To me, a detector (which follows a world line) is the only thing that can observe. Coordinate labels and simultaneity are abstractions, not observables. So any detector that can observe light emitted inside the EH observes red or blueshift following normal GR rules. The general GR rule is to parallel transport the emitter 4-velociy to the receiver world line along the light path followed. Then express the transported emitter 4 velocity and local light path tangent vector in the local frame basis of the reception event, and use SR doppler formula. This formulation covers every combination of kinematic and curvature and cosmological contribution to redshift in one general method.According to me, a worldline cannot observe. Anyway, I still wonder why the emission frequency would be positive and real (from a distant perspective of course) instead of imaginary as with the approximate solution:
The concept of a gravitational redshift 'field' as in that wikipedia formula is a special case that applies only to static space time. It is doubly nonsensical to apply it across the event horizon:
1) There are no emitter and receiver possible from the inside to outside the EH. Therefore it is applied outside its domain of validity.
2) You can't even write an alternative equivalent formula for inside the horizon because the region inside the horizon is not static, so there is no such thing, even approximately, of a gravitational potential. The concept of gravitational redhsift factored out separately from general redshift as I described above is possible only for static spacetime.
One final key point: for emitter outside EH and receiver inside EH, you can compute the red/blue shift using the standard method I describe and get a perfectly normal result - because there is a light path from emitter to receiver in this case. The result cannot be summarized into a redhift as a function of position for the very reason above: the light path goes from a static region of spacetime to a non-static region.
I'm sorry if all of these realities of GR are ignored in wikipedia level treatments, but I can assure you they were well understood by 1960.
There is no such thing as a global frame in GR. There are local frames and global coordinate systems. So your statement asks for something GR states does not exist.I have followed some of those threads without participating: the parts that I saw exactly avoided addressing such issues as this one. And it looks as if you formulated it better (more precisely) last time:
"They can deduce that the object has reached the singularity. Specifically, they can compute when to send a signal such that the infaller will receive it an instant before reaching the singularity. [..] "
If you transform that description of yours to our distant ECI frame, I suspect that you will find that to our reckoning, those signals of us will reach that person shortly before t=∞ so that "an instant before reaching the singularity" transforms to "perhaps after the end of this universe". But if that is wrong, please clarify why.
The best you can do in GR is define a family non-intersecting spacelike 3-surfaces of simultaneity that you parametrize with a timelike parameter. SC coordinates do this in a way that covers only the exterior region of a BH. Lemaitre and Kruskal coordinates do this in a way that covers the whole spacetime. Each provides a specific (and different) answer to what events inside the horizon are simultaneous with a given event outside.
Each of these coordinate systems (Lemaitre and Kruskal) provide a finite well defined answer to 'when', for an outside observer, a signal reaches a given inside detector.
Short answer: your question reflects fundamental misunderstanding of GR; corrected in the only way I know, your conclusion is wrong.
As above, GR says this question is wrongly formulated - there is no preferred definition of global earth coordinates. There are at least two common global coordinates that answer this question - each differently. SC coordinates don't answer it for the trivial reason they don't cover enough of spacetime. (Note: SC coordinates provide two separate coordinate patches, one for interior region, one for exterior. If you want a simultaneity convention between spacetime regions, you need a single coordinate patch that includes both regions. Thus SC coordinates are simply inadmissable for answering your question).That may be well the case. I think that there will be even much less misunderstanding if you or someone else would be so kind to express those events in "earth coordinates", which would clarify if in theory this topic relates to something that can ever happen. But as this is the fourth consecutive post in which I ask this, I'll leave it at this if it still doesn't get addressed.
[EDIT: Perhaps this coordinate free description will help ... or not; we'll see. You can't treat causally connected events as simultaneous. The relations of backward and forward going light cones defines the causal structure of spacetime. In the SC geometry, every event outside the EH is in the causal past of a set of events inside the EH. The causal future of any internal event includes only internal events. For a given event, anywhere, you can choose to consider any event between its past and future light cones as simultaneous with it. For an outside event, this means there are a set of interior events in its future light cone that cannot be considered simultaneous. Any interior event 'before' this future light cone can be considered simultaneous with the chosen external event. Thus not only is it possible to choose interior events simultaneous with external event, there are infinite possible choices. Similarly, given an internal event, there is a set of external events in its past light cone; any external event outside of this past light cone is a possible choice for a simultaneous external event]