Still working on the magic Killing tensor ##K_{\mu\nu}=a^2\left(g_{\mu\nu}+U_\mu U_\nu\right)## in FLRW spacetime:

- We use ##K_{\mu\nu}## to show that interstellar gas cools in an expanding universe.
- Then we use it to show that as the universe expands the observed frequency of a photon will decrease.
- Cosmologist measure the redshift, ##z##, of distant objects and we show how this tells us the scale factor ##a## when the object emitted the light being measured.
- I then looked up some galactic distances and redshifts to compare them (pictured). Extrapolating the graph indicates the size of the visible universe is ~15 Gly.
- Carroll then seems to ramble a bit and goes back to a non-relativistic redshift. The history of Doppler and Hubble, which he does not mention, is quite interesting.
- Getting back on track Carrol shows that the nearby universe can be thought of as flat and we are able to use proper distance meaningfully and derive Hubble's law. I think he is overcomplicated.
- Next step other distances!

Read it here Commentary 8.5 Redshifts and Distances.pdf (9 pages)