This is the main difference between general relativity and the newtonian notion of gravity: GR says gravity is just spacetime curvature, and by following an inertial (geodesic) path through spacetime, it appears to be acceleration to certain observers, but is in fact inertial. Newtonian mechanics says instead that gravity is a force and you're indeed accelerating, but it's a view that breaks down at high mass/speed, and GR makes correct predictions where Newtonian mechanics does not, so we generally prefer the GR interpretation.

But the falling person's frame isn't the only one available. Someone on the ground would rightfully describe the falling person as accelerating towards the ground (or them) (at least until they reach terminal velocity).

The most important lesson of relativity is that no reference frame is privileged. That means both descriptions are valid, provided you attach the reference frame to the description.

No, the most important lesson of relativity is that only invariants are physically meaningful. IIRC Einstein once commented that he wished his theory had been called the "theory of invariants" since that would have been a better description of its most important feature.

The second most important lesson of relativity is that yes, you can use whatever frame you wish, because invariants are the same no matter what frame you use to calculate them. So you will come up with the same physically meaningful quantities no matter what frame you choose.

This is the current predominant belief among physicists, yes, but that doesn't mean it's established fact.

> the truth is we don't know what happens within a black hole, and probably never will

This is way too pessimistic. The predictions of GR for this regime are precise and unequivocal. Even if you believe GR breaks down when spacetime curvatures reach the Planck scale (which is the current predominant belief, as I said above), that still leaves most of the black hole's interior within GR's domain of validity.

Thanks, that's what I (as a layman) supposed: that inside the event horizon isn't some space that you can't think about; it's qualitatively not distinguishable from outide.

Upthread, someone mentioned that the singularity wasn't a place in space, it was a location in time. That made me think; the mass of a BH isn't the mass of a volume of space, it's the mass of the "singularity", which (presumably?) is located at the center of the space described by the event horizon. So there must be a place where the singularity is located, and there must be a direction towards it.

Someone else suggested there are two things: a mathematical singularity and a gravitational singularity, and that they aren't the same. That hadn't crossed my mind.

That was me.

> That made me think; the mass of a BH isn't the mass of a volume of space, it's the mass of the "singularity"

This is not correct. The mass of the hole is not somehow "concentrated" at the singularity. There is no matter anywhere inside a black hole. The "mass" of the hole is a global property of its spacetime geometry.

> "singularity", which (presumably?) is located at the center of the space described by the event horizon.

Your presumption is wrong. There is no "center" of the black hole in the ordinary sense. Its spacetime geometry simply doesn't work that way; it is not like the spacetime geometry inside an ordinary spherical object.

> So there must be a place where the singularity is located

Wrong. See above.

> and there must be a direction towards it.

There is such a spacetime direction, but it's a timelike direction--towards the future--not a spacelike direction.

> Someone else suggested there are two things: a mathematical singularity and a gravitational singularity, and that they aren't the same.

That is not correct.

I'm not sure how that relates to falling into a BH, bit I don't know; there's a lot of this that I'm struggling to understand. I'm not a physicist, and not clever, just very interested.

The special relativistic effect you describe has nothing whatever to do with falling into a BH.

Not in your rest frame. In your rest frame, the ground accelerates towards you.

However, the acceleration described in both frames (yours and the Earth's) is coordinate acceleration, which is frame-dependent, and in relativity frame-dependent quantities don't have any direct physical meaning. The quantity that has physical meaning in terms of acceleration is proper acceleration--what an accelerometer strapped to you would read--and in free fall that is zero.

In a very real sense (although not a useful one for a skydiver), the answer is yes. This is a key insight of the Equivalence Principle and the curved-spacetime view of gravity. You (in free-fall) are on an inertial "straight line" path through spacetime towards the center of the Earth. When you reach the ground, it pushes on you, messily accelerating you off of that path.

This is barely useful for humans since we are very used to 'standing still' actually involving the force of your weight, so we consider that distinct from acceleration. But physics gets much easier if you don't make that distinction.