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The Relativistic Doppler Effect
The relativistic Doppler effect gives us a means of measuring the relative velocity of two observers.
This effect is also discussed (without naming it) on the pages Measuring Relative Velocity (1) and (2).

We will consider two inertial observers, A and B, having a relative velocity of magnitude v.
Let A send a continuous beam of light (or other electro-magnetic radiation) towards B.
We will assume that both A and B possess the means of measuring the frequency of that radiation.
For now, we will assume they are moving away from each other.
In that case, it is clear that each wave crest will have a little further to go to reach B than the preceding wave crest.
So the time period of received waves will be slightly greater than the time period of the transmitted waves.
We therefore have an effect which is very similar to the Doppler effect observed using sound waves.

 At this point it should be noted that there is a significant difference between the Doppler effect and the relativistic Doppler effect. In the mathematical analysis of the Doppler effect we consider the motion of observer and source relative to the medium (the air) through which the sound travels. The velocity of source and observer relative to the air can, of course, vary. It is now generally accepted that the "aether" does not exist and that all observers measure the velocity of electro-magnetic waves to be a constant, c. Hence, the mathematical analysis of the relativistic Doppler effect is fundamentally different from the Doppler effect in sound waves.

When trying to find a method of measuring relative velocities (see here) we defined a constant, k to be

Remembering that

to use the same notation here, we will define k to be

and, it has been shown that the relation between k and v is

The Doppler effect is usually expressed by comparing the change in frequency (often called the Doppler shift) with the transmitted frequency.
This gives the relative Doppler shift, defined as

which means that we can write

A slight rearrangment this gives

and if v << c (which is likely to be the case) this can be approximated by

so, finally, we have, to a very good approximation, in most practical cases

which is rather pleasingly simple... and can, of course, be used to measure the relative velocity of the two observers.
Notice that a negative relative Doppler shift (that is, a decrease in frequency) corresponds to bodies which are moving away from each other, and a positive shift... well, finish the sentence for yourself!

The relativistic Doppler effect is of importance in cosmology as the "red shift" of the light from distant galaxies gave observational evidence for the expanding universe which was predicted by Einstein's General theory of relativity.
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