Why plane wave approximation in curved spacetime "exactly" determines the causal structure?
In Phys. Rev. <PERSON> while working out the light propagation in gravitational background the author had to solve $$D_{\mu}F^{\mu\nu}+\xi^2R^{\mu\nu}{\hspace{7pt}\sigma\tau}D{\mu}F^{\sigma\tau}=0$$ where $\xi^2=\frac{\alpha}{90\pi m_e^2}$ and the background spacetime satisfies $R_{\mu\nu}=0$. I'm having understanding this passage
In order to derive the equation for the characteristics of photon propagation we make the simplest geometrical-optics plane-wave approximation. We should emphasize that although this is an approximation for the actual solution of the field equation for the photon, it exactly determines the characteristics of propagation and hence the causal structure of the solution.
Now my doubt is how come this approximation gives us the exact characteristics of light propagation. In this approximation we can set $F_{\mu\nu}=f_{\mu\nu}e^{i\theta}$ where $f_{\mu\nu}$ is a constant tensor and all the time-varying properties is encoded in $\theta$, along with the direction of propagation. 651
These are the two scenarios I'm thinking which are causing me the issue:
1. In <PERSON> spacetime when the light propagates there is no backreaction on the spacetime but in the case of curved spacetime there will be backreaction (The $\vec{E}$ causes a change in $g_{\mu\nu}$ and then the $g_{\mu\nu}$ metric affects $\vec{B}$ though this effect is order 2 above the order we are doing perturbation calculation) and this will change the direction of propagation ultimately changing the causal structure. Although this calculation might make a tiny change in causal structure but due to this we can't use the word "exact" there "almost" seems more suitable. At the face value, this reasoning seemed flawed since I'm talking about order calculation while the author uses plane wave approximation but depending on the structure of $F_{\mu\nu}$ we may have different change in causal strucutre.
2. When we study gravitational waves we have to define region in spacetime where we can approximate our metric to get the propagating wave. So do we have to define region of spacetime here as well? But the causal structure is defined everywhere on the spacetime manifold so I think we have to solve the wave equation exactly. 651