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 electromagnetic 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
electromagnetic 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.

