Time of flashlight crossing the event horizon as seen from an external observer 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?
 A: In addition to the answer by Ben Crowell, I would like to add one point.
If we model the flashlight and the observer somewhat realistically, than flashlight would have a finite power that it radiates and observer would have a finite threshold for the detection of such radiation. As the flashlight crosses  horizon in a finite time by its own clock it would radiate only a finite amount of energy. Observer outside at finite radius would see that the frequency of light and its power start to drop exponentially. While the intensity of the signal from the flashlight as absorbed by an observer may remain positive for all times, quite soon it will drop below any theoretically feasible threshold of detection.
So in summary, observer will see the flashlight for a finite amount of time, the number of polarization flips (and total number of photons emitted by flashlight) would remain finite.
A: This is also a note on Ben Crowell's answer (it started as a comment but I think it makes a point which isn't in the other two answers).
As Ben says, the total number of flips you observe is finite, say $n$.  But the intervals you observe between the flips are not constant.  In particular, if you adjust the timing mechanism on the torch suitably before dropping it, you can arrange things so the time you have to wait (your proper time) before you see the $n$th (last) flip is arbitrarily large.
As A.V.S. also points out, the total amount of energy emitted by the torch before it crosses the horizon is finite.  A consequence of this is that it becomes increasingly hard to observe the flips, and in particular the last flip: the light you see becomes very faint and very red-shifted.  Especially if you assume that the torch emits a stream of photons, you quite quickly (quickly in your proper time) reach the point where the probability of you being able to observe any more photons from the torch is vanishingly low.  At that point there can be no real observational difference whether the torch has crossed the horizon or not: there is always a chance that you might detect another photon, but that chance is vanishingly small.
In fact see A.V.S.'s comment below: the photons pretty rapidly drop below the background Hawking radiation from the BH in fact, so it really is the case that the infalling object becomes undetectable, even in theory.
A: You would see flip $n$, which was the last to occur outside the horizon, and you would never see flip $n+1$.
A: I guess my own answer to the question would be: for any frequency of flipping, there will be one last flip before it can never be observed again (because the next flip would be beyond the event horizon on flashlight's proper time). But the higher the frequency, the more it will take for you to observe the last flip. If there is no limit on how high that frequency can be, there is no upper limit on how long the last visible flip can take to happen.
Please correct me if I am wrong.
