Change in travel time due to localised spacetime perturbation

[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?

 

[PeterDonis]

You might want to Google "Shapiro time delay".

 

[cicero]

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).

 

[Ibix]

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.