Relativistic Doppler Calculator
The relativistic Doppler effect governs the observed frequency (or wavelength) of electromagnetic radiation emitted by a source moving at relativistic speeds. For a source moving directly toward the observer at velocity beta = v/c, the observed frequency is f' = f * sqrt((1 + beta)/(1 - beta)). For recession (moving away), replace beta with -beta or use f' = f * sqrt((1 - beta)/(1 + beta)). Unlike the classical Doppler formula, the relativistic version is symmetric between source and observer motions and includes a time dilation correction. It applies to all electromagnetic radiation including light, radio waves, and X-rays, and is essential for astronomical redshift measurements, GPS synchronization, and radar speed measurements at very high velocities.
Relativistic Doppler formula
Approaching (beta > 0): f' = f * sqrt((1 + beta) / (1 - beta))
Receding (beta < 0): f' = f * sqrt((1 - |beta|) / (1 + |beta|))
Redshift z = f / f' - 1 = lambda' / lambda - 1
For recession, z is positive (wavelength increases, frequency decreases). For approach (blueshift), z is negative. At small beta, f' = f * (1 + beta) recovers the classical result.
Doppler shift examples
- Source receding at 0.1c: observed frequency = 0.905 * f (9.5% redshift).
- Source approaching at 0.5c: observed frequency = 1.732 * f (blueshift).
- Source receding at 0.9c: observed frequency = 0.229 * f (z = 3.36, large cosmological redshift).
- Transverse Doppler (90 deg, no formula): f' = f / gamma (always redshift).
Relativistic Doppler effect: frequently asked questions
What is the relativistic Doppler effect?
The relativistic Doppler effect describes the change in observed frequency of a wave when the source moves relative to the observer at relativistic speeds. For motion directly toward the observer (approaching): f' = f * sqrt((1 + beta)/(1 - beta)), where beta = v/c. For recession: f' = f * sqrt((1 - beta)/(1 + beta)). These formulas differ from the classical Doppler formula because of time dilation.
How does the relativistic Doppler formula differ from classical?
Classical Doppler: f' = f * (c + vo) / (c - vs) where vo and vs are observer and source speeds. Relativistic Doppler for source approaching: f' = f * sqrt((1 + beta)/(1 - beta)). The relativistic formula includes a time dilation factor (the square root involving gamma) not present classically. Both agree at low velocities but diverge significantly at relativistic speeds.
What is cosmological redshift?
Cosmological redshift is not a Doppler shift but arises from the expansion of spacetime itself. However, for nearby galaxies the distinction is small, and the recession redshift z = (lambda_observed - lambda_emitted) / lambda_emitted = sqrt((1 + beta)/(1 - beta)) - 1 closely approximates the cosmological redshift for moderate recession velocities.
What is blueshift?
Blueshift occurs when the source approaches the observer: the observed frequency is higher (wavelength shorter, shifted toward blue). Redshift occurs for receding sources (lower frequency, longer wavelength). The Andromeda Galaxy is approaching the Milky Way at about 110 km/s (beta = 3.67 x 10^-4) and shows a blueshift.
What is the transverse Doppler effect?
The transverse Doppler effect is a purely relativistic phenomenon: when a source moves perpendicular to the line of sight (theta = 90 degrees), classical Doppler predicts no frequency shift, but relativistic time dilation still causes a redshift: f' = f / gamma. This effect has been confirmed experimentally with atomic beams and is a direct confirmation of special relativity.
Official sources
- OpenStax University Physics Vol. 3: Relativistic Velocity Transformation.
- NIST: Speed of Light in Vacuum (CODATA 2018).
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.