A thought experiment: imagine we have a clock, falling into Schwarzschild black hole. Obviously, any real clock would have some non-zero size in all space dimensions. Here we will be concerned with just two: r and any orthogonal one. So for simplicity let the clock be a simple rubber-like oscillating ring with a fixed k and infinite resistance to tearing. (You could also take infinite k, but I'd argue that would not be physically meaningful in this setup)
As the ring is closing to the r = 0, its oscillations will slow down and come to a halt due to physical stretching along the r dimension. What I am trying to say is that maybe these oscillations make more sense as the measure of time for the ring people, than what a numerical value of proper time tells us. In a similar way the time singularity at the horizon is nothing special for a freely falling observer.
I am unsure how to interpret the fact, that the number of oscillations per proper time unit is going down though. Seems to be quite the opposite of my original note about the volume, yet something is ringing.
