Is hyperbolic space consistent with homogeneity?

[center o bass]

The FRW metric is usually expressed as
$$ds^2 = -dt^2 + a(t)^2 ( \frac{dr^2}{1-kr} + r^2 d\Omega^2))$$
where $k=-1,0,+1$ respectively for a hyperbolic, flat or spherical space. The spatial part of this metric can be derived by considering a 3-sphere embedded in a four-dimensional flat space -- any sphere obviously being homogeneous.

Similarly, the k=-1 case can be derived by considering a hyperboloid embedded in a flat four dimensional space. Now, the hyperboloid is only a homogeneous space when embedded in a flat minkowskian space -- in Euclidean space the point at the tip of the hyperboloid (corresponding to r=0) is certainly special.

Since it is the euclidean distance we measure when measuring distances to other galaxies, it seems like the k=-1 case is not consistent with homogeneity. Is this correct, or is my thinking wrong?

 

[PeterDonis]

Since it is the euclidean distance we measure when measuring distances to other galaxies, it seems like the k=-1 case is not consistent with homogeneity.

The distance we measure is not the Euclidean distance; it's the distance in a spacelike slice of constant cosmological time (i.e., $dt = 0$ in the metric you wrote down), given the metric of the spacelike slice. (Note that we don't directly "measure" this distance either; we calculate it based on other "distance" observations.) None of this has anything to do with how, or whether, you can embed that spacelike slice in a higher-dimensional Euclidean or Minkowskian space. The only requirement for homogeneity is that no point is "special" with reference to the intrinsic metric of the spacelike slice. The $k = -1$ hyperbolic space satisfies that requirement.

 

[center o bass]

The distance we measure is not the Euclidean distance; it's the distance in a spacelike slice of constant cosmological time (i.e., $dt = 0$ in the metric you wrote down), given the metric of the spacelike slice. (Note that we don't directly "measure" this distance either; we calculate it based on other "distance" observations.) None of this has anything to do with how, or whether, you can embed that spacelike slice in a higher-dimensional Euclidean or Minkowskian space. The only requirement for homogeneity is that no point is "special" with reference to the intrinsic metric of the spacelike slice. The $k = -1$ hyperbolic space satisfies that requirement.

I agree that the spatial distance measure is not euclidean; its Minkowskian (see Weinberg Gravitation p. 391) -- so the space is homogeneous only with respect to this metric. However, I would argue that we measure distances in the universe by the Euclidean distance measure, and thus, to us, I would not think that it appears homogeneous regarding the distribution of galaxies etc.

 

[center o bass]

So I agree that the space is homogeneous with regards to the intrinsic metric. However, as we do not use this metric (which has signature -1), my argument is that we do not __see it__ as being homogeneous?

 

[PeterDonis]

I would argue that we measure distances in the universe by the Euclidean distance measure,

Why do you think that? The "Euclidean distance measure" is a measure in a higher-dimensional embedding space that doesn't even exist (it's just a mathematical fiction used by some people for modeling).

However, as we do not use this metric (which has signature -1)

I'm confused. What "intrinsic" metric do you think we are not using? (If you think we are using the Euclidean metric, see above.)