This question is closely connected to this old question of mine. The reason for this new one is that I realize part of the issue with that old question is that its main point - the idea of throwing a particle into a black hole - seems very ill formulated. Let me elaborate.
Classical Discussion
Consider the Schwarzschild spacetime and suppose a massless particle is sent radially on the direction of the horizon. Considering the exterior region of the black hole with its usual Schwarzschild coordinates the metric becomes $$g=-f(r)dt^2+f(r)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2),\quad f(r)=1-\frac{2M}{r}.$$
The particle being thrown will have a trajectory $\gamma(\lambda)$ obeying the geodesic equation. We may solve the equation by demanding the curve be radial and ingoing. Doing so we find that $r$ itself works as an affine parameter and that $t(r)$ reads $$t(r)=-\left(\lambda+2M\ln\frac{|r-2M|}{2M}\right)+C.$$
As $r\to 2M$ we get $t(r)\to +\infty$. In this sense: a Schwarzschild observer never sees the particle crossing $r = 2M$.
We may get out of this, though, by noticing that $r = 2M$ is approached in finite affine parameter of the geodesic. In that sense we may say that the particle does indeed crosses the horizon.
If the particle were massive this last point would mean that in finite propertime it crosses the horizon, so it certainly does even if the Schwarzschild observer can't see this.
Quantum Discussion
Now let's take quantum mechanics into consideration. I want to consider the particle as quantum of a field. So let us take a massless Klein-Gordon field $\phi$.
How do we formulate and discuss the idea of "a quantum of $\phi$ falling through the horizon"? Honestly I have no idea and I see lots of problems:
The idea of particles is ill-defined in curved spacetimes because (1) distinct observers disagree on what particles are, and (2) the background may create particles. In that sense, even if we say that in the far past an observer has thrown a particle in the direction of the horizon, talking about the particle falling in seems ill-defined from start.
The particle no longer has a well-defined geodesic with an affine parameter. In the classical case even though the Schwarzschild observer could not acknowledge the particle falling into the black hole, the fact that along its geodesic the horizon is crossed in finite affine parameter shows that indeed it has gone into the hole. Here we can't do this analysis.
Taking (1) and (2) into account, it seems we could never speak of a quantum mechanical particle falling into a black hole. I see no way of "the particle acknowledging it" like with the affine parameter thing, and I see no way of taking one observer to acknowledge it as well.
This is tremendously puzzling to me. First because it obviously makes sense to me that things may fall into a black hole. Second because I see in several occasions people talking about particles falling into black holes as if it were the most common thing to consider. I won't get one extensive list on this, but take this paper, in the conclusions they say:
Our results may have implications for the black hole information loss problem. Virtually all discussions of information loss in the black hole context rely on the possibility of localizing particles - from throwing a particle into a black hole to keeping information localized.
So, how on Earth can we talk about throwing a quantum particle into a black hole in the context of QFT on which we see particles as quanta of fields? How can this be turn into a well defined idea?
