# Does the homogeneity of space imply that the expansion of the universe is uniform?

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?

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.

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