This Insight is part of my attempt to develop a formal definition of ‘large-scale isotropy’, a concept that is fundamental to most cosmology, but that is nowhere that I have seen properly defined.
The definitions of isotropy are as precise as one could wish, but the ‘large-scale’ bit is in every case I have seen just a hand-wave. It turns out that it’s quite messy to try to make that ‘large-scale’ notion precise.
I made a thread on it, which is here.
It is possible that everything below is in that thread, in more up-to-date and better versions. But I am not sure, and as I don’t have time to check before the blogs are deleted, I’m posting the material below just in case.
If you’re interested in this topic, I’d suggest going to the linked thread first, and then only coming back to this post if there are links in the thread to it.
First, define ##\Sigma_t## as the hypersurface of constant time t in the foliation.
##d(t,P,Q)## is the length of the shortest path in ##\Sigma_t## from...