Relativistic Doppler Shift
Compare light frequency and wavelength shifts for approaching and receding sources without introducing a propagation medium.
By Riano · Published · Updated · Report a correction
Relativistic Doppler Shift
In special relativity, light propagates in vacuum at speed c in every inertial frame without requiring a mechanical medium. For one-dimensional radial motion with signed relative recession parameter β = v/c, combining kinematic wavefront travel time with Lorentz time dilation γ = 1/√(1 − β2) yields the observed wavelength λobs = λrest√((1 + β)/(1 − β)) and reciprocal frequency ratio fobs/frest = √((1 − β)/(1 + β)).
Worked Example
Consider a light source receding at β = 3/5 = 0.6 with a source-rest wavelength of λrest = 500 nm:
- Calculate the Lorentz factor: γ = 1 / √(1 − β2) = 1 / √(1 − 0.36) = 1 / 0.8 = 1.25.
- Calculate the wavelength ratio: λobs / λrest = √((1 + 0.6) / (1 − 0.6)) = √(1.6 / 0.4) = √4 = 2.
- Calculate the frequency ratio: fobs / frest = 1 / 2 = 0.5.
- Determine the observed wavelength: λobs = 500 nm × 2 = 1000 nm.
- Find the fractional wavelength shift: z = (λobs − λrest) / λrest = 2 − 1 = 1 (Receding redshift).
Experiment
Explore the relativistic Doppler shift by comparing light frequency and wavelength ratios for approaching and receding sources in the Relativistic Doppler Shift module:
- Set Recession beta to 3/5 and Rest wavelength nm to 500 to observe a receding redshift where observed wavelength doubles to 1000 nm.
- Change Recession beta to −3/5 with Rest wavelength nm set to 400 to observe an approaching blueshift where the wavelength ratio drops to 0.5 (200 nm).
- Compare 4/5 and −4/5 at 600 nm to verify reciprocal wavelength ratios (3 vs 1/3) while the Lorentz factor remains γ = 5/3 ≈ 1.66666666667.
Limitations
This model evaluates purely radial (collinear) relative motion between an inertial source and observer in flat spacetime vacuum, restricted to |β| ≤ 4/5 away from the light-speed singularity. It does not model off-axis transverse Doppler shifts, acoustic Doppler shifts in a medium, gravitational redshift, or cosmological metric expansion.
References
The derivation and physical interpretation of the special-relativistic Doppler shift for light in vacuum are detailed in:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Relativistic Doppler Shift module.
Try Relativistic Doppler Shift on Riano →