Mike S. said:
My question as asked only addresses the space outside the event horizon
Yes, but you are also relying on Schwarzschild coordinates as a "frame". They're not, at least not in the sense you are using the term.
Mike S. said:
Your objection to "relative to the frame" surprises me. When learning special relativity, there are few things so common as calculating the time it takes a particle to decay or a twin to age.
Yes. Are you aware that you can do such calculations without using
any "frame" at all, inertial or otherwise? That, for example, the "twin paradox" can be solved without any use of a frame?
If you are not aware of these things, I strongly suggest spending some time with a textbook that emphasizes invariants over "frames". I first got this viewpoint from the classic textbook on GR by Misner, Thorne, and Wheeler, but that book is a heavy lift. Taylor & Wheeler's
Spacetime Physics used to emphasize this at least somewhat in its old edition (which was the book I first learned SR from), but I don't know if it still does in its new edition.
As I said before, you can, of course, calculate invariants in any coordinates you like, as long as they cover the relevant region of spacetime. If you are indeed only concerned about what happens outside the horizon, you can indeed calculate all the required invariants in Schwarzschild coordinates. But you are going to have to suppress plenty of intuitions while doing so, because if you take those coordinates to have the sort of physical meaning that presentations of SR often ascribe (wrongly, IMO) to inertial frames, you are going to be led down some very tempting but wrong paths of reasoning. Your previous posts in this thread have already shown some examples of this.
Mike S. said:
Is it accurate to say that the event horizon of a black hole is stationary
Not with the meaning of "stationary" that you appear to be using. The horizon is a null surface, which is composed of light rays (radially outgoing light rays, in this case), and light rays can never be stationary in this sense. To put it another way, the horizon is not a "place": any "place" must be described by a timelike curve (or a set of them), not a null curve (or a set of them).
There is a meaning of "stationary" that (almost) applies to the horizon, but it doesn't have the implications that the ordinary meaning of "stationary" does. This meaning is technical: a curve can be said to be "stationary" if it is an integral curve of a timelike Killing vector field. (I won't explain that further here, but it should be easy to find a definition of this term online, or in any number of GR textbooks.) The worldlines of observers that are "hovering" at constant altitude above the black hole, and have no tangential velocity (i.e., they are not in orbits about the hole, just "hovering" radially) are stationary in this sense. The horizon is composed of integral curves of the same Killing vector field as those worldlines, but they are null, not timelike, so they are not stationary in this sense, but they "almost" are since they are the limits of such timelike curves as the radial coordinate ##r## goes to ##r_s##, the Schwarzschild radius of the hole.
Mike S. said:
or that its velocity is zero, relative to the outside observer on the statite?
No. In a curved spacetime, there is no valid concept of "relative velocity" between things that are not co-located. The only observers who can assign any meaningful "velocity" relative to them to the horizon are observers who are falling through the horizon, and they all assign to it a relative velocity of ##c##, radially outgoing.