How does the following brief thought experiment fail to show that general relativity (GR) has a major problem in regards to black holes?
The full thought experiment is in my blog post. The post claims that GR violates its own equivalence principle at the horizon of a black hole. The principle says that the laws of physics in any sufficiently small, freely falling frame are the same as they are in an inertial frame in an idealized, gravity-free universe. Here's a condensed version of the thought experiment:
In an arbitrarily small, freely falling frame X that is falling through the horizon of a black hole, let there be a particle above the horizon that is escaping to infinity. A free-floating rod positioned alongside the particle and straddling the horizon couldn't be escaping to infinity as well, or else it'd be passing outward through the horizon. However, if instead the rod didn't extend as far down as the horizon, then in principle it could be escaping, possibly faster than the particle beside it. In an inertial frame, unlike in X, a body's freedom of movement (in principle and if only relative to other free objects in the frame) doesn't depend on the body's position or extent. Then a test of the laws of physics can distinguish X from an inertial frame. If X was equivalent to an inertial frame, I wouldn't be able to tell whether the rod could possibly be passing the particle in the outward direction, by knowing only whether the rod extends as far down as an imaginary boundary (the horizon) within the frame. If X was equivalent to an inertial frame, the rod could in principle be passing the particle in the outward direction regardless of its extent within X.
The thought experiment above takes place completely within X, which is arbitrarily small in spacetime (arbitrarily small both spatially and in duration). That is, the experiment is completely local. That the particle is escaping to infinity is a process occurring within X; it tells us that the particle won't cross the horizon during the lifetime of X. The particle needn't reach infinity before the experiment concludes.
It isn't necessary to be able to detect (by some experiment) that a horizon exists within X. It's a given (from the givens in the thought experiment) that a horizon is there. Likewise, I am free to specify the initial conditions of a particle or rod in relation to the horizon. For example, I am free to specify that the rod straddles the horizon, and draw conclusions from that. The laws of physics in X are affected by the presence and properties of the horizon regardless whether an observer in that frame detects the horizon.
It seems to me that the only way the equivalence principle is satisfiable in X is when in principle the rod can be escaping to infinity regardless of its initial position or extent in X, which would rule out black holes in a theory of gravity consistent with the principle. Otherwise, it seems the bolded sentence must be incorrect. If so, how? In other words, how can I not tell whether the rod can possibly be passing the particle in the outward direction, by knowing only whether it extends as far down as the horizon?
I'd appreciate hearing from Ted Bunn or other experts on black holes. A barrier to getting a satisfactory answer to this question is that many people believe the tidal force is so strong at the horizon that the equivalence principle can't be tested there except impossibly, within a single point in spacetime. An equation of GR (see my blog post) shows that a horizon isn't a special place in regards to the tidal force, in agreement with many texts including Ted Bunn's Black Hole FAQ. In fact the tidal force can in principle be arbitrarily weak in any size X. To weaken the tidal force in any given size X, just increase the mass of the black hole. (Or they might believe it's fine to test the principle in numerical approximation in a frame larger than a point, but not fine to test it logically in such frame anywhere. Kip Thorne disagrees, in a reference in my blog post.) Note also that the Chandra X-ray Observatory FAQ tells us that observations of black holes to date aren't confirmations of GR, rather they actually depend on the theory's validity, which is to say the existence of black holes in nature isn't proven.
Edit to add: I put a simple diagram, showing GR's violation of its own EP, at the blog post.
Edit to add: I'm awarding the bounty to dbrane, whose answer will likely retain the lead in votes, even though it's clearly incorrect as I see it. (In short, the correct answer cannot be that an infinitesimally small frame is required to test the EP. It is in fact tested in larger labs. The tidal force need only be small enough that it doesn't affect the outcome. Nor is the horizon a special place in regards to the tidal force, says GR.) I do appreciate the answers. Thanks!
Edit to add: this question hasn't been properly answered. The #1 answer below made a false assumption about the question. I've beefed up the question to address the objections in the answers below. I added my own answer to recap the objections and reach a conclusion. Please read the whole post before answering; I may have already covered your objection. Thanks!