I've got a problem setting up a general relativity (GR) thought experiment. Thanks for your help!
I'm the observer, in a lab that is falling through the event horizon of a black hole, observing a cloud of freely falling test particles that initially straddles the horizon. The particles above the horizon are escaping to infinity in formation, as GR allows. My lab is small enough, or the black hole massive enough, that the tidal force is negligible throughout this experiment.
Here's my problem: I want all the particles to move in formation as I measure. GR's equivalence principle (EP) tells me that the laws of physics are the same in my lab as in any other inertial frame. In another inertial frame I could let all the particles move in formation just by giving them all the same velocity as I measure. But GR also tells me that this cloud must be splitting apart! GR demands that the particles below the horizon move inexorably inward, toward the black hole's singularity, whereas the particles above the horizon move ever outward, away from the black hole. Then how can I let all the particles move in formation, so that the EP is true?
If I simply let all the particles have the same velocity as I measure (the velocity of the escaping particles), how can they move toward multiple destinations? If instead my lab was falling toward the ground on Earth, with me observing a cloud of particles of which half are escaping to infinity and the other half are falling toward the ground, they won't all have the same velocity as I measure; they won't be moving in formation. This seems to be a real paradox. GR won't let all the particles move in formation, but its own EP demands that they can. What am I missing?
(I'll say in advance that I won't be satisfied with an answer that invokes the tidal force, since I've declared that to be negligible as GR allows. Humans and their labs can in principle survive a fall through an event horizon, without being worse for wear. You can assume that whether the cloud is splitting apart can be detected by me in an arbitrarily short time on my clock, before the particles have moved so much as a nanometer as I measure. Giving credit where credit is due, this question was inspired by an article you can find by searching for "no black holes finbot". It's not mine.)