Longitudinal Doppler Shift

Equivalent forms

f_{obs} = f_{src}\sqrt{\frac{1-\beta}{1+\beta}}\ (\text{receding})

f_{obs} = f_{src}\sqrt{\frac{1+\beta}{1-\beta}}\ (\text{approaching})

A single formula covers redshift and blueshift by the sign of beta, and reduces to the classical Doppler shift plus a second-order time-dilation correction.

Symbol	Name	SI unit	Dimension	Range
$fobs$	Observed Frequencyoutput Frequency measured by the observer	Hz	T^-1	0 – 2000000000000000
$fsrc$	Source Frequency Frequency emitted in the source rest frame	Hz	T^-1	1000000000000 – 1000000000000000
$beta$	Speed Ratio v/c, positive when receding	dimensionless	1	-0.95 – 0.95

Dimensional analysis

f_obs -> [T^-1]

f_src * sqrt(dimensionless) -> [T^-1]

Frequency.

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Discovery

Christian Doppler / Albert Einstein · 1905

Doppler described the classical shift in 1842; Einstein's 1905 relativity added the time-dilation factor, giving the exact square-root formula confirmed by Ives and Stilwell in 1938.

Research status: stable

Real-world applications

Radial-velocity spectroscopy of stars and exoplanets
Police/weather radar (low-beta limit)
Relativistic beam diagnostics

Common misconceptions

The classical Doppler formula 1/(1+beta) is not exact — it misses the gamma time-dilation factor
Redshift here is genuine relative motion, distinct from cosmological (expansion) redshift
Sign conventions matter: define beta>0 as receding and stay consistent

Experimental verification

The Ives-Stilwell experiment (1938) measured the second-order (time-dilation) part by comparing forward and backward emission from a fast hydrogen beam; modern storage-ring versions confirm the relativistic factor to parts in 10^9.

Derivation

In the observer frame the source recedes at v.

Successive crests are emitted a proper period T_src apart, but the source clock is time-dilated, so in the observer frame they are gamma T_src apart, and each crest travels an extra v*gamma*T_src to reach the observer.

The observed period is gamma T_src (1+beta), giving

f_obs = f_src / [\gamma (1+\beta )] = f_src \sqrt((1-\beta )/(1+\beta ))

.

Limiting cases

beta << 1⟶

f_obs \approx f_src (1 - \beta )

Recovers the classical linear Doppler shift; time dilation is a second-order correction.

beta -> 1 (receding)⟶ f_obs -> 0Infinite redshift as the source approaches light speed away from you.

beta < 0 (approaching)⟶ blueshiftFrequency increases; sign flip gives the approaching formula.

What if…

What if the motion is sideways (transverse)?

The longitudinal factor vanishes and only time dilation remains: $f_obs = f_src/\gamma$ , the transverse Doppler effect.

What if

\beta = 0.6

approaching?

$f_obs = f_src \sqrt{1.6/0.4} = 2\,\mathrm{f}_src$ — the frequency doubles (strong blueshift).

1

Star receding at 0.1c

Given ·

fsrc:: 500000000000000
beta:: 0.1

Find · f_obs

Steps

$\sqrt((1-0.1)/(1+0.1)) = \sqrt{0.818} = 0.9045$
$f_obs = 5e14 * 0.9045 = 4.52e14\,\mathrm{Hz}$

Result ·

f_obs = 5e14 * \sqrt{0.9/1.1} = 5e14 * 0.9045 = 4.52e14\,\mathrm{Hz}

— redshifted from green toward red.

Relativistic Doppler Effect→Transverse Doppler Effect→Cosmological Redshift→Lorentz Factor→