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From Wikipedia:

Universe in an expanding sphere. The galaxies farthest away are moving fastest and hence experience length contraction and so become smaller to an observer in the centre.

Does length contraction really apply to the far galaxies moving due to expansion? If the galaxies near the edge of the observable universe are moving away faster than light, then how to apply the length contraction formula? And can we observe this contraction effect, considering that it's in a direction perpendicular to our observation?

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    $\begingroup$ Please note: the quote you provide concerns the "Milne model", which uses just special relativity, and where there is no general relativistic expansion. In most cosmological models there is no "edge" or "centre". $\endgroup$ – Rob Jeffries Apr 23 '18 at 18:25
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The material on WP is describing a cartoonish illustration, not making a claim about what is actually observed. Note that the cartoon is drawn from an imaginary point of view outside the universe, which isn't actually possible.

There are at least two reasons why this part of the article is misleading.

(1) Even in special relativity, length contraction isn't what we actually see in optical observations. The time for propagation of light from different parts of an object to the observer is different, so what we see is more complicated. Objects can appear elongated rather than contracted, and there can be a rotation as well.

(2) It's an oversimplification to say that distant galaxies are moving away from us at some speed.

In reality, I don't think it's feasible to detect relativistic distortions of the shape of a galaxy optically. For galaxies close enough that it makes sense to use special relativity, the velocities $\ll c$. For distant galaxies, the motion (to the extent that it makes sense to call it motion) is almost purely radial. And in any case we don't necessarily have an undistorted shape to compare with, since galaxies aren't simple geometrical shapes.

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