We have been told several times that for an observer any object will seem to move slower and slower the closer it comes the event horizon. In the end, it will seem as if it is standing completely still.
But I assume we are just talking about the information available for us. After all, it would be impossible for a black hole to grow in size if objects never crossed the event horizon.
So even if it looks like an object has stopped moving for an outside observer, how much time will actually have passed outside the black hole’s gravity field when an object has finally crossed the event horizon?
“How long does it take for the astronaut to cross the event horizon outside of the gravity” : well if you’re talking about the actual event, and not its image, it happens immediately.
The thing is : time is relative so the question doesn’t really mean anything.
Let say you’re the far away observer and you have an impossible magical telephone with face time that doesn’t obey the law of physics and you can talk live to the astronaut. You’re going to see them go into the black hole immediately when it happens. If you look through your normal telescope they look red and suspended forever. In your magical FaceTime they’ll be gone.
If your question is : for how long the red phantom image will stay visible : a very very long time. Yes some math can be done to calculate that very long time. Yes the math will take into account the mass of the black hole. But it’s going to be a very very long time anyway.
The concept of a magical telephone, or a videophone, works good. In science fiction it’s called an ansible I think. But let’s adjust it to the different time experiences. If a group of astronauts are travelling so close to the speed of light that they are experiencing time just half as fast as someone on earth, having a conversation with them will feel like they are talking and moving with just 50% of your own speed. The astronauts on the other hand, will see you talk and move twice as fast as them.
If we use the same videophone when we’re talking with the astronaut about to cross the event horizon, how much slower will he talk and move before he crosses over?
I hope that makes the question clearer.
If the magic telephone works this way (it’s magic but still obey to physics when it comes to time dilation), then :
When they approach the event horizon the astronauts will talk slower and slower until they just froze in place.
So it won’t be “much slower” it’ll be “frozen and in place (almost) forever”
From the astronauts point of view tho, they’ll see us get old and die in a few milliseconds.
But at this point we don’t need a magic telephone: this is what would happen with a normal telescope as well.
(To give you another image let say in a galaxy there’s one super nova each century. When the astronaut is going to get close to the event horizon when he’ll watch the closest galaxy he’ll see it sparkle like a Christmas tree with dozens of super nova each seconds.)