Resolving the Black Hole Paradox: A Perspective from Different Reference Frames

[Ray Eston Smith Jr]

An observer is firing his rockets in order to hover over a black hole event horizon. With his super-infrared vision, he sees all the mass of the black hole "frozen" in slow time just above the event horizon. Then he briefly cuts off his rockets so that he is in free-fall. In the free-falling frame, he sees no event horizon. By the conventional internet wisdom, he very quickly sees all the mass of the black hole black-out as it actually crosses the Schwarzschild radius. Then before he himself crosses the radius, he turns on his rockets, so he is once again somewhere above the event horizon. Now what does he see? A black hole with all the mass inside the event horizon? That's not supposed to happen in finite time.

Resolution of the paradox:
When the observer cut off his rockets, he did not see the black hole mass crossing the Schwarzschild radius. Instead he saw all the mass already well past the radius and already clustering around the singularity, where it's "frozen" by the higher gravity there . When he turned his rockets back on, he saw the same mass "frozen" near the event horizon. The event horizon IS the singularity -- as viewed from a different reference frame.

 

[PeterDonis]

Then he briefly cuts off his rockets so that he is in free-fall. In the free-falling frame, he sees no event horizon.

This is not correct. The event horizon is an invariant property of the spacetime; it is not observer-dependent and it is not there only for certain observers.

By the conventional internet wisdom, he very quickly sees all the mass of the black hole black-out as it actually crosses the Schwarzschild radius.

I don't know where you're getting this "conventional internet wisdom", but you should not expect to learn actual physics from it. You need to look at actual textbooks and peer-reviewed papers. They will tell you that an observer above the horizon who is free-falling towards it sees much the same things that an observer hovering above the horizon sees; the only difference is the exact redshift/blueshift of the light coming from objects closer to the horizon that are falling in.

Then before he himself crosses the radius, he turns on his rockets, so he is once again somewhere above the event horizon. Now what does he see?

Still much the same thing. See above.

A black hole with all the mass inside the event horizon? That's not supposed to happen in finite time.

If you are claiming that a black hole cannot form in a finite time, that is not correct. We have a series of Insights articles on Schwarzschild spacetime that discuss this and other common misconceptions:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-1/

Resolution of the paradox:

There is no paradox to resolve. Your claims are based on a misunderstanding of how black holes actually work.

When the observer cut off his rockets, he did not see the black hole mass crossing the Schwarzschild radius. Instead he saw all the mass already well past the radius and already clustering around the singularity, where it's "frozen" by the higher gravity there . When he turned his rockets back on, he saw the same mass "frozen" near the event horizon. The event horizon IS the singularity -- as viewed from a different reference frame.

None of this is correct. In particular, the event horizon and the singularity are perfectly distinct features of the spacetime. The Insights series linked to above discusses all this.