This is a follow-up to my previous question: Can something (again) ever fall through the event horizon?
Consider the following thought experiment: I am again on my rocket at height $h$ over a black-hole event horizon. I am stationary with respect to the black-hole because my thrust perfectly counters the gravity.
I have a special flashlight: its light is polarized, and every second it flips the light polarization by 90°. I also have a telescope capable of detecting the polarization of light with arbitrarily long wavelength.
I turn the flashlight on and drop it from the rocket into the black-hole, which is big enough so the flashlight won't be spaghettified as it approaches the event horizon. I keep counting how many times the polarization flips as it falls. For a mass $M$ black-hole, with a drop of height $h$, how many times will the polarization flips before reaching the event horizon?
It must be a finite number, right? Otherwise the flashlight would never cross the event horizon from its own frame of reference. If so, it possible for the external observer to see the time the flashlight crossed the event horizon? Shouldn't the flashlight fall forever, never crossing the event horizon for the external observer?