Relativistic Doppler Calculator
Enter source frequency, speed and observation angle to compute observed frequency f'=f·√(1−β²)/(1−β·cosθ). Approaching θ=0, β=0.6 → f'=2f (blue); receding 180° → f'=0.5f (red).
Input Data
Results
At a glance:Relativistic Doppler effect (Einstein 1905 special relativity): the observed frequency of a moving source is f'=f·√(1−β²)/(1−β·cosθ_obs), where β=v/c and θ_obs is the angle between the source-velocity direction and the observer-line of sight in the observer frame. Derivation: combine Lorentz time dilation (source intrinsic period T₀→observer T=γT₀) and the optical-path change (source moves vT·cosθ per period) to get f'=f·√(1−β²)/(1−β·cosθ). Special cases: (1) longitudinal approach θ=0 → f'=f·√((1+β)/(1−β)) (blue, f'>f); (2) longitudinal recede θ=180 → f'=f·√((1−β)/(1+β)) (red, f'<f); (3) transverse θ=90 → f'=f·√(1−β²) (transverse Doppler redshift, pure time dilation, no classical counterpart). Redshift parameter z=(f−f')/f', positive on recede. History: predicted by Einstein 1905; verified transversely by the Ives–Stilwell experiment in 1938. Applications: (1) cosmic redshift (Hubble law, though cosmic expansion is general relativity); (2) accelerator radiation spectra; (3) relativistic plasma diagnostics; (4) GPS satellite Doppler correction; (5) Doppler cooling (laser cooling).
Formula
Observed frequency: f' = f·√(1−β²) / (1 − β·cosθ)
β = v / c
Longitudinal approach: f' = f·√((1+β)/(1−β)) (blue)
Longitudinal recede: f' = f·√((1−β)/(1+β)) (red)
Transverse θ=90: f' = f·√(1−β²)
$$f' = \frac{f\sqrt{1-\beta^2}}{1 - \beta\cos\theta}, \quad \beta = \frac{v}{c}$$How to Use
- Enter source frequency f (Hz).
- Enter speed v (m/s; significant near c).
- Enter observation angle θ (°, 0=approach, 180=recede, 90=transverse).
- The tool gives observed f', the ratio and red/blue verdict.
Case Studies
Cosmic redshift and GPS correction
A galaxy receding at β=0.1 (θ=180) → f'=f·√((1−0.1)/(1+0.1))≈0.953 f, z≈0.049 (mild redshift).
Distant quasars at β≈0.9 show huge redshift, the evidence for an expanding universe.
GPS satellites at ~3.8 km/s need a relativistic Doppler + time-dilation correction (~38 μs/day) to keep positioning accurate.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.