I have read other questions concerning this subject, and by now I believe that in order to solve a gravitational problem in GR, one has to basically abandon the notion of frames of reference. However, I cannot get my intuition to align with this, and I would like someone could explain me the logic and the intuition behind (part of) an answer to the following problem:
Two free falling masses are to be employed to measure the effect of a gravitational wave by monitoring the time delay in successive laser pulses going from one mass to the other. What is the maximal time delay?
Of course, the question provides a diagonal metric describing this wave pertubatively. Its exact form, though, is irrelevant here. The part of the solution that bothers me is the following: we want to integrate the trajectory of the photon, and the first step is to impose the "infinitesimal" spacetime interval to be lightlike. Now, given that we ask that these masses lie on an $xy$ plane perperdicular to the propagation direction $z$ of the wave, we can impose that "$dz^2 = z= 0$".
We want to integrate the trajectory of the photon, and the first step is to impose the "infinitesimal" spacetime interval to be lightlike. Now, given that we ask that these masses lie on an $xy$ plane perperdicular to the propagation direction $z$ of the wave, we can impose that $dz^2 = 0 = z$.
This is what bothers me. Intuintively I can easily see why the photon should travel along a geodesic with constant $z$, and that this could be easily attributed the coordinate value $0$, but how do we formally justify this? Is the statement "$dz=0$" a rigorous one, or rather a shorthand notation meaning that the $z$ component along the geodesic is constant? Can we justify this fact a priori? How? If the geodesic deviation equation in this case tells us that the separation vector between the geodesics followed by each mass has zero $z$ component, can we employ this to justify a priori that the path followed by the light pulse has constant $z$?
