Multiple Infalling to Black Hole Horizon: Alice, Bob & Charlie

[_PJ_]

It seems that every description providing ways to visualise or form analogies or presernt a simple model for infalling to Black Hole causal event hortizons rely on two observers, one aporoaching and potentially 'crossing' the horizon, whilst the other 'observes' from a "safe" distance.

It makes sense of coursr, to have just Alice and Bob as the two intrepid BH experimentalists, for the most simplistic and readily understandable/accessible picture of the potentially paradoxical consideration of when or even if the infalling event occurs at all.

Yet despite adding a necessary amount of complexity, considerring just oner extra observer should reveal a much richer description.

For instance, since the time dilation effect increases to limit on approach to the Horizon, then even an accelerating Charlie trying to catch up with Bob as the latter continues towards the Horizon, will never reach either since Bob will always be in the distance despite slowing down - To me, and this may require correction and/or further clarification, the distance to the Horizon will appear to increase regardless ofg Charlie's motion towards Bob?
Alice will 'see' Bob redshifted and slowing until fading from view before the horizon, but will she notice Charlie catching up to Bob or never getting closer? Admittedly, this is similar to the consideration of just observing as Bob heads to the Horizon, but I think with Chartlie's frame of reference considered and a cave that possibly even for Bob's perspective, there will always be something (especially light) that will be moving to the horizon beforehand which I am under the impression should necessitate a 'prevention' so that nothing can actuially experience crossing the Horizon.

This 'road block principle' also helps eradicting certain paradoxes, but without a holograpghic BH would mean that BH couldn't actually accept any more matter!

 

[PeterDonis]

the potentially paradoxical consideration of when or even if the infalling event occurs at all.

This is not a "paradox" at all. It's just a misunderstanding (unfortunately a common one). The correct answer is that the event does occur.

since the time dilation effect increases to limit on approach to the Horizon, then even an accelerating Charlie trying to catch up with Bob as the latter continues towards the Horizon, will never reach either

Wrong. Charlie will catch up with Bob, and will also cross the horizon.

Alice will 'see' Bob redshifted and slowing until fading from view before the horizon, but will she notice Charlie catching up to Bob or never getting closer?

If Charlie catches Bob before either of them crosses the horizon, Alice will see Charlie catch up to Bob. If Charlie doesn't catch up to Bob until after they have both crossed the horizon, then what Alice sees depends on the initial conditions--how far Bob has fallen when Charlie starts after him.

This 'road block principle'

Is just a misunderstanding on your part.

 

[_PJ_]

What measurement do you make to determine B crosses the Horizon?

 

[PeterDonis]

What measurement do you make to determine B crosses the Horizon?

You observe the readings on his clock. Alice will see B's motion appear to slow down--but she will also see his clock appear to slow down, so much so that there is a particular finite value for B's clock reading that she will never see. That value is the reading on B's clock when he crosses the horizon. Alice can compute this value by taking appropriate limits based on her observations.

 

[_PJ_]

That doesn't make sense.
Who can read the clock?
Bear in mind that even if Bib tried to signal anything,the time delay for A would increase.
The limit you refer to is infinite- to Alice Bob will never cross the horizon and as he slows down, before he gets there any light will be redshifted so much that a receiver comparable to the horizon radius would be needed to detect! Alice would die of old age long before.
Calculating position and velocity of B to extrapolate where A thinks he should end up is not a measurement.

 

[PeterDonis]

Who can read the clock?

Alice can see it in the light she receives from Bob.

The limit you refer to is infinite

No, it isn't. Look up the math.

It is true that Alice would have to live indefinitely long to see all the light signals Bob emits before he reaches the horizon, but this is a thought experiment; it's not bound by the limitations of our current technology or biology.

Calculating position and velocity of B to extrapolate where A thinks he should end up is not a measurement.

It's not where A "thinks" B should end up; it's where B will end up if no other factor comes into play. We do the same thing when predicting when space probes will arrive at particular planets. And A can, in principle, observe B to make sure no other factor comes into play, right up to the point where B reaches the horizon.

I am closing this thread because the misunderstandings you are exhibiting are not new, and they have already been addressed in many, many threads on this forum, not to mention in most textbooks on GR. You should consult mainstream sources to improve your understanding.