I Minkowski metric beyond the event horizon

[disregardthat]

Far away from a mass and if A and B are traveling 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.

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.

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.

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?

Then I'm confused about what scenario you are asking about. I understood you to be asking about a star that collapses to a black hole.

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.

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.

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.

 

[PeterDonis]

how do you mean that it works fine as a measure of distance?

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 thought A and B were posed as free falling.

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).

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).

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).

 

[PAllen]

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).

I think you’re right about this. The stretch would still be finite when one hits the singularity.