Question of three clocks

[Markus Hanke]

1 hour ago, joigus said:

Now, one thing that I think should not be overlooked is the fact that such a trajectory --interpreted as "letting go" of an object with initial zero radial velocity--, is only consistent with doing so from spatial infinity, not from a finite distance.

Yes, but if one starts from the geodesic equations, then this can be fixed by supplying different initial/boundary conditions when solving them. I don’t know off hand what expression that would yield, but I do seem to remember that such a frame (free fall from finite distance) is called a ‘drip frame’ (as opposed to rain frame for free fall from rest at infinity, and hail frame for from finite distance with initial velocity v>0). In general though, the separation between events along a purely radial time-like in-fall geodesic from rest should simply be

\[\displaystyle{\tau =\int _{r_{1}}^{r_{2}}\left(\frac{d\tau }{dr}\right) dr}\]

The devil in the details of how to find that expression under the integral.