Source emits at rest frequency f₀ and moves at v = βc; observer at angle θ measures f_obs = f₀ √(1 − β²) / (1 − β cos θ). Animated lab-frame wavefronts, observer at any angle, and a 380–700 nm spectrum strip showing the apparent colour shift of a 555 nm reference line — including the purely relativistic transverse Doppler (θ = 90°) red-shift f₀/γ.
About this model
A source emits at rest frequency f₀ while moving at v = βc; an observer at angle θ measures f_obs = f₀ √(1 − β²) / (1 − β cos θ). The formula combines ordinary Doppler geometry with time dilation. Animated lab-frame wavefronts show compression ahead and stretching behind; a 380–700 nm spectrum strip maps the shift of a 555 nm reference line into apparent colour. At θ = 90° the denominator is 1, leaving the purely relativistic transverse Doppler red-shift f_obs = f₀/γ. Idealizations: inertial motion, monochromatic rest frequency, no gravitational redshift, and a single observer angle. You vary β, θ, and f₀ to explore longitudinal blueshift/redshift and the transverse case.
Who it's for: Special relativity and astrophysics courses covering relativistic Doppler and transverse redshift.
Key terms
relativistic Doppler effect
transverse Doppler
redshift
blueshift
Lorentz factor
wavefront animation
How it works
Relativistic Doppler shift f_obs = f₀ √(1−β²) / (1 − β cos θ). Animated wavefronts from a source moving at v = βc, an observer placed at angle θ, and a live spectral-line shift on a 380–700 nm strip. Captures the longitudinal blue/redshift asymmetry and the purely relativistic transverse Doppler effect at θ = 90°.
Frequently asked questions
Why is there a Doppler shift at θ = 90°?
Classical acoustics predicts no first-order shift for a source moving transversely at the emission instant, but relativity still includes time dilation: f_obs = f₀/γ < f₀. That transverse redshift is a hallmark of special relativity and has been confirmed with beams and rotating sources. Mistaking θ = 90° for “no shift” is a common classical intuition error.
How does the formula reduce at small β?
Expanding for β ≪ 1 recovers the classical longitudinal factor ≈ 1 + β cos θ plus higher-order terms that include the γ contribution. Head-on approach (θ = 0) blueshifts; recession redshifts. The simulator’s spectrum strip makes even modest fractional shifts visible as colour changes of the 555 nm line.
Is this the same as cosmological redshift?
No. Cosmological redshift comes from expanding space between emission and observation; this lab is special-relativistic Doppler for a moving source in flat spacetime. Both can stretch wavelengths, but the physics and formulas differ. Gravitational redshift is also omitted here.