Last updated: 2026-06-16

Relativistic Doppler Effect

The relativistic Doppler effect describes how the frequency and wavelength of light change when a source and an observer move relative to one another at speeds approaching the speed of light $c \approx 3 \times 10^8\ \text{m/s}$. It generalises the familiar Doppler shift of sound to electromagnetic waves, where there is no medium and time dilation must be taken into account.

This calculator takes a source frequency $f_s$ and a relative velocity $v$, and returns the observed frequency $f_o$, the observed wavelength $\lambda_o$, and the velocity ratio $\beta = v/c$ for a source that is either receding or approaching along the line of sight.

How the relativistic Doppler effect works

For sound, the Doppler shift depends on motion relative to the air. Light carries no medium, so only the relative velocity between source and observer matters. Special relativity adds the Lorentz time-dilation factor, which slows the source's emission as measured by the observer. Combining the geometric path-length change with time dilation gives the longitudinal Doppler formula.

A source moving away stretches successive wavefronts, lowering the frequency and lengthening the wavelength — a redshift. A source moving toward the observer compresses the wavefronts, raising the frequency and shortening the wavelength — a blueshift.

Formula

As $\beta \to 0$ the shift vanishes and $f_o \to f_s$. As $\beta \to 1$ a receding source's frequency drops toward zero, while an approaching source's frequency grows without bound.

Worked example

A radio source emits at $f_s = 100\ \text{MHz}$ and recedes at v = 0.6c, so β = 0.6.

Step 1 — Doppler factor:

Step 2 — Observed frequency:

The observer measures 50 MHz, half the emitted frequency — a strong redshift. Had the source been approaching at the same speed, the factor would invert to 2 and the observed frequency would be 200 MHz. Enter these values in the calculator to reproduce the result.

Shift at different speeds

Real-world relevance

Astronomers use the relativistic Doppler shift to measure how fast stars and galaxies move along the line of sight. The systematic redshift of distant galaxies, where recession speed grows with distance, is central evidence for the expanding universe. The effect also matters in particle physics and in precision tests of relativity such as the Ives–Stilwell experiment, which confirmed the time-dilation contribution to the shift.

Limitation: longitudinal motion along the line of sight

This calculator models motion directly toward or away from the observer, where the shift is largest. For motion perpendicular to the line of sight there is a smaller transverse Doppler effect, $f_o = f_s\sqrt{1 - \beta^2}$, arising purely from time dilation. General-relativistic gravitational shifts near very massive bodies are a separate effect.