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Obviously, homogeneity implies that the density is the same everywhere at any time. However, does this imply that the expansion is uniform? By uniformity, I mean that if I pick three galaxies to form a triangle, then the ratio of the side lengths will never change over time.

EDIT: I have forgotten to add this: if both homogeneity and isotropy are assumed, can we prove that the expansion is uniform?

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No, homogeneity does not implies that the expansion is uniform. Homogeneous expansion could be anisotropic which would lead to different changes in length depending on orientation.

A simple example to demonstrate this is the Kasner metric which is homogeneous but anisotropic. For a $(3+1)$ spacetime this metric could be written in the following form: $$ ds^2 = - dt^2 +t^{2p_1} dx^2 +t^{2p_2} dy^2 +t^{2p_3} dz^2. $$

Now let us assume that we have three galaxies at a moment $t=1$ first at origin $(0,0,0)$, second at a point with spatial coordinates $(a,0,0)$, third at a point $(0,b,0)$.

At the moment $t=\tau$ these galaxies would have the following spatial coordinates: first $(0,0,0)$, second $(\tau^{2p_1} a,0,0)$, third $(0,\tau^{2p_2} b ,0)$.

We see that if $p_1\ne p_2$ then the ratio of the distances $d_{1-2}/d_{1-3}$ would be different at different times.

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  • $\begingroup$ Thank you a lot! What about having BOTH homogeneity AND isotropy assumed? Can we deduce uniform expansion in this case? $\endgroup$ – Ma Joad Oct 14 '19 at 22:38
  • $\begingroup$ @MaJoad: You might want to ask that as a separate question. $\endgroup$ – user4552 Oct 14 '19 at 23:11
  • $\begingroup$ @BenCrowell OK. Here is the new thread: physics.stackexchange.com/questions/508640/… $\endgroup$ – Ma Joad Oct 17 '19 at 8:00

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