Relativistic Doppler Effect Calculator

Calculate observed frequency including both Doppler effect and time dilation at relativistic speeds.

Light: ~10¹⁵ Hz, Radio: 10⁶-10⁹
0.5 = half light speed
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Relativistic Doppler Shift Includes Time Dilation

For light, the Doppler shift at high relative speed must include special relativity rather than the classical sound-wave formula. For motion directly along the line of sight, the frequency ratio can be written fobs/fsrc=√((1−β)/(1+β)) for a receding source, where β=v/c. Reversing the sign describes approach and produces a blueshift.

Unlike sound, light needs no medium, so there is no separate source-versus-observer speed relative to an ether. The result is symmetric between inertial observers. At low speed, the relativistic formula approaches the familiar first-order Doppler shift, but at large β the time-dilation contribution becomes essential.

Receding: fobs=fsrc√((1−β)/(1+β)),   β=v/c
SymbolMeaningWhy it appears / units
βSpeed as a fraction of cDimensionless; must satisfy |β|<1 for massive observers.
fsrcEmitted frequencyHz in the source rest frame.
fobsObserved frequencyHz measured by the receiver.

A receding source has a lower observed frequency and longer wavelength; an approaching source has a higher frequency and shorter wavelength. For arbitrary viewing angle, use the full relativistic angular Doppler relation rather than the one-dimensional form.

The relativistic Doppler factor should reduce to 1 at zero relative speed. An approaching source should give a factor greater than 1 for observed frequency, while a receding source should give a factor below 1. Those limits quickly expose a reversed numerator or denominator.

Worked Examples

Example 1: Galaxy receding at v=0.5c, f₀=6×10¹⁴ Hz
f=6e14×√(0.5/1.5)
Result: f=3.46×10¹⁴ Hz — redshifted to IR
Relativistic redshift from cosmic recession
Example 2: Star approaching at v=0.8c
f=f₀×√(1.8/0.2)=f₀×3
Result: 3× frequency — strong blueshift
Relativistic jets show extreme blueshifts
Example 3: Receding at 0.6c
β=0.6 → fobs/fsrc=√(0.4/1.6)
Result: 0.5
The observed frequency is exactly half the emitted frequency for this line-of-sight speed.
Example 4: Approaching at 0.6c
Reverse the radial direction
Result: fobs/fsrc=2
Approach gives the reciprocal factor, showing the relativistic symmetry of the longitudinal shift.

Common Mistakes

⚠️
Using the classical sound Doppler equation for light

Classical formulas depend on speeds relative to a medium and fail as v approaches c.

⚠️
Mixing wavelength and frequency shift directions

Redshift means frequency decreases while wavelength increases; blueshift means the opposite.

⚠️
Using the longitudinal formula for transverse viewing

At 90° there is still a relativistic transverse Doppler shift from time dilation, requiring the angular formula.

Frequently Asked Questions

Classical vs relativistic Doppler?
Classical: f=f₀(1±v/c). Relativistic: f=f₀√[(1±β)/(1∓β)]. At v=0.5c: classical gives 1.5× but relativistic gives 1.73×. The difference is time dilation — moving clocks run slow, affecting source frequency.
Cosmological redshift?
Galaxy recession is not classical Doppler — it's space itself expanding (metric expansion). For z<0.1 Hubble law applies. For z>0.1 need full cosmological model. CMB photons have z≈1100.
What happens to the formula at low speed?
For |v| much smaller than c, expanding the square root gives approximately fobs/fsrc≈1−v/c for recession, matching the expected first-order Doppler behavior.
Can light from a receding source ever be blueshifted?
For purely radial recession in flat spacetime, no. More complicated situations involving gravity or cosmological expansion can combine multiple redshift and blueshift contributions.
What is the transverse Doppler effect?
When relative motion is perpendicular to the line of sight at the instant of observation, classical Doppler theory predicts no shift, but special relativity predicts a redshift associated with time dilation.
Is relativistic Doppler shift the same as cosmological redshift?
No. Relativistic Doppler shift describes relative motion in special relativity, while cosmological redshift arises from expansion of spacetime and is treated with cosmological models.
How does the sign convention affect relativistic Doppler shift?
Approach and recession must be distinguished explicitly. For light, an approaching source is blueshifted to higher observed frequency and shorter wavelength; recession produces redshift. Different algebraic forms use different signs for β, so identify the convention before substituting a signed velocity.

Formula Explorer connections

Interpretation: This relationship connects frequency, wavelength, speed, phase, intensity or resonance in an oscillating system. Assumption: Identify the medium, boundary conditions and reference frame. Linear waves, small amplitudes, nondispersive media or ideal resonance may be assumed.

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