[antognini]

I don't completely understand the argument you're making, so please correct me if I'm misunderstanding you, but I think I see the problem.

It is true that in an inertial frame you can send a signal between two points. But if you have an inertial frame which is falling into a black hole any signal sent from beneath the event horizon will not reach any other point until that point has also fallen beneath the event horizon.

Remember that the inertial frame is in free fall, so points which are fixed in the inertial frame are moving according to a distant observer.

[fargolime]

> But if you have an inertial frame which is falling into a black hole any signal sent from beneath the event horizon will not reach any other point until that point has also fallen beneath the event horizon.

Problem is, the escaping particle never falls beneath the horizon. So an inertial frame in which that particle is at rest cannot extend below the horizon at all. That's a violation of law K for a frame falling into a black hole. By violating law K, the latter frame cannot be inertial. Note that law K is about allowing inner frames be extended to fill all of an outer frame.

Let a cloud of particles straddle the horizon. Let all the particles above the horizon be escaping to infinity. According to GR, such cloud must be splitting apart. The particles below the horizon must move inexorably inward, toward the singularity at the center of the black hole, whilst the particles above the horizon move ever outward, away from the black hole. Then a frame falling through the horizon of a black hole cannot be inertial, for in an inertial frame the cloud needn't be splitting apart (since law K applies). For example, in an inertial frame in Earth's atmosphere you can have a cloud of particles in which half the particles are given to be escaping to infinity, and the cloud needn't be splitting apart (just let all the particles escape in formation).