What is the relativistic Doppler effect?

It is the change in observed frequency (and wavelength) of a wave when source and observer move relative to each other at speeds comparable to the speed of light. Unlike the classical Doppler effect, the relativistic version accounts for time dilation and gives exact results at any speed.

What is β (beta) in this context?

β = v/c is the velocity of the source as a fraction of the speed of light c = 299,792,458 m/s. A value of 0.1 means the source moves at 10% of c. β must be between 0 and 1 (exclusive) for a massive source.

What is the difference between blueshift and redshift?

Blueshift occurs when source and observer approach each other — observed frequency increases and wavelength decreases. Redshift occurs when they recede — observed frequency decreases and wavelength increases.

How accurate is the classical Doppler formula?

The classical formula f_obs = f_source × (1 ± β) is a first-order approximation. At β = 0.01 (1% of c) the error is about 0.005%. At β = 0.1 the error grows to about 0.5%, and at β = 0.5 it exceeds 15%. Use the relativistic formula for any velocity above roughly 1% c.

Does this apply to light and sound?

The relativistic Doppler formula applies exactly to light (and any massless wave). For sound in a medium, the classical Doppler formula using source and observer speeds relative to the medium is usually appropriate, since sound propagation speeds are far below c.

What is the Doppler factor k?

The relativistic Doppler factor k = √((1 + β) / (1 − β)) for approach and k = √((1 − β) / (1 + β)) for recession. Observed frequency = source frequency × k (approach) or ÷ k (recession). Observed wavelength is the reciprocal.

Relativistic Doppler Calculator

Name: Relativistic Doppler Calculator
Availability: InStock
Author: Nham Vu

Enter source frequency (or wavelength) and velocity as a fraction of c to get observed frequency, wavelength, and Doppler shift for approach and recession.

Source & Velocity

Input type

Source frequency

Relative speed β = v/c

Enter a value between 0 and 0.9999 (fraction of c).

Approaching (blueshift) Source moving toward observer

k = √((1 + β) / (1 − β))

Doppler factor k

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Shift

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Observed freq.

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Observed wavelength

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Receding (redshift) Source moving away from observer

k = √((1 − β) / (1 + β))

Doppler factor k

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Shift

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Observed freq.

—

Observed wavelength

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Classical vs. Relativistic Comparison

Case	Classical f_obs	Relativistic f_obs	Difference

Classical formula: f_obs = f_source × (1 ± β). Accurate only at β ≪ 1.

Doppler Factor Reference — Approach (Blueshift)

β (v/c)	Approach k	Recession k	Classical k (approach)	Context

Formulas

β = v / c — velocity as a fraction of the speed of light (c = 299,792,458 m/s)

f_obs (approach) = f_source × √((1 + β) / (1 − β))

f_obs (recession) = f_source × √((1 − β) / (1 + β))

λ_obs = c / f_obs — observed wavelength from observed frequency

The approach and recession factors are reciprocals of each other. Wavelength shifts by the inverse of the frequency shift.

Summary

Enter source frequency (or wavelength) and velocity as a fraction of c to get observed frequency, wavelength, and Doppler shift for approach and recession.

How it works

Select whether to enter source frequency (Hz, kHz, MHz, GHz, or THz) or source wavelength (nm, µm, mm, or m).
Enter the source value in the chosen unit.
Enter the relative speed as a fraction of the speed of light (β = v/c), between 0 and 0.9999.
Click Calculate. The tool applies the relativistic Doppler formula for both approach and recession.
Results show observed frequency, observed wavelength, percentage shift, and the relativistic Doppler factor for each case.
A comparison row shows the classical Doppler prediction so you can see where the approximation breaks down.

Use cases

Find the observed frequency of a star moving toward Earth at 10% of c.
Calculate the redshift of a galaxy receding at 0.5c to compare with spectroscopic measurements.
Verify special-relativity homework problems involving Doppler frequency shifts.
Compare relativistic and classical Doppler predictions at various speeds to understand when relativistic corrections matter.
Determine the observed wavelength of a laser source on a fast-moving spacecraft.

Frequently Asked Questions

Last updated: 2026-07-23 · Reviewed by Nham Vu