I haven't done the maths, but I expect the repeats would be longer and longer, as they are in the Newtonian regime. The question is: why? Is it because the distance between the ships is increasing? Or is it because the spacetime geometry is changing between the ships as they fall in?

Outside the event horizon you can answer the question by finding some observers for whom local spacetime geometry is not changing. These are hovering observers, and we can use their notion of "at the same time" to define a notion of "space" and separate out the time change due to distance-in-space.

Inside the event horizon, however, there can be no hovering observers. In fact, there are no observers for whom spacetime is not changing around them. So there is no one to tell you "x% of the time delay is due to distance, y% due to time dilation/spacetime curvature" (formally: there are no time-like Killing vector fields inside the event horizon). You are free to attribute all of the change in pulse return time to the differing spacetime geometry between A and B, or all of it to changing distance, or any mix thereof.

That's my understanding at least. If Peter or pervect tells you different, believe them...

What do you mean by a Newtonian regime? I am not sure one can approximate the answer to that with Newtonian mechanics. Especially if the mass of S is large.

The question as presently stated does not require a third frame of reference, I believe. It refers exclusively to a system of two particles and a gravitational body in the frame of reference of A.

No. This is a case where you really need to look at the math; your intuitions are leading you astray.

I'm confused; you also said this:

The latter is the correct answer, since you can't understand black holes from an SR perspective. But that means your previous attempt at doing that won't work.

This won't work, because there isn't a well-defined "frame of reference of S", and there isn't a well-defined "gravitational force", at or inside the horizon of a black hole.

It's worse than that. You can't even define a "reference frame" at all that covers A and B. The best you can do is find a global coordinate chart that covers both their worldlines. But you won't be able to use that chart to construct a "reference frame" that allows you to define "distances" the way @disregardthat is trying to do.

@PeterDonis Like I said, it was my motivation for my question, and did not involve black holes at all (S is not a black hole singularity in that example). Since I do not know general relativity, I merely attempted to look at what my (limited) knowledge of special relativity could tell me in such a situation, which was not much. I am eager to see your comment on the situation where there is not reference to any reference frame other than A (and if it even then can not be meaningfully be applied to the situation of a black hole, then I would like to know the answer when S is simply a non-black hole gravitational body).

Far away from a mass and if A and B are travelling slowly then general relativity looks very like Newtonian gravity. Remember that the GR correction for Mercury's orbital precession is only 43 seconds of arc per century. You can simply disregard all of the complexity and use Newton to work out what happens if you drop two objects into the Sun on any timescale less than a decade. In this case, normal intuition applies and the distance between A and B grows and the pulse return time grows. All of the complexity we've been discussing applies, strictly speaking, but the error from ignoring it is tiny. You have a slight question mark about how light behaves, but as long as the velocities of A and B relative to the Sun are low, any plausible answer doesn't actually make a qualitative difference to the outcome.

Of course, if you have a denser or more massive body, you can no longer disregard the worries about changing curvature. But I suspect that you will find that the pulse return time continues to grow.

It depends what you want to know. Does the pulse return time grow? You can answer that one in a coordinate independent way, yes. As I said I expect it continues to grow, although I have not confirmed that mathematically.

Where you need to introduce coordinates (not a reference frame! That's strictly an SR concept, as Peter points out) is if you want to ask "is the pulse return time changing because the distance between A and B is changing?" And you can't really answer that from A's point of view because the spacetime geometry around him is changing so the concept of "distance as defined by A" isn't definable. So to answer this question - which was where you started, I believe - you do need to introduce a coordinate system that is static. And you can do that outside the event horizon by pegging the coordinates to hovering observers. But there cannot be hovering observers inside the event horizon.

This works fine as a definition of "distance", but it requires that it be possible for light to make repeated round trips between A and B. If one of the two is inside the horizon of a black hole, and the other is outside, this is not possible.

As long as the second one free-falls in after the first one it'll work, I think. Although the fact that it just fails completely if the second one aborts its fall after the first has crossed the horizon is a rather dramatic illustration of the point.

But how do you mean that it works fine as a measure of distance? Are you just naively multiplying the return time by c/2, as per Dolby and Gull's radar time methodology? Isn't that pretty much a reasonable-but-completely-arbitrary decision?

I thought A and B were posed as free falling. Then they can continue exchanging signals until A ‘dies’. If they started out close enough, these radar coordinates will be near Minkowski until both are well inside the horizon. Signal intervals between them will grow, and a geometric argument can be given that this growth is primarily a distance growth as long as they remain reasonably close. This argument is that if they had only radial separation when both were just outside the horizon, then when both are inside, their separation along the extra spacelike killing direction (besides the two angular killing directions which hold constant for them) will be increasing (ultimately without bound). Since this distance is measured along a killing symmetry, I think it is reasonable to attach some physical significance to it. This is also the direction in which any body will be stretched, while being compressed in the angular directions.

The velocities are low, but so is the velocity differential. The difference in acceleration for two objects A and B with a tiny separation d is small, and I don't quite see why it should be sufficiently larger than any resulting relativistic effects.

I see, so I gather that "frame of reference" in GR is basically a concept which is only meaningful for a point with approximately flat spacetime. Is there no meaningful way to speak of a "frame of reference" for a particle A in curved space?

That second question was part of my discussion with (EDIT: you!). I explicitly pointed out a non-black hole point mass for that scenario, and I feel like there should be no cause for confusion in my post on this particular issue.

Will this definition of distance work if both A and B are inside the event horizon, and how will it behave? I'm also interested in the situation where A and B are falling towards a non-black hole.

I mean you can define "distance" this way and it will give you consistent results. The word "distance" is an ordinary language word and does not have a single precise meaning; the round-trip light travel time method is one way of giving it one, but of course not the only way (and the different possible ways are not all consistent with each other).

I'm no longer sure exactly what scenario the OP is asking about. In the case that A and B are both free-falling, yes, they can continue to exchange light signals until one hits the singularity, provided they are not separated too far initially (if their initial separation is large enough, A will hit the singularity before even one round-trip light signal).

I'm not sure about the "ultimately without bound". It seems to me that their separation along the other spacelike Killing direction should approach a finite limit. More precisely, if for any radial coordinate outside the horizon, the events at which the two objects cross that radial coordinate are separated by a finite Killing time (since the fourth Killing field is timelike outside the horizon), then the limit of their separation along the fourth Killing field inside the horizon, as both approach the singularity, will be finite (although of course it can be made arbitrarily large by allowing the time separation at which the objects cross a particular radial coordinate outside the horizon to be arbitrarily large).