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When you say that spacetime itself is extremely stiff, I can't help myself but to visualize a lattice of discrete points through which the wave propagates through. I understand this would be the wrong way about thinking about it, but is there another way?

This isn't really a wrong picture to imagine. In a way, this is what the experiments are trying to measure. You have a mirror and a laser at two points of your lattice (the space itself), and the "region" between them compresses and expands, which makes the laser take a slightly different time to bounce back.

Far away in linear approximation (which is what we saw here), the lattice picture is quite appropriate as a first order analogy.

That's an open problem right now. Lots of proposals, insufficient evidence to resolve the question.

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