Relativistic Doppler Shift

The normal Doppler shift for waves such as sound which move with velocities v much less than c is given by the expression

where the plus sign is taken for waves traveling away from the observer. For light and other electromagnetic waves, the relationship must be modified to be consistent with the Lorentz transformation and the expression becomes

Here v is the relative velocity of source and observer and v is considered positive when the source is approaching.
DerivationCalculationLow velocity limit (v/c small)
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Doppler Effect, Electromagnetic Waves

The Doppler effect is observed with visible light and all other electromagnetic waves. Just as in the case of sound waves, the wavelength in the direction of the source motion is shortened to

where all quantities here are measured in the observer's frame. To relate this to the source frequency, it must be expressed in terms of by using the time dilation expression

This is used to get the frequency.
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Fractional Wavelength Change

For purposes of determining recession speed of stars and galaxies with the Doppler effect by observation of the red shift of spectral lines, it is convenient to express the Doppler effect in terms of the shift in wavelength compared to the source wavelength. For these purposes it is more convenient to define a receding velocity as positive in the wavelength relationship:

The fractional wavelength change is defined as the z parameter for characterizing red shifts:

In terms of the usual relativity symbols, this becomes

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Doppler Frequency

From the Doppler shifted wavelength, the observed frequency is

Rearranging gives the convenient form

where the relative velocity vs is positive if the source is approaching and negative if receding. In terms of the usual relativity symbols, this becomes

Relativistic Doppler effectCalculationComparison with soundLow speed approximation
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