Relativistic Doppler Shift Calculator
Computes the observed frequency of light or radio waves when the source and observer move relative to each other at relativistic speeds. Essential for astronomy, radar physics, and understanding redshift in cosmology.
About this calculator
The classical Doppler formula breaks down at speeds near the speed of light. The relativistic Doppler effect accounts for both the change in wave compression and time dilation. For a source and observer moving directly toward each other, the observed frequency is: f_obs = f_s × √((1 + β) / (1 − β)), and for recession: f_obs = f_s × √((1 − β) / (1 + β)), where β = v/c is the ratio of relative velocity to the speed of light and f_s is the source frequency. When the source approaches, frequencies are blueshifted (higher); when receding, they are redshifted (lower). Astronomers use redshift measurements to determine how fast distant galaxies are moving away from us, directly supporting the expanding-universe model.
How to use
Suppose a star emits light at f_s = 600 THz (green light) and is moving toward Earth at v = 0.5c. Enter source frequency = 600×10¹² Hz, velocity = 0.5c, direction = approaching. The calculator applies: f_obs = 600×10¹² × √((1 + 0.5) / (1 − 0.5)) = 600×10¹² × √(1.5 / 0.5) = 600×10¹² × √3 ≈ 1,039 THz. This shifts the light well into the ultraviolet range — a dramatic blueshift caused by the star's relativistic approach speed.
Frequently asked questions
What is the difference between classical and relativistic Doppler shift?
The classical Doppler formula f_obs = f_s × (c ± v_obs)/(c ∓ v_src) works well at everyday speeds but ignores time dilation. The relativistic formula incorporates the Lorentz factor, making it accurate at any speed up to c. At low velocities (v ≪ c) the two formulas give nearly identical results, but at v > 0.1c the relativistic correction becomes significant. For precise work in astrophysics or particle physics, the relativistic version is always required.
How is relativistic Doppler shift used to measure the expansion of the universe?
Astronomers measure the redshift of spectral lines from distant galaxies — specific frequencies of light emitted by hydrogen or other elements that are well-known in the lab. By comparing the observed frequency to the rest frequency using the relativistic Doppler formula, they calculate the recession velocity of each galaxy. Hubble's Law then connects recession velocity to distance, allowing cosmologists to map the expansion rate of the universe. Extremely high redshifts (z > 1) from quasars and early galaxies require the full relativistic treatment.
Why does a relativistic source moving perpendicular to the observer still show a frequency shift?
This is called the transverse relativistic Doppler effect, and it has no classical analog. Even when the source moves at 90° to the line of sight, time dilation causes the observed frequency to be lower than the source frequency by the Lorentz factor: f_obs = f_s / γ. This purely relativistic redshift was experimentally confirmed by Ives and Stilwell in 1938 using fast-moving hydrogen atoms. It is direct evidence that moving clocks run slow, independent of the direction of motion.