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It is said the farthest galaxy we can see in one direction is beyond the cosmic horizon of the farthest galaxy we can see in the opposite direction; they can't see each other. Doesn't this contradict the law of addition of velocities? The sum of two velocities cannot be greater than $c$ in our frame, and so the mutual velocity also cannot be greater than $c$, using the addition law again.

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Special Relativity is not applicable here, because the expanding universe is described by the Friedmann metric, not by the flat Minkowski metric. Check out https://arxiv.org/abs/astro-ph/0310808v2 (in particular, figure 2 on page 7).

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