Caption: A comparison of the Doppler effect (or shift) and the cosmological redshift for electromagnetic radiation (EMR). The observer is the Sun.

Features:

1. The upper case in the image illustrates Doppler effect: to be exact, the relativistic Doppler effect (in vacuum) which is needed for electromagnetic radiation (EMR) since EMR (in vacuum) is always relativistic since it always moves at the vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns. It's redshift/blueshift for recession/approach.

   The Doppler effect is a shift in wavelength/frequency depending on the relative motion of two observers in a single inertial frame. It's treatment when the two observers are accelerated with respect to each other can be somewhat tricky (see Wikipedia: Transverse Doppler effect).

2. The two lower cases in the image illustrate cosmological redshift: the upper lower case is a negative cosmological redshift: i.e., a blueshift.

3. The cosmological redshift is the scaling up of EMR wavelength (therefore also a scaling down of frequency) with the expansion of the universe.

   The expansion of the universe is a growth of space itself, NOT an ordinary motion.

   So the cosmological redshift is NOT a Doppler effect though it is a related phenomenon. One can, in fact, derive the cosmological redshift from the Doppler effect by considering the continuous Doppler shift as EMR propagates across the continuum of local inertial frames making up expanding universe: i.e., the comoving frames of the expanding universe (Li-38--39). So there is an argument for calling the cosmological redshift a Doppler effect, but yours truly doesn't buy it.

   Two observers at rest in their respective comoving frames observe the cosmological redshift. If they are in motion relative to their comoving frames, there are superimposed Doppler shifts.

   We do the derivation of the cosmological redshift from the Doppler effect in the extended file Cosmology file: cosmological_redshift_doppler_shift_4.html which could be this file in which case the derivation is below. There is also a direct general relativity derivation (Li-127--129).

4. The general formulae for the Doppler effect and the cosmological redshift are different and involve quantities with different meanings though they agree to 1st-order in v/c where ordinary velocity v and recession velocity v (the growth of space rate) also agree to 1st-order.

   Yours truly thinks this 1st-order agreement shows that growth of space and free-fall motion in the classical limit do become the same thing somehow in the classical limit. Yours truly could be wrong.

5. Just to go a bit further, the EMR 1st-order Doppler effect formula is

     Δλ/λ = v/c , 

   where λ is initial or final wavelength (there is no 1st-order difference), Δλ is the change in wavelength, v is the line-of-sight relative velocity counting increasing/decreasing separation as positive/negative relative velocity, and vacuum light speed c = 2.99792458*10**8 m/s.

   The 1st-order formula is valid in the limit that v/c << 1.

   Note the 1st-order formula is NOT a differential equation and CANNOT be used for recover the full relativistic Doppler effect (see Wikipedia: Relativistic Doppler effect: Relativistic longitudinal Doppler effect).