Cosmological vs Doppler redshift

Milne Universe. Flat and expanding.

I want to compare Carroll's section 3.5 where he "demonstrates the conceptual distinction between  the cosmological redshift and the conventional Doppler effect" and Orodruin's Physics Forums Insight were he concludes "I have seen many instances where people in popular texts make a very strong claim that cosmological redshift is fundamentally different from Doppler shift. The computations above clearly show that this is not the case, instead cosmological redshift and Doppler shift are two sides of the same coin, just viewed in different coordinates."

Let's see whether Carroll is one of the guilty ones writing popular texts or if his "conceptual distinction" is just a matter of reference frames.

My first attempt at this was in September 2019 and I got told by Orodruin to read his Insight on Physics Forums. I was (correctly) daunted and postponed the reading. My next attempt was in November 2019, while travelling, but time and place defeated me. At last after nearly 1½ years I have won.

I followed most of Orodruin's insight. His approximation for the mapping from Minkowski to FLRW coordinates was the most difficult part for me. I make some effort to check its validity. With that under his belt he briskly derives Hubble's law in Minkowski coordinates showing that there are (at least) two ways of looking at the expansion of the universe. The punch line comes at the end with his pathological example (a Milne universe, pictured) where he shows, without approximations, that the only distinction between cosmological and Doppler redshift is the frame of reference used. 

Carroll's thought experiment showing a "conceptual distinction" between them involved some very unreal events: Galaxies are started and stopped in a Minkowski frame and FLRW expansion is turned on and off. I have not pinned down exactly what is wrong with that but Orodruin's method is much more real. I am convinced.

On the way I learnt 

• more about Taylor series which I find peculiar.
• that coordinates are orthogonal if the metric is diagonal and I now almost understand the notation ##e_\tau=\partial_t##.
• about proper distance and simultaneity conventions, which I had never heard of before.
• I learnt about the varying speed of light!

I made notes about all these. It was an enlightening three weeks.

Read it all at Commentary 3.5 Expanding Universe Revisited #3.pdf.