The FRW metric is usually expressed as(adsbygoogle = window.adsbygoogle || []).push({});

$$ds^2 = -dt^2 + a(t)^2 ( \frac{dr^2}{1-kr} + r^2 d\Omega^2))$$

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?

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# Is hyperbolic space consistent with homogeneity?

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