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We know that an observer at infinity cannot see a star forming into a black hole as the matter will take progressively longer and longer time to compress (from this observer's point of view).

Is there an observer, outside of the Schwarzschild radius, who is able to observe this taking place in a finite amount of time?

PS: I am aware of this question Can black holes form in a finite amount of time? I am not asking the same thing, as the answer there was given in terms of proper time. I am interested purely in real observers, but not necessarily sitting at infinity, the observer can be arbitrarily close to the Schwarzschild radius.

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Any observer outside the Schwarzschild radius sees the same thing: matter approaching the Schwarzschild radius at slower and slower (asymptotically zero) speed, forming a thin shell around the event horizon. The matter takes an apparently infinite time to collapse, and infinity is infinitely larger than a large finite the same way it's infinitely larger than a small finite.

The apparent horizon of a black hole is affected by more than simply the mass of the black hole - notably, all observers see the horizon as somewhere "below" them (i.e. between them and the singularity) even if they've passed the Schwarzschild radius on their way in. What this means is that even for free-falling observers already inside the Schwarzschild radius, the black hole appears to be surrounded by a shell of asymptotically un-moving matter at the horizon, since the apparent horizon for a given observer has the same observed properties as the actual horizon.

Taking that into account, the only reason the Schwarzschild radius is important at all is mathematical/theoretical: it is the radius of the apparent horizon observed by a stationary (relative to the black hole) observer at infinite distance.

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  • $\begingroup$ So other than optically, can they perform some experiment that will show that matter has actually passed the event horizon and that there is a black hole even though they can't see it? $\endgroup$ – SMeznaric May 2 '16 at 6:44
  • $\begingroup$ @SMeznaric Not really. The best we can really do is measure the mass and radius of the body and make an educated guess, since a thin shell of matter slightly larger than its Schwarzschild radius is not as likely as a black hole that only looks like a massive thin shell. $\endgroup$ – Asher May 2 '16 at 13:20
  • $\begingroup$ If the theory predicts that nobody is able to experimentally verify that a black hole exists, then how do we know that they actually exist? $\endgroup$ – SMeznaric May 2 '16 at 14:08
  • $\begingroup$ @SMeznaric Because we observe behavior that matches what we expect from a black hole, such as the orbits of stars near the center of our galaxy appearing to orbit Sagittarius A*, or the gravitational waves that LIGO detected last year. That's the only way to "know" something "exists" in physics. $\endgroup$ – Asher May 2 '16 at 15:20
  • $\begingroup$ But doesn't that then imply that there are experiments we can perform that detect that a black hole has formed? $\endgroup$ – SMeznaric May 3 '16 at 18:47
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No, an observer never observes the event horizon even begin to form. However, they observe regions of space closer and closer to the event horizon keep forming. When an object falls into a blackhole, we see it getting closer and closer to the event horizon. That doesn't contradict the fact that we never observe the event horizon begin to form because we never observe it keep up to region of space just outside the event horizon we just observed form. Anything falling into a blackhole we observe keeping on getting more red shifted without bound until we can't see it anymore. For that reason, black holes appear black so it makes sense to say the star has already formed into a black hole.

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