There are many questions on this forum about objects falling into black holes. Most of them describe that to distant observers, the objects never quite reach the event horizons and appear to be frozen just outside. For example: How can anything ever fall into a black hole as seen from an outside observer?

Answers often refer to the Schwarzchild metric, even though that describes a static situation which isn't strictly applicable to in-falling objects. We know from studies of black-hole mergers (Revisiting Event Horizon Finders ) that in more realistic scenarios, event horizons can start growing before objects actually cross them.

This seems to suggest (perhaps naively) that outside observers could possibly see objects disappear behind event horizons in a finite time, since the event horizon may be larger than previously assumed.

That view might be countered by noting that the gravitational time dilation would also be changing along with the event horizon, with the result that the disappearance would still take an infinite time.

I don't expect that such a complicated situation could have a definitive answer without resorting to numerical relativity calculations, but would like to know what conclusions have been reached about the effects of event horizon growth.

It has been suggested in comments, that this question is a possible duplicate of Does an expanding event horizon “swallow” nearby objects? However, that question involves the merger of two black holes, and therefore has two event horizons, while this question is for the somewhat simpler case of aa object falling into a single black hole. I'm not sure if the answers would be different, but I believe this is still a separate question.

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    $\begingroup$ Possible duplicate of Does an expanding event horizon “swallow” nearby objects? $\endgroup$ – John Rennie Apr 16 '19 at 19:06
  • $\begingroup$ Actually you might have a point: I think the definition of horizon (null closed hypersurface) does not imply that the time-time component of the metric would be zero in a non static case. I will think about it. First I will check the possible duplicate of the question posted above. $\endgroup$ – AoZora Apr 16 '19 at 19:47
  • $\begingroup$ "event horizons can start growing before objects actually cross them" - Correct. "This seems to suggest (perhaps naively) that outside observers could possibly see objects disappear behind event horizons in a finite time" - Incorrect. Everything "frozen" at the horizon moves out with it when the horizon expands. This is self evident, if you consider a collision of a static rock with a fast flying black hole. The rock will be pushed by the moving horizon and will move with the horizon. $\endgroup$ – safesphere Apr 17 '19 at 3:58
  • $\begingroup$ @John Rennie: I had forgotten about that question & might not have posted this one if I had remembered it. Reading it again however, I think I would call it a related question, not an exact duplicate. It's actually a comical situation. According to a comment from the OP, his question involved merging black holes, but you wrote your answer assuming it was about an object falling into a black hole (like this question). Then, even though he didn't seem to agree with your answer, he accepted it anyway. $\endgroup$ – D. Halsey Apr 17 '19 at 14:54
  • $\begingroup$ @D.Halsey yes it was a bit of a mix up :-) $\endgroup$ – John Rennie Apr 17 '19 at 14:56

In relativity one needs to make a distinction between “an outside event happening at a time $t$ by the clock of an observer $\beta$” and “an observer $\beta$ sees an event happening at time $t$ by her clock”. The first situation means that event happens within the slice of a constant time $t$ in the observer's frame of reference, while the second implies that there is a null geodesics connecting the event with the observer at time $t$.

With that in mind,

  • By taking the backreaction into consideration (in this situation it means that the horizon would be growing) an object would cross the event horizon in finite time by the clock of an outside observer (with any reasonable scheme of defining slices of constant outside time in the vicinity of black hole, such as Fermi–Walker frames)

  • Signals propagating with the speed of light from the event of “object crosses the event horizon” would by definition never reach an outside observer. Signals from events immediately preceding would reach the outside observer arbitrarily late and extremely redshifted, so in practice an observer would only see an object outside the event horizon for some time until the signals from it get redshifted beyond any possibility of detection.

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  • $\begingroup$ I was under the impression that it was only in SR that observations of events were corrected for the travel-time of light between the event & the observer. Are you saying that is also appropriate in GR? $\endgroup$ – D. Halsey Apr 17 '19 at 14:21
  • $\begingroup$ @D.Halsey: Special relativity is a special case of general relativity (flat spacetime plus restricted choice of coordinates). So relativity of simultaneity is not only present in GR but much enhanced since causal structure could be much more complex. $\endgroup$ – A.V.S. Apr 17 '19 at 15:39
  • $\begingroup$ @ A.V.S. :OK, that makes sense. But in this case, where the time dilation may be infinite, can you still make sense of any notions of simultaneity? $\endgroup$ – D. Halsey Apr 17 '19 at 15:44
  • $\begingroup$ @D.Halsey: That's where the growth of the event horizon comes in. If BH is growing, then EH crossing does not happen at the surface of infinite time dilation but strictly outside it. So simultaneity there could still have some ambiguity but there are no infinities. In more technical terms, apparent horizon is the surface of infinite dilation of “outside time”, while for a black hole that is gaining mass, event horizon would be strictly outside it. $\endgroup$ – A.V.S. Apr 17 '19 at 16:00
  • $\begingroup$ @ A.V.S. : +1 That's the kind of thing I was hoping to learn. $\endgroup$ – D. Halsey Apr 17 '19 at 16:10

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