Does Isotropy Necessarily Imply Homogeneity in the Universe?

[ChrisVer]

quating from Cosmological_principle

Homogeneity means that the same observational evidence is available to observers at different locations in the universe ("the part of the universe which we can see is a fair sample"). Isotropy means that the same observational evidence is available by looking in any direction in the universe ("the same physical laws apply throughout"). The principles are distinct but closely related, because a universe that appears isotropic from any two (for a spherical geometry, three) locations must also be homogeneous.

How can one see that if the universe appears isotropic from any two locations it must also be homogeneous?
And why would we need three points for a sphere?
Thanks.

 

[wabbit]

The converse is easy - a universe which is spherically symmetric wrt one observer, but inhomogeneous (composed on concentric shells of different densities for instance) will be isotropic from the center of symmetry - and only from there if it is flat, but also from the antipodal point is it is globally spherical (which are the only two points of isotropy in that case).
Not sure how to prove there are no other possibilities though.

 

[PeterDonis]

How can one see that if the universe appears isotropic from any two locations it must also be homogeneous?
And why would we need three points for a sphere?

The second question is easy: because as wabbit showed, you can have two points of isotropy in a universe which is spatially a 3-sphere, without having it be homogeneous; just make the two points antipodal points. Another way of saying this is, for the two point requirement to be sufficient, we must restrict consideration to an open, non-compact spacelike hypersurface as the "universe" (more particularly, the universe at one instant of time).

Given that restriction, wabbit's argument should also be enough to show you why isotropy about two points is sufficient: because isotropy about one point, combined with lack of homogeneity, rules out isotropy about any other point. So if we know we have isotropy about two points, we must have homogeneity.