a cosmological redshift: space itself is expanding, and the light waves are stretched along with it as they travel — the wavelength grows by exactly the factor the Universe has expanded during the journey. The galaxies aren't so much hurtling through space as being carried apart by it. That's why the most distant objects can show redshifts implying "speeds" above c without breaking relativity: no object is overtaking a light beam locally; it's the stretching of space in between that piles up. The Doppler picture on this page is the exact, correct story for a source moving through space; the cosmological version is its grand generalisation once gravity and an expanding Universe enter.

Don't use the classical sound-Doppler formula for light, and don't forget which way a redshift goes. Two frequent slips:

The light formula \sqrt{(1\mp\beta)/(1\pm\beta)} is not the sound formula 1/(1\pm\beta). They agree only to first order in \beta; the extra \gamma (time dilation) matters once \beta isn't tiny. Sound also lets you tell "source moving" from "observer moving" (there's a medium); for light, only the relative velocity matters.
Recede → red (lower frequency, longer wavelength); approach → blue. It's easy to flip. Anchor it on the word: a source running away looks red. And a redshift lowers frequency but raises wavelength — the two always move oppositely.