The Relativistic Doppler Effect
When a light source moves relative to an observer, the observed frequency and wavelength differ from those in the source's rest frame. This is the Doppler effect. In special relativity, this effect has both classical and purely relativistic components.
The General Formula
For a source moving at velocity v = βc at angle θ relative to the line of sight, the observed frequency f' is related to the rest-frame frequency f by:
f' = f × γ(1 - β cos θ)
where γ = 1/√(1-β²) is the Lorentz factor. This single formula encompasses both the classical Doppler effect (from changing distance) and time dilation.
Longitudinal Doppler (θ = 0° or 180°)
Approaching (θ = 0°)
When the source approaches head-on (cos θ = 1), the formula becomes:
f' = f × γ(1 - β) = f × √[(1-β)/(1+β)] × √[(1+β)/(1-β)] = f × √[(1+β)/(1-β)]
This is the blueshift formula. Light from an approaching source has higher frequency (shorter wavelength, shifted toward blue). At β = 0.6, this gives a factor of 2.0, doubling the frequency.
Receding (θ = 180°)
When the source recedes (cos θ = -1):
f' = f × γ(1 + β) = f × √[(1-β)/(1+β)]
This is the redshift formula. Light from a receding source has lower frequency (longer wavelength, shifted toward red). This is why distant galaxies appear redshifted — they're receding due to cosmic expansion.
Transverse Doppler (θ = 90°)
The most surprising case is when the source moves perpendicular to the line of sight (cos θ = 0). Classically, there should be no Doppler effect at this instant, since the distance isn't changing. But relativistically:
f' = f / γ
The observed frequency is always lower than the rest-frame frequency, regardless of direction. This is pure time dilation: the moving source's clock runs slow, so its oscillations appear stretched out. At β = 0.8 (γ = 1.67), the transverse redshift is 40%.
This effect has no classical analog. It's a direct consequence of time dilation and was verified experimentally in 1938 by Ives and Stilwell using hydrogen canal rays.
Aberration and Beaming
At high velocities, relativistic aberration causes photons to be "beamed" forward. Even if a source emits isotropically in its rest frame, most photons appear concentrated in a narrow cone in the forward direction when viewed from the lab frame.
The half-angle of this cone is approximately θ ≈ 1/γ. For β = 0.99 (γ ≈ 7), photons are confined to a ~8° cone. This is the searchlight effect or relativistic beaming.
Color Visualization
In the visualization above, the observed color corresponds to the Doppler-shifted wavelength. Green light (λ = 500 nm) from an approaching source (β = 0.6) shifts to blue (~400 nm). From a receding source, it shifts to red (~625 nm).
At extreme velocities, visible light can shift completely out of the visible spectrum. A source emitting green light at β = 0.99 receding produces infrared (λ ≈ 3500 nm), while approaching it produces ultraviolet (λ ≈ 70 nm).
Real-World Applications
Astrophysical Jets
Relativistic jets from active galactic nuclei and gamma-ray bursts move at β > 0.99. Due to beaming, we only observe jets pointed nearly toward us. The Doppler factor for these jets can exceed 10, making them appear far brighter than they would be at rest.
Cosmic Redshift
While cosmological redshift is fundamentally different from Doppler shift (it's due to expanding space, not motion through space), the formulas are similar. A galaxy at z = 1 has doubled wavelengths, corresponding to β ≈ 0.6 if it were a Doppler shift.
Particle Colliders
In the LHC, protons at β = 0.999999991 produce synchrotron radiation that's Doppler-shifted from the infrared/microwave regime to X-rays due to the enormous γ ≈ 7460. This radiation must be accounted for in beam diagnostics.
GPS Satellites
GPS satellites orbit at ~4 km/s (β ≈ 10⁻⁵). The transverse Doppler shift causes their clocks to run slow by ~7 microseconds per day. Combined with gravitational time dilation (+45 μs/day), the net effect is +38 μs/day, which must be corrected for GPS accuracy.
Things to Try
- Symmetric Redshift/Blueshift: Set θ = 0° and β = 0.6. Note the wavelength. Now set θ = 180° with the same β. The redshift is not symmetric with the blueshift! k(approaching) × k(receding) = 1, but they're not reciprocals.
- Transverse Redshift: Set β = 0.8 and θ = 90°. There's a 40% redshift purely from time dilation. Now vary the angle slightly — notice the asymmetry between θ = 85° and θ = 95°.
- Extreme Velocity: Use the "Extreme" preset (β = 0.99, θ = 0°). Green light (500 nm) becomes deep ultraviolet (71 nm) — a factor of 7 blueshift. Now set θ = 180° and watch it redshift to far infrared (3536 nm).
- Color Shifts: Start with red light (λ = 650 nm) and β = 0.6 approaching. It blueshifts to green. Start with violet (λ = 400 nm) and β = 0.6 receding — it redshifts to orange. The visible spectrum is narrow!
- Beaming Effect: Set β = 0.9 and sweep θ from 0° to 180°. Notice how the Doppler factor drops sharply near θ = 0°. Most of the photon energy is concentrated in a narrow forward cone.
Mathematical Derivation
The Doppler formula comes from the Lorentz transformation of 4-momentum. A photon with frequency f has 4-momentum (E, p) = (hf, hf/c) in the direction of motion. Transforming to a frame moving at velocity β:
E' = γ(E - βp_x)
p'_x = γ(p_x - βE)
For a photon traveling at angle θ to the x-axis, p_x = (E/c) cos θ. Thus:
E' = γE(1 - β cos θ)
Since E = hf, this immediately gives f' = f × γ(1 - β cos θ). The angle-dependent term (1 - β cos θ) is the classical Doppler effect; the factor γ is pure time dilation.
Connection to Other Phenomena
The Doppler effect is intimately connected to aberration (how angles transform under boosts) and the relativistic beaming of radiation. Together, these effects explain why pulsars appear to flash (beaming from rotation), why superluminal jets appear to move faster than light (projection effects + Doppler), and why the cosmic microwave background has a dipole anisotropy (our motion through the CMB rest frame).
The transverse Doppler effect, in particular, is a direct window into time dilation. Unlike length contraction (which involves simultaneity and is more subtle), the transverse redshift is unambiguous: moving clocks tick slower, period. This was one of the first quantitative tests of special relativity.