https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

the spatial part is flat.

I try to imagine the following experiment:

First create two rigid shells at two coordinates r1 and r2 outside of the event horizon.

The observer on the outside shell has a long ribbon. The ribbon is slowly spooled down. After each millimeter or so of downspooling the ribbon, the outer shell observer sends at r2 a signal to the observer at r1 to ask whether the ribbon has reached r1 or not. When the signal comes back, the next millimeter is spooled from the ribbon, until the ribbon reaches r1. At this point, the totally spooled length is noted at r2 and defined as distance between r2 and r1.

Using the Schwarzschild metric, the distance ds will obviously be greater than (r2-r1) due to the "funnel shape".

The same result for ds should come out using the GP coordinates (same amount of ribbon spooled), but due to the exchange of light signals and the slow (could be arbitrarily slow) process of spooling of the ribbon, I do not see how the dr dT-term in the GP metric enters the calculation to yield the same result, since dT can basically be arbitrary. Somehow the difference in the time coordinate at the two shells must enter here, but I am not sure what the correct way to understand/calculate this is considering that after the ribbon reaches from the outer to the inner shell, everything is stationary.

Probably I'm making a very simple mistake here, but I'm not able to find it, and previous forum entries, although touching the subject, did not really clarify things.