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A clock is sent towards a black hole. My understanding is that because of time dilation, the clock will appear to tick more and more slowly to an observer on Earth, to the point where the time dilation factor diverges to infinity as the distance between the clock and the event horizon converges to 0. Correct?

In the frame of the clock, it will take a finite amount of time to cross the event horizon and reach the singularity. We suppose the black hole is so massive that the clock may cross the event horizon without being crushed by tidal forces. After 1 second measured in the clock's frame inside the black hole (after being past the event horizon), how much time would have elapsed on Earth?

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    $\begingroup$ There is no such thing as global frames of reference in general relativity. The question asked simply does not have any physically meaningful answer. $\endgroup$ – mmeent Apr 12 at 12:22
  • $\begingroup$ Could you elaborate please? I know that no such frame of reference exists in general relativity, but I can't see what in my question refers (explicitly or implicitly) to such concept. $\endgroup$ – Alfred F. Apr 12 at 12:32
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    $\begingroup$ I'll eloborate by asking a question: What do you mean by "how much time would have elapsed on Earth"? How would you define this interval? $\endgroup$ – mmeent Apr 12 at 12:39
  • $\begingroup$ I think I start to see where this is going. When I ask what happens on Earth while the clock is inside the BH, I implicitly refer to what is going on simultaneously within those two frames, which is pointless. In order to acknowledge time dilation, the clocks need to be reunited in the same frame, which is impossible here since one was captured by the BH. Correct? $\endgroup$ – Alfred F. Apr 12 at 12:47
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    $\begingroup$ @safesphere that is incorrect. Simply compute the trajectory of the clock in coordinates such as Eddington-Finkelstein or Kruskal-Szekeres, which work perfectly well at the horizon. $\endgroup$ – Javier Apr 13 at 13:23
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I may try answering by asking another question: in Earth referrence frame, will the clock ever cross the event horizon?

Now, as you mentioned time dilation would diverge as the clock approaches the event horizon, therefore I would say that in Earth reference frame the clock will never be inside the black hole.

This makes me think that you cannot compare the time measured in Earth reference frame with the one measured in the clock reference frame once the clock is inside the black hole; in fact I think you cannot compare the time measured in any inertial reference frame that is outside the black hole, because of the time dilation you mentioned.

To me the only way your question may have a (very speculative) point is making the clock exist directly inside the black hole, being there since the origin of everything without having crossed the event horizon in order to be inside the black hole.

GZ

EDIT

The answer to this question about whether M87 black hole is an actual black hole might be interesting to you, as the example there made can be compared and used to your question: time dilation does not imply that the clock never crosses the event horizon, but there is no way to communicate with said clock.

In fact, Earth reference frame would receive signals the clock reference frame sent before crossing the event horizon, and those signals would take up to infinite time to reach Earth reference frame.

On the other side, Earth reference frame would not be able to send signals to the clock reference frame once the clock crosses the event horizon and viceversa.

According to this, then, a confrontation between the elapsed time in those two reference frame is not possible

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    $\begingroup$ Yes the clock will never cross the Event Horizon (EH) in a finite amount of time from an outside frame. But my point is that in its frame, the clock will. Let me add some precisions. EH is an abstract boundary: the clock experiences continuity of motion, time and existence as it crosses it. It seems reasonable to wonder what happens to the physical world outside EH. I suppose nothing prevents the clock to collide with a particule that crossed EH after it did? Before crossing EH, where was that particule in the clock's frame, and with what proper time relative to the clock's own proper time? $\endgroup$ – Alfred F. Apr 12 at 11:58
  • $\begingroup$ You are right and I did not understand your question properly (feel free to tell me to delete my answer since it is not an actual answer to your question). EH is in fact an abstract boundary but is a boundary of no communication: one cannot know what happens beyond that boundary, therefore to my knowledge there is no answer to your question nowadays. Sorry for not being much helpful. $\endgroup$ – gianmarcozanardi Apr 12 at 12:05
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    $\begingroup$ @AlfredF. The question you are asking in the comment above is different from your original question. This answer correctly answer the original question - there is no correlation between the time inside and outside, they are not linked, but independent of each other. In your comment you are essentially asking if the in-falling observer can still see the outside world and what happens to it. Yes he can, but only the light that was emitted shortly after his fall, because light is slowed down the same way by the same time dilation: physics.stackexchange.com/questions/436274#436324 $\endgroup$ – safesphere Apr 13 at 4:08

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