I Calculating Relative Change in Travel Time Due to Spacetime Perturbation

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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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You might want to Google "Shapiro time delay".
 
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PeterDonis said:
You might want to Google "Shapiro time delay".
I am aware of the Shapiro time delay, though in my books I always had it down as the particular case of light traveling around some central mass like a star.

I guess what my question was more targeted at was how to calculate something like this in general (and not just for the Schwarzschild metric, as for the Shapiro time delay).
 
Generally speaking, I think you would keep the source and mirror at specified coordinates in the asymptotically flat region. Then you find a null geodesic connecting the two for the outbound journey and one connecting them on the return journey (the latter is trivial in a static spacetime but not in a general spacetime). Then you compute the proper time along the emitter's worldline between the emission and return events.

It's easier if you mean a weak perturbation, when spacetime is nearly flat and you can write ##g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}## and neglect higher order terms in ##h##. But the principle is the same.
 
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