I think you may be having it backwards. When a gravitational wave hits LIGO, the length of the arm increases slightly, as does the time it takes for light to traverse that length. The photon worldline therefore remains a null geodesic, but the total path traveled and the total time it takes to travel that path both increase. Hence, the phase delay which the interferometer observes.

V.T.

Thanks for your response. However, the LIGO detector is based on the assumption that the speed of light is always the same. However, the Shapiro Time Delay shows that the speed of light varies with the curvature of space-time. A gravitational wave will cause the length of the arms to stretch and shrink rhythmically. When the arm stretches, so does the space itself stretch between the mirrors, causing the photon to travel slightly faster. When the arm shrinks, the photon will travel slightly slower, as proven by the Shapiro Time Delay. This results in no change in the phase delay of the two returning light beams.

Peter, Shapiro Time's Delay has to be differently interpreted. The speed of light is "always" the same. It is uncorrect to state that the photon slows down. This effect states that due to the "unexpected" curvature of spacetime (respect to a straight line that we would imagine a photon would follow) a photon has to travel "a longer distance":  the time measured for a round trip will be therefore longer and the discrepancy between the hypothesized time (that along a straight line, assuming spacetime is never curved) and the actually measured time (after the photon's trip in the curved spacetime) corresponds to the time delay described by Shapiro. So, simply more time because of a longer path.

There's also another matter which seems unclear: you say that the length of the 4km arms would slightly decrease because of the space curvature in proximity of the star.

We know that length contraction is a relativistic effect of "special relativity" due to speed: when a body is moving at almost "c" its length is "perceived" shortened (only perceived by the observer, not really shortened in its reference system) because the time available for photons to bring the visual information to the observer's eyes is critically small (photons travel at "c" and the object at almost "c").

This results in a shortening at the object's extremities, since their points are more distant from the observer.

On the other hand in General relativity the space curvature describes the geodesic (the path a body would follow in space-time according to the distribution of the massive bodies which deform it), that is spacetime deformation, not bodies' structural deformation.

Maybe you want to consider these arguments and re-think your question? Bye.

I have been worried about the same problem as you Peter, though I have not ever mentioned it.  It is known near a gravitational source light will slow down.  This is the nature of an event horizon or why light tends to change direction (gravitational lens).  It isn't exactly the same as light slowing down in a dielectric but it is similar.  Its not frequency dependent to my knowledge.  Distance will also contract to infinity in an event horizon but locally if your at an event horizon the light looks like it still has the same velocity but non-locally (away from near the horizon) light has slowed down, and distance has also contracted.  If you are local then it appears there is no change in space or time for the flight of light regardless of the change in space and time.  All measurements are taken locally so we have a problem? 

You could argue that non-locally you observe the arm length contracting but then you would also have to say you see light and time slowing down (negating any change in phase with distance).  A local observation would have to parallel a non-local observation, with regard to the outcome of an event. 

Любое вещество  расширяется поперёк направления движения, а не сокращается вдоль. Отсюда равенство времён хода лучей интерферометра. Остальное - ошибки теоретиков.

на английский

Любое вещество расширяется поперёк направления движения, а не сокращается вдоль. Отсюда равенство времён хода лучей интерферометра. Остальное - ошибки теоретиков.

Any substance extends across the direction of motion, and not decreased along. Hence the equality of times the light beam path of the interferometer. Else - error theorists.

Here peter. I found this. This seems to set my mind a bit more at ease at how it works though I don't claim to fully understand it. They acknowledge that indeed the gravitational wave does stretch the light waves but give the reason why it still detects it. See the video at time 5:20 https://www.youtube.com/watch?v=iphcyNWFD10

I guess what I don't get is why they think when the space stretches the laser light still travels at the speed of light in that stretched space instead of traveling slightly faster. Or in contracted space why they still think the laser light still travels at c instead of v=c/K. I think the closest explanation I have heard is that there is a discrepancy between the passage of time and of change in space such that the two compounded lead to a detectable change. I guess one would need to look at the physics, math, that describes the compounded effect.