Relativistic Doppler Effect

Equivalent forms

f = f_0 \sqrt{\frac{c - v}{c + v}}

1 + z = \sqrt{\frac{1 + \beta}{1 - \beta}}

Unites time dilation and wave propagation — the foundation of cosmological redshift.

Symbol	Name	SI unit	Dimension	Range
$f$	Observed Frequencyoutput Frequency seen by the observer	Hz	T^-1	0 – 10000000000000000
$f0$	Source Frequency Frequency in source rest frame	Hz	T^-1	0 – 10000000000000000
$beta$	v/c Source velocity fraction (positive = receding)	dimensionless	1	-0.999 – 0.999

Unit systems

Symbol	SI	CGS	Imperial
$f$	Hz	Hz	Hz
$f0$	Hz	Hz	Hz
$beta$	1	1	1

Where it holds

For radial motion. Transverse Doppler effect requires

f = f_{0}/\gamma

. Cosmological redshift needs general relativity for large distances.

Dimensional analysis

f \to [T^-1]

f_{0}\cdot \sqrt((1-\beta )/(1+\beta )) \to [T^-1]\cdot [1] = [T^-1]

Both sides frequency.

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Discovery

Albert Einstein · 1905

Einstein derived the relativistic Doppler effect combining time dilation with the classical Doppler shift.

Try this

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.

Research status: stable

Real-world applications

Measuring galactic recession velocities
Radar gun corrections at extreme speeds
GPS timing
Spectroscopy of relativistic jets

Common misconceptions

Classical Doppler only uses $1 \pm \beta$ , missing the gamma factor
Transverse motion still produces a shift (purely relativistic)
Cosmological redshift is stretching of space, not pure kinematic Doppler

Experimental verification

Ives–Stilwell experiment (1938) confirmed the relativistic correction; modern atomic clocks measure transverse Doppler to 10^-15.

Derivation

In the source frame, successive wave crests are emitted with period

T_{0} = 1/f_{0}

.

In the observer frame the period is

\gamma \cdot T_{0}\cdot (1 + \beta )

(time dilation plus classical wavefront stretching), giving

T = T_{0}\cdot \sqrt((1 + \beta )/(1 - \beta ))

and

f = 1/T = f_{0}\cdot \sqrt((1 - \beta )/(1 + \beta ))

.

Limiting cases

\beta \to 0

⟶

f \to f_{0}

No shift at rest.

\beta \to 1

⟶

f \to 0

Extreme redshift — energy vanishes as source approaches c outward.

\beta \to -1

⟶

f \to \infty

Blueshift diverges as source approaches c inward.

What if…

What if motion is purely transverse?

$f = f_{0}/\gamma$ — the transverse Doppler effect, purely from time dilation.

What if the source is non-inertial?

Instantaneous formula applies, but acceleration adds gravitational-type corrections.

What if

\beta = 1

?

Observed frequency drops to zero — source redshifted out of existence.

1

Galaxy at 0.5c

Given ·

f0:: 600000000000000
beta:: 0.5

Find · f

Steps

$\beta = 0.5$
ratio $= \sqrt{0.5/1.5} \approx 0.5774$
$f = 6e14 \times 0.5774 \approx 3.46e14\,\mathrm{Hz}$

Result ·

f = 6e14\cdot \sqrt{0.5/1.5} \approx 3.46e14\,\mathrm{Hz} (\lambda \approx 866\,\mathrm{nm})

.

2

Approaching quasar

Given ·

f0:: 1000000000000000
beta:: -0.9

Find · f

Steps

$\beta = -0.9$ (approaching)
ratio $= \sqrt((1 - (-0.9))/(1 + (-0.9))) = \sqrt{1.9/0.1}$
ratio $\approx 4.36 \to f \approx 4.36e15\,\mathrm{Hz}$

Result ·

f = 1e15\cdot \sqrt{1.9/0.1} \approx 4.36e15\,\mathrm{Hz}

— strong blueshift.

Time Dilation→Lorentz Factor→Relativistic Velocity Addition→