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Since LIGO's first detection of gravitational waves, I have been searching for an intuitive way -as long as intuition can be useful in relativity, which often it isn't- to understand how the detector works. More specifically, a way to understand how gravitational waves can be detected even when the measuring ruler (laser light pulses with a fixed wavelenght) is stretched and shrunk as anything else affected by the wave. I've read the standard answer provided by the LIGO team, and the heuristic answer of Peter Saulson, which are great. Both explanations rely mostly on time, which led me to a tentative interpretation that goes as follows:

Let's say that the big interferometer is not a ruler with two arms, but rather two light clocks, in which the ticks are replaced by reflections on the mirrors. In absence of gravitational waves, both clocks are synchronized. When a gravitational wave crosses the perpendicular arms, though, the distance between the mirrors -whose detection might be problematic as stated before- changes, but so does the time between reflections, in order to keep invariant the relativistic interval (which for light motion is zero). The gravitational wave affects the two arms -light clocks- differently, so they get out of sync producing an interferometric effect when the beams are reunited in the detector.

I haven't found any literature about LIGO with this interpretation, so I guess is either self evident or simply wrong. What do you think?

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Your explanation in terms of light travel time is correct (and actually I would say better than the usual explanation in terms of changing lengths).

However, a key point is that gravitational waves stretch space for inertial (freely-falling) particles. The particles in a ruler are not freely falling, they are bound together by electromagnetic forces. Therefore, a gravitational wave will stretch a meter stick differently than it would stretch the distance between two freely floating particles that were 1 meter apart before the gravitational wave passed. The fact that there is an interplay between the "strain force" of a gravitational wave and the internal forces holding an object together is the basis of resonant bar gravitational-wave detectors. Anyway, this is all to say that you could in principle use a ruler to measure the displacement of freely falling particles due to a passing gravitational wave, because they are not stretched by the same amount.

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