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We will derive both the relativistic and non-relativistic Doppler shift. The easiest way of doing this is to derive the relativistic formula and then take the non-relativistic limit, i.e., .
Consider a light wave with frequency and wavelength in the frame. Then the period of the wave is given by
Suppose the light wave is traveling in the -direction. Now consider a frame, , which is moving with velocity in the -direction relative to . There might be - and -components to the velocity, but these will not concern us for the moment. What does the light wave look like in this frame? In particular, what period is measured in the frame?
Suppose there is an observer in the frame who is sitting at . In the frame, this observer will appear to be moving on a trajectory given by . First, analyze the situation in the frame; then transform the result to the frame. Suppose the moving observer is at when the front of the light wave reaches him. At , the back of the wave will reach . But by this time, the observer is no longer at . Instead, he is at . If is positive, then the back of the wave still has farther to go to reach the observer. If is negative, then the back of the wave will already have reached the observer. Let denote the time at which the back of the wave reaches the moving observer. Then
The first term is the time it takes for the back of the wave to get to (remember the clock started when the front of the wave got to ); the second term is the time it takes to get from to the observer’s new position. Note that the second term can be negative. Then we have
This is the time between the arrival of the front of the wave and the arrival of the back of the wave at the moving observer (remember that the front of the wave reached the observer at ).
Now, is the Doppler-shifted period as seen in . What we really want is the period in which we will denote by . From the time-dilation formula (see Lorentz transformations),
where, as usual, and is now the full 3-dimensional velocity of relative to .
More generally, we can think of the -direction as the line-of-sight. So is just the velocity along the line-of-sight away from the observer. Let’s replace with . Then, since the wavelength is proportional to the period,
This result is valid in the framework of special relativity and depends not only on the motion along the line-of-sight but also on the perpendicular components of the velocity through . Since , the relativistic effects always contribute to a blue-shifting of the spectrum. This boils down to a time-dilation effect. The astronomer is in the same spatial location for both events (the arrival of the front and back of the wave), and so the astronomer’s clock measures the proper time. The proper time is always smaller than the time interval measured in any other reference frame, so the period of the wave is smaller for the astronomer which is a blue-shift. This effect can be overcome by a large velocity in the direction away from the astronomer. This is the contribution of the other term in the denominator which comes from the stretching of the wavelength due to recession.
It is often useful to look at the non-relativistic limit: . Expanding to first order in yields
Here we see that the dependence on the perpendicular components has disappeared. In this limit, a receding object will always appear red-shifted, and an approaching object will always appear blue-shifted.