Relativistic Doppler Shift Calculator

Calculate the observed frequency and wavelength of light from a relativistically moving source.

Last reviewed: June 2026
Built and maintained by · Last updated June 2026
THz
545 THz = green light (~550 nm)
c
Positive = approaching (blueshift), Negative = receding (redshift)
Observed Frequency
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Source Wavelength
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Observed Wavelength
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Frequency Ratio
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Doppler Factor
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Educational model: This calculator models line-of-sight relativistic Doppler shift for light from the values entered. It does not replace spectroscopy software, instrument calibration, radar/lidar analysis, or cosmological-redshift modeling.

What this relativistic Doppler calculator does

This calculator estimates how the frequency and wavelength of light change when a source moves directly toward or away from the observer at a chosen fraction of the speed of light. Enter the source frequency in THz and beta, where beta is velocity divided by the speed of light. The page then shows observed frequency, source wavelength, observed wavelength, the frequency ratio, the Doppler factor, and an approximate color swatch for visible wavelengths.

The sign convention is calculator-specific: positive beta means approaching and produces a blueshift; negative beta means receding and produces a redshift. That is why the default formula uses a larger observed frequency when beta is positive.

Formula used

For a light source moving along the line of sight, the implemented frequency relationship is:

f_obs = f_src * sqrt((1 + beta) / (1 - beta))

where beta = v / c, f_src is the source frequency, and f_obs is the observed frequency. The calculator uses c = 299,792.458 nm*THz for wavelength conversion, so wavelength_nm = 299,792.458 / frequency_THz.

Worked examples

With the default 545 THz source and beta = 0, the observed frequency remains 545 THz. The source and observed wavelengths both round to about 550.08 nm, which is visible green light.

If beta is changed to 0.10, the Doppler factor is about 1.105542. The observed frequency rises to about 602.52 THz and the observed wavelength shortens to about 497.57 nm, a blueshift toward cyan-blue.

If beta is changed to -0.10, the Doppler factor is about 0.904534. The observed frequency falls to about 492.97 THz and the observed wavelength lengthens to about 608.14 nm, a redshift toward orange-red.

Common mistakes

  • Using the sound-wave formula: this calculator is for light. Sound Doppler calculations depend on the medium and separate source/observer velocities.
  • Reversing the sign: on this page positive beta means approaching, even though some textbooks define positive velocity as receding when writing the wavelength form.
  • Expecting cosmological redshift: this is a special-relativity line-of-sight model. It does not model expansion of space, gravity, transverse Doppler shift, or cosmological distance.
  • Reading the color swatch too literally: the swatch is an approximate visible-color aid. Spectral line identification and instrument response require more than a simple RGB mapping.

Sources and further reading

For the relativistic Doppler relationship, see the OpenStax University Physics key equations for relativity. For astronomy context on spectral shifts, see NASA's Webb explainer on spectroscopy, redshift, and blueshift.

Frequently Asked Questions

Positive beta means the source is approaching the observer, which increases observed frequency and shortens observed wavelength. Negative beta means the source is receding, which lowers observed frequency and lengthens observed wavelength.
For light along the line of sight, the calculator uses f_obs = f_src * sqrt((1 + beta) / (1 - beta)). It then converts source and observed frequencies to wavelengths using c = 299,792.458 nm*THz.
No. This page is for relativistic light or electromagnetic radiation. Sound in air uses a different classical formula because the medium matters and source and observer motion are handled separately.
Beta is velocity divided by the speed of light. Massive objects cannot reach or exceed beta = 1, and the relativistic Doppler factor diverges as beta approaches 1.