Does detector LIGO work on freewill only?

[Alex K]

What do you consider a "small laboratory" in this case?... And at which point does the interferometer look at the color of the laser anyway?

1) the 4 km of LIGO is much, much smaller length, than the period of the gravitational wave (otherwise the theory does not work).

That's true, indeed, and I concede that LIGO is small in the sense of the equivalence principle - compared to the long waves recorded there, and higher order effects showing the violation of the E.P. probably don't play an essential role.

Where you go wrong in your application of the equivalence principle to LIGO is that the equivalence principle only states that the results are the same in all (small) laboratories if one considers the exact same experimental setup. The opposing mirrors in a LIGO arm 1. without a gravitational wave and 2. with a gravitational wave trough or peak passing through, are however not the same distance. When the wave enters the interferometer arm, due to the change in spacetime, the mirrors now suddenly have a different distance, and therefore the interference pattern changes.

The experiment would only show the same results with the wave present if the mirrors were the same distance as measured in the old spacetime without wave and the new spacetime with wave. To achieve that, they would have to be glued to the (on the timescale of the wave sufficiently rigid) interferometer arms instead of hung freely, because ideal rigid interferometer arms keep constant length when the wave comes in because their length is determined by the atomic constants. Then the experiment would show the same when the wave comes in as you demand it from the equivalence principle.

HOWEVER, since the mirrors are hung freely, their distance will change when the wave comes in due to the local expansion and constriction of space between them, the numerical parameters of the setup change (namely the spatial distance of the mirrors), and a changing interference pattern is not in contradiction with the equivalence principle.