Change in travel time due to localised spacetime perturbation

• I
cicero
TL;DR Summary
Knowing the return time of a signal traveling between locations A and B in flat space, what is the change in this return time due to a localised perturbation of flat space between A and B?
Suppose you have the following situation:

We have a spacetime that is asymptotically flat. At some position A which is in the region that is approximately flat, an observer sends out a photon (for simplicity, as I presume that any calculations involved here become easier if we consider a massless object). At some point B which again is in a region where the spacetime can be considered approximately flat, that photon is reflected ("the spaceship turns around"), and returns to A. From previous experiments, the travel time ##\Delta\tau_0## between A and B in Minkowski spacetime is known (to the observer at A, so in proper time for that observer).

Now suppose this experiment is performed but not in Minkowski spacetime but instead a localised perturbation of the flatness of spacetime far enough away from A and B not to affect them meaningfully has appeared. Clearly, this is going to change the travel time ##\Delta\tau## of the photon as observed at A (again, in proper time for A). From the perturbed metric ##g_{\mu\nu}##, how would I calculate ##\Delta\tau/\Delta\tau_0##, so the relative increase/reduction in travel time?

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Mentor
You might want to Google "Shapiro time delay".