Reading MTW Gravitation book, to me is unclear the point related to the Schild's argument about spacetime curvature. It is basically the following:
Consider two observers at rest in the gravitational field of the Earth, one at height $z_1$ and the other at height $z_2>z_1$, in the context of Special Relativity where the frame at rest with the Earth acts as a global Lorentz frame.
The lower observer sends two successive light pulses to the upper observer. This defines four events in spacetime as follows: $E_1$ and $E_2$ are the emissions of the two light pulses by the lower observer, and $R_1$ and $R_2$ are the receptions of the two light pulses by the upper observer. In the spacetime diagram related to the global Lorentz frame, these four events form a parallelogram--it must be a parallelogram because opposite sides are parallel. The lower and upper sides, $E_1E_2$ and $R_1R_2$, are parallel because the two observers are at constant heights; and the light pulse sides, $E_1R_1$ and $E_2R_2$, are parallel because the spacetime is static, so both light pulses follow exactly identical paths--the second is just the first translated in time, and time translation leaves the geometry of the path invariant.
So, in principle, $E_1R_1$ and $E_2R_2$ paths could be not represented as straight lines in the diagram, what is really needed is just the "congruence" between them.
Now my question is: if we admit the above about $E_1R_1$ and $E_2R_2$ paths, it would mean the speed of the light would be not isotropic and constant in the global Lorentz frame violating the really basic feature of a Lorentz frame !
What's wrong in the above argument ?