Gravitational collapse to form a black hole

[Finbar]

Ok so if we have some matter(a star or whatever) and under classical laws of gravity(GR) this matter is going to form a black hole. If we're in the frame A of the collapsing matter then there will come a time(measured in that frame) that the matter crosses the event horizon after which no events that occur can be seen by an observer outside this horizon. But now if we are in the frame B of an observer outside the horizon how long will it take for the matter to cross the horizon? Surely we never see the matter cross the horizon because if we did we would be seeing an event which happened behind the horizon.

I concede that as the matter approaches the horizon it gets more and more red shifted but if we continue t use measuring equipment that can detected ever increasing wavelengths in theory we can still detect the matter.

In this sense we never see the full formation of a black hole before we in fact fall in black hole ourselves.

Now if we take into account hawking radiation on the horizon this surely implies that the horizon will shrink as observed from B. Eventually the black hole will get very small and thus even to the outside observer the matter(which is always present at the event horizon) will approach the origin.

If we go back to frame A all this evaporation has occurred while the matter is just about to cross the horizon but yet the horizon is shrinking so the matter is in a sense falling to the origin. At some point of coarse we need a theory of quantum gravity to know what happens next.

I'm just wondering what is wrong with my reasoning here?

I'm I correct in thinking that for an outside observer will never be a time at which the matter actually crosses the horizon?

I'm I also correct that according to Hawking's theory there will be a time at which the horizon approaches the Planck length?

 

[Finbar]

Any GR/Black hole experts out there?

 

[xantox]

If we go back to frame A all this evaporation has occurred while the matter is just about to cross the horizon but yet the horizon is shrinking so the matter is in a sense falling to the origin.

We should say that photons emitted by the infalling body infinitesimally before it crossed the horizon are received by B with a diverging delay. The frozen image of A is thus not falling toward the center, as it has already been emitted before the start of the Hawking evaporation and is just undergoing a slower travel onto curved spacetime as measured by B's clock.

I'm I correct in thinking that for an outside observer will never be a time at which the matter actually crosses the horizon?

Correct, if we neglect evaporation. If we consider evaporation, it will cross the horizon when the black hole has fully evaporated.

I'm I also correct that according to Hawking's theory there will be a time at which the horizon approaches the Planck length?

Yes, in principle the evaporation should lead to the black hole completely disappearing (it is also sometimes considered that a Planck scale remnant could form). The details remain speculative without a full theory of quantum gravity.

 

[Finbar]

Correct, if we neglect evaporation. If we consider evaporation, it will cross the horizon when the black hole has fully evaporated.

This seems like a bit of a contdictory statement. If the black hole has completely evpourated then there is no horizon for the matter to cross. Never the less it makes sense in the sense that the matter can no longer be in the outside region of spacetime. Clearly the final evpouration process can't be described classicaly.

But you essentially agree with me?

 

[xantox]

This seems like a bit of a contdictory statement. If the black hole has completely evpourated then there is no horizon for the matter to cross. Never the less it makes sense in the sense that the matter can no longer be in the outside region of spacetime. Clearly the final evpouration process can't be described classicaly.

There is no contradiction, as "seeing a body crossing the horizon" means that the last photon which was emitted by the body before crossing the horizon is now reaching the observer. This cannot happen after the last Hawking massless particle reaches the observer.

 

[Finbar]

There is no contradiction, as "seeing a body crossing the horizon" means that the last photon which was emitted by the body before crossing the horizon is now reaching the observer. This cannot happen after the last Hawking massless particle reaches the observer.

Yeah this seems like a good defintion. So your reasoning is that at what ever point the last quanta of energy can be emmited from the black hole this is the point that the black hole has formed in the sense that no more information can be extracted from the collapsing matter.

But when would you say this would be? My reasoning is that this point could be once the black hole has got to order a Planck mass via hawking radiation. Is there reason to believe that this would happen much sooner?

If we assume that I'm right then Quantum gravity should dictate what happens to the collapsing matter. What I mean is that we cannot assume the matter has crossed the horizon until we expect the classsical/semi-classical laws to break down. Also one should expect any statistical physics laws to break down once we approach the Planck scale so ultimatly the black body lcharacter of hawking radiation should also break down.

 

[xantox]

But when would you say this would be? My reasoning is that this point could be once the black hole has got to order a Planck mass via hawking radiation. Is there reason to believe that this would happen much sooner?

The last photon emitted by the infalling body will be emitted at some distance from the horizon. Depending on very small variations on this distance, it can be received by the distant body exponentially sooner.