Doppler Shift
"I have looked further into space than any human being did before me."
~ Sir William Hershel[7]
Doppler shift comes from the difference in observation from two observers in separate reference frames. The easiest way to think about this is the sound of a fire truck (or any other vehicle with a siren) siren. When the siren is coming towards you it sounds higher pitched than when it is stopped and conversely for when the fire truck is going away from you. Yet if you are in the fire truck, the siren sounds the same no matter how fast or what direction you are going. The reference frames are, one, the person in the firetruck and, two, the person outside the firetruck.
We will call the reference frame of the earth K and we call the reference frame of the star we are observing K' which are moving at a velocity v relative to each other. we can relate the position of objects between the frames as such
[2]
where x refers to the position in the K frame, x' refers to the position in the K' frame and t refers to time which is independent of reference frame. The phase of the light waves must be the same in reference frames because it is simply proportional to the number of wave crests that have passed an observer; not dependent on any spacial coordinates. We can then relate the phase, φ, between the reference frames as such
[2]
where ω is the angular frequency, c is the speed of light, and n is the direction of propagation. Now, since time is invariant, t = t'. subbing in x' - vt for x, we get
[2]
This must be true for all times and positions in our K' reference frame, so we find
[2]
now looking at the ω equation, if we rearrange it so that the angular frequencies are all on one side, and relating the angular frequencies to the wavelengths we are left with
[3]
which is the familiar form of the Doppler shift formula. Astronomers often define Δλ / λo as the redshift, z[3].