Missing step in the derivation of the Robertson-Walker metric

[PAllen]

Thats actually not true, you can put a dtdr term, it does not violate spatial isotropy or homogeneity. dtdφ of course does, which is why I did not put in the metric I mentioned in my original post.

This is false. An irremovable r dependence violates homogeneity, an irremovable angular dependence violates isotropy. Thus they are on equal footing.

Just as an aside, homogeneity (everywhere) does not imply isotropy, but isotropy (everywhere) implies homogeneity.

 

[PAllen]

Just to tie up a couple of my earlier posts:

(#21) https://www.physicsforums.com/threads/missing-step-in-the-derivation-of-the-robertson-walker-metric.980564/#post-6262656 links a paper which is based on and subsumes Weinberg's theorem. This paper actually implements and makes rigoruous the approach I described (#16) https://www.physicsforums.com/threads/missing-step-in-the-derivation-of-the-robertson-walker-metric.980564/#post-6262586. It first (beginning of proof of theorem 5.1) establishes that synchronous coordinates are achievable in a neighborhood without any homogeneity or isotropy assumptions. Then, in several steps, reaches the global further simplified metric as the most general form given isotropy and homogeneity.