I have a question on cosmological redshift which I have just learned about from Sean Carroll. After calculating it for an expanding universe he does a thought experiment to show that it is different to Doppler redshift which would be detected if two galaxies were flying away from each other in a flat (therefore not expanding) universe.

We have flat universe L on the left with two galaxies separated by distance ## s##. A photon is emitted from galaxy 1, galaxy 2 is quickly propelled to a separation of ##2s##, galaxy 2 stops and the photon arrives. Since the galaxies are now not relatively moving there would be no Doppler redshift.

On the right the galaxies are also separated by ## s## but, instead of moving a galaxy, the universe expands by a factor of 2 (it briefly gets a metric like ## ds^2=-dt^2+t^2dx^2##), then stops expanding and then the photon arrives. According to the cosmological redshift formula, the photon has a redshift.

This implies that the galaxies in universe R are still separated by ## s##, because rulers would expand along with everything else. One can also check this by drawing out and back light paths before and after expansion.

This is spooky. It also implies that in our 'expanding' universe distant galaxies are not really moving away! One also wonders how we tell that it's cosmological not Doppler redshift.

Have I got the picture roughly right? The next step will be to compare these to real values like the Hubble constant.

On PP at https://www.physicsforums.com/threads/have-i-got-the-right-picture-for-cosmological-redshift.977816/

"This example underlines the main message of this Insight: That the assignment of properties and interpretations based on an assumed set of preferred coordinates is not necessarily coordinate invariant and we need to be careful not to impose any coordinate interpretation as absolute truth.

I have scanned the article (thus getting to the punch line). It deserves further study, especially I know more about the Robertson–Walker (RW) universe which are the subject of chapter 8 section 2.

On the right the galaxies are also separated by ## s## but, instead of moving a galaxy, the universe expands by a factor of 2 (it briefly gets a metric like ## ds^2=-dt^2+t^2dx^2##), then stops expanding and then the photon arrives. According to the cosmological redshift formula, the photon has a redshift.

This implies that the galaxies in universe R are still separated by ## s##, because rulers would expand along with everything else. One can also check this by drawing out and back light paths before and after expansion.

This is spooky. It also implies that in our 'expanding' universe distant galaxies are not really moving away! One also wonders how we tell that it's cosmological not Doppler redshift.

Have I got the picture roughly right? The next step will be to compare these to real values like the Hubble constant.

On PP at https://www.physicsforums.com/threads/have-i-got-the-right-picture-for-cosmological-redshift.977816/

### The great Orodruin replied:

He wrote about this in January 2018 here. It's about 10 pages and the punch line is:"This example underlines the main message of this Insight: That the assignment of properties and interpretations based on an assumed set of preferred coordinates is not necessarily coordinate invariant and we need to be careful not to impose any coordinate interpretation as absolute truth.

**In particular, 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.**I also have to admit to being among the set of people who did this error until I actually performed these calculations myself."I have scanned the article (thus getting to the punch line). It deserves further study, especially I know more about the Robertson–Walker (RW) universe which are the subject of chapter 8 section 2.

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