## Tuesday, 9 March 2021

### Redshifts and Distances

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!