Playground
Expanding wavefronts from a moving source showing relativistic Doppler shift; beta slider controls source velocity.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Observed Frequencyoutput Frequency seen by the observer | Hz | T^-1 | 0 – 10000000000000000 | |
| Source Frequency Frequency in source rest frame | Hz | T^-1 | 0 – 10000000000000000 | |
| v/c Source velocity fraction (positive = receding) | dimensionless | 1 | -0.999 – 0.999 |
Deep dive
Derivation
In the source frame, successive wave crests are emitted with period T0 = 1/f0. In the observer frame the period is gamma·T0·(1 + beta) (time dilation plus classical wavefront stretching), giving T = T0·sqrt((1 + beta)/(1 - beta)) and f = 1/T = f0·sqrt((1 - beta)/(1 + beta)).
Experimental verification
Ives–Stilwell experiment (1938) confirmed the relativistic correction; modern atomic clocks measure transverse Doppler to 10^-15.
Common misconceptions
- Classical Doppler only uses 1 ± beta, missing the gamma factor
- Transverse motion still produces a shift (purely relativistic)
- Cosmological redshift is stretching of space, not pure kinematic Doppler
Real-world applications
- Measuring galactic recession velocities
- Radar gun corrections at extreme speeds
- GPS timing
- Spectroscopy of relativistic jets
Worked examples
Galaxy at 0.5c
Given:
- f0:
- 600000000000000
- beta:
- 0.5
Find: f
Solution
f = 6e14·sqrt(0.5/1.5) ≈ 3.46e14 Hz (λ ≈ 866 nm).
Approaching quasar
Given:
- f0:
- 1000000000000000
- beta:
- -0.9
Find: f
Solution
f = 1e15·sqrt(1.9/0.1) ≈ 4.36e15 Hz — strong blueshift.
Scenarios
What if…
- scenario:
- What if motion is purely transverse?
- answer:
- f = f0/gamma — the transverse Doppler effect, purely from time dilation.
- scenario:
- What if the source is non-inertial?
- answer:
- Instantaneous formula applies, but acceleration adds gravitational-type corrections.
- scenario:
- What if beta = 1?
- answer:
- Observed frequency drops to zero — source redshifted out of existence.
Limiting cases
- condition:
- beta → 0
- result:
- f → f0
- explanation:
- No shift at rest.
- condition:
- beta → 1
- result:
- f → 0
- explanation:
- Extreme redshift — energy vanishes as source approaches c outward.
- condition:
- beta → -1
- result:
- f → ∞
- explanation:
- Blueshift diverges as source approaches c inward.
Context
Albert Einstein · 1905
Einstein derived the relativistic Doppler effect combining time dilation with the classical Doppler shift.
Hook
A galaxy recedes at 0.5c — by how much is its 500 nm light redshifted?
Use f = f0·sqrt((1 - beta)/(1 + beta)) with beta = 0.5, giving f/f0 ≈ 0.577 and λ ≈ 866 nm.
Dimensions:
- lhs:
- f → [T^-1]
- rhs:
- f0·sqrt((1-beta)/(1+beta)) → [T^-1]·[1] = [T^-1]
- check:
- Both sides frequency. ✓
Validity: For radial motion. Transverse Doppler effect requires f = f0/gamma. Cosmological redshift needs general relativity for large distances.