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Imagine yourself and anything you're able to perceive to be drawn onto a rubber band. Any means of determining distances would be affected by the stretching/contracting of the rubber band. The relation between any two lengths would remain exactly the same because every length is increased/decreased by the same factor as the band stretches/contracts. According to my understanding, that is what the dilation/compression of space is like for us and why it should not lead to any change in distance.
This also applies to relational measurements of length as they are performed, for example, by devices like LIGO. Although the deformation (e.g. gravitational waves) propagates only in one of two directions of measurement, the relation between both lengths should not be affected because none of the individual lengths should be affected.
Accordingly, anything that travels the four kilometers in LIGO should take the exact same amount of time, no matter whether space is currently stretched or compressed. The speed of light being constant should be irrelevant in this this case.
If gravitational waves stretch space itself, how can they have an effect on the interference of two laser beams? According to my understanding, compression or dilation of space itself (as opposed to something in space) should not affect the distance between two points.
Disclaimer: Obviously, the above is not true. It's just how I thought is was. Hopefully, it helps to see where my understanding went awry.
In this question ( If space is "expanding" in itself - why then is there redshift? ), the problem I had has actually been solved. It comes down to the fact that atoms (because of the non-gravitational forces holding them together) are in fact not affected by spacetime deformations. This makes these deformations detectable.