Relativistic Doppler effect

Frequency shifts from relative motion

The Doppler effect is the change in observed frequency caused by relative motion between source and observer. For light, the effect must be treated relativistically because all observers measure light speed as $c$.

If source and observer move toward each other, observed light is blueshifted: frequency increases and wavelength decreases. If they move apart, light is redshifted: frequency decreases and wavelength increases.

Longitudinal relativistic Doppler shift

For motion directly along the line of sight, define eta=v/c. If source and observer are receding from each other, the observed frequency is

f_{obs}=f_{emit}sqrt{rac{1-eta}{1+eta}}

The observed wavelength is

lambda_{obs}=lambda_{emit}sqrt{rac{1+eta}{1-eta}}

For approaching motion, replace $v$ by $-v$, giving a frequency increase.

Why relativity changes the formula

The classical Doppler effect for sound depends on motion relative to a medium. Light in vacuum has no medium. The relativistic formula combines two effects: changing separation between wave crests and time dilation of the moving source or observer.

Time dilation contributes a factor that has no classical sound-wave analog in the same form.

Redshift parameter

Astronomers often use redshift $z$, defined by

1+z=rac{lambda_{obs}}{lambda_{emit}}

For relativistic recession along the line of sight,

1+z=sqrt{rac{1+eta}{1-eta}}

Positive $z$ means redshift. Negative $z$ corresponds to blueshift.

Low-speed limit

For $vll c$, the fractional shift is approximately

rac{Delta f}{f}approx-rac{v}{c}

for recession. This agrees with the familiar first-order Doppler shift. Relativistic corrections become important at high speeds.

Transverse Doppler effect

Special relativity predicts a transverse Doppler effect even when motion is perpendicular to the line of sight at closest approach. Classically, one might expect no shift at exactly transverse motion. But time dilation makes the moving source's clock run slow, so observed frequency is reduced:

f_{obs}=rac{f_{emit}}{gamma}

This effect is a direct test of relativistic time dilation.

Applications

Relativistic Doppler shifts are used in astronomy, accelerator physics, radar, spectroscopy, and astrophysics. Light from jets near black holes, rapidly moving stars, and distant galaxies can show significant shifts. In cosmology, redshift also includes expansion of space, which requires general relativity for full interpretation.

The big idea

The relativistic Doppler effect describes frequency and wavelength shifts of light caused by relative motion. Unlike sound, it does not rely on a medium and includes time dilation. Longitudinal shifts produce redshift or blueshift, while the transverse Doppler effect reveals time dilation directly.