In a solid sphere there is always force, and it always points to, and is lowest, at the center.
Or more accurately oscillating back and forth in the middle, but not touching the wall.
If there was any vapor in the middle (which seems likely to me, considering it would be a perfect vacuum at pretty extreme temperatures), then the gas would eventually slow down the oscillations and it would end up exactly in the middle.
BTW, Sci fans of yesteryear were disappointed when it was discovered that Dyson Spheres and Ring Worlds gravitationally unstable...
It won't. The first time it touches (or comes near) the wall it will take what it can get, but it will never go farther than that. There is nothing that will "push" it closer to the walls.
So this means that inside the cave the black hole will just move in one direction, until it hits the wall. Then it will suck up some mass and be pulled back to the center. Then it will suck up some mass on the other side, etc.
The only way the black hole will remain in the cave is if you put it in the cave with exactly zero velocity to begin with. As soon as one atom falls into it it will start to eat up the earth.
You are missing a point.
Yes, inside the cave there is no force. But the black hole can hit the wall just once. After that it can not go any farther.
If passes thought/inside the wall it will be pulled back to the center, and the momentum exchange with the earth will slow it down (so next time it won't go as far). If it was just touching the wall, then that is as far as it can possibly go.
So once it's inside it's little cave, it's done. It can't eat anymore. (Unless of course there is no cave and the earth feeds it, but we are assuming a cave.)
You are missing three points: one is that if there is no cave, and it's oscillating through the earth, it will be pulled to the center. Two is that it looses momentum each time through the earth. And three is that once it's inside it's hollow cave it can no longer reach the wall.
> You are missing three points: one is that if there is no cave, and it's oscillating through the earth, it will be pulled to the center.
Agreed.
> Two is that it looses momentum each time through the earth.
I'm not so sure about that. Mass is added to the black hole, and perhaps this causes the total momentum to increase.
> And three is that once it's inside it's hollow cave it can no longer reach the wall.
If it is stopped inside the cave. But there is no force to stop it.
Imagine a tiny black hole arriving on earth. It grabs some mass. The black hole is moving, but the mass it ate is not. So rather than free falling, it looses momentum. As it falls through the earth it keeps doing that.
So it will not rebound back up to the top at the same height it fell. It will be lower. The second time through the same thing will happen.
Each time it oscillates it will get lower and lower.
Next. Say the black hole is a bit bigger. And it eats some mass on one side of the earth - now there is an imbalance. There is more mass on the other side. So the forces will cause it to slow down more than it otherwise would.
When it finally settles down enough to make a cave, it will "swing" as much as it can. But each swing can not be any higher than the one before. Over time, with random atoms falling into the cave it, it will come to rest at the exact center.
There is no force on the black hole, so if it moved at all it would crash into the wall. Only when it actually ate some of the wall would it turn around. So each swing would eat more and more.
[EDIT] Hmm I think I see the point. The resulting force would be proportional to the ratio (black hole mass / earth's mass). Very small indeed.
For a solid sphere, there is always force toward the center. (And the closer to the center the lower the force, reaching 0 in the middle.)