Imagine we have three observers, A, B and C, all standing on a platform hovering just outside the event horizon of an ideal Schwarzschild black hole$^1$. At time $t=0$, as measured from the platform, A jumps off the platform radially in toward the black hole. At time $t=1$, as measured from the platform, B jumps in after A.
My general question is, How does B observe A's trajectory and how does A observe B's, specifically thinking about when each observes themself and the other crossing the event horizon? I've only taken a half semester of GR, years ago so I'm not sure how to even approach this mathematically. However, I am doing my PhD in condensed matter physics so a mathematically rigorous answer should be fine for me to parse.
My (likely silly and uninformed) thoughts on the matter:
I remember learning that C will never see A or B cross the event horizon. I believe C will see A and B slowly converge on each other approaching the event horizon while being red-shifted to oblivion. Please correct me if this is wrong.
Extending the intuition about what C observes, I would guess that B cannot observe A crossing the horizon until B also crosses. Otherwise an outside observer would observe an infalling object cross the event horizon in finite time. I am very suspicious about this conclusion though since 1) B is accelerating w.r.t. A and 2) this suggests B catches up to A. In A's own reference frame, A should cross the event horizon in finite time (perhaps fast enough that in A's frame, A hasn't even observed B to have jumped yet) and so B should not be able to catch up to A. So what is it that A and B actually observe? After A crosses the horizon (in A's frame), when if ever will A observe B also enter the interior of the black hole?
Any intuition, math or explanation about this would be much appreciated!
$^1$ Assume whatever is necessary for this to be reasonable e.g. super powerful rockets to keep the platform stationary, a massive enough black hole where tidal forces / Hawking radiation is survivable, ...etc.
