This would in fact turn into a black hole since a black hole’s radius is proportional to its mass. Also, it would collapse due to compressive forces since the force between any pair of hemispheres is proportional to r^2 but the surface to bear the force is only proportional to r.
Yeah, I had already begun to worry about that second point, that the compressive pressure might be greater than I had imagined, but I couldn’t see an easy way to calculate it. Your first point, that a black hole will form, is an easier way to see that this arrangement will fail. In fact I think the two points are different ways of looking at the same thing; for a black hole to form the shell must be pushed beyond the limit of its compressive strength.
What about a circular ring rather than a spherical shell? The mass is proportional to the radius, so it looks like it has a hope of working. But no, using the results of the paper here (behind a paywall, sorry) the pressure inside the ring rises with log(r), so it will eventually break.
:-(
Can we prove that all sufficiently large structures collapse?
Sketch of proof: you proved that a stick collapses (compression scaling as Log(L)).
Well every connected object is either a stick, a curvy stick, or one of those things plus some extra atoms. So—prove that making things curvy or adding atoms doesn’t help (enough). So, e.g. thickening the stick in the middle won’t save it since you’d need infinite thickness.
for a black hole to form the shell must be pushed beyond the limit of its compressive strength.
hmmm… we’ve been talking as if in a space without dark energy. But with dark energy, a sufficiently large shell could be balanced by the antigravity of the dark energy within it, the acceleration caused should scale linearly with the radius. So that would be able to be under no stress at all. But I’m not sure it wouldn’t form a black hole at large radius. As the interior gets bigger and bigger, eventually you get a cosmological event horizon forming so the interior forms a white hole—so light can’t leave the shell to the interior. Since it’s balanced, for symmetry reason’s I’d expect the same to apply on the exterior. So you have this shell black hole between an interior and exterior both ‘outside’ the black hole.
Of course, this shell would actually be crushed by the stress in the radial direction, it’s only not under stress circumferentially. But, now that we’ve got this example, we can extend to an ultralight aerogel (space-o-gel?) that balances the dark energy everywhere. I’d expect this to look externally the same as the shell example, so it should also eventually form a black hole. These are just guesses though—not actually calculated.
Edit: I’m now very suspicious of this analogy between the sphere and space-o-gel and will have to think about it more.
I agree with what you’ve said. I’ll have to think about it for a while.
I did look at the case of a spinning ring and it seems stable. But I’m only using the Newtonian approximation, and you have to spin it at a speed that increases like sqrt(log(r)) so eventually that’s invalid.
This would in fact turn into a black hole since a black hole’s radius is proportional to its mass. Also, it would collapse due to compressive forces since the force between any pair of hemispheres is proportional to r^2 but the surface to bear the force is only proportional to r.
Yeah, I had already begun to worry about that second point, that the compressive pressure might be greater than I had imagined, but I couldn’t see an easy way to calculate it. Your first point, that a black hole will form, is an easier way to see that this arrangement will fail. In fact I think the two points are different ways of looking at the same thing; for a black hole to form the shell must be pushed beyond the limit of its compressive strength.
What about a circular ring rather than a spherical shell? The mass is proportional to the radius, so it looks like it has a hope of working. But no, using the results of the paper here (behind a paywall, sorry) the pressure inside the ring rises with log(r), so it will eventually break.
:-(
Can we prove that all sufficiently large structures collapse?
Sketch of proof: you proved that a stick collapses (compression scaling as Log(L)).
Well every connected object is either a stick, a curvy stick, or one of those things plus some extra atoms. So—prove that making things curvy or adding atoms doesn’t help (enough). So, e.g. thickening the stick in the middle won’t save it since you’d need infinite thickness.
hmmm… we’ve been talking as if in a space without dark energy. But with dark energy, a sufficiently large shell could be balanced by the antigravity of the dark energy within it, the acceleration caused should scale linearly with the radius. So that would be able to be under no stress at all. But I’m not sure it wouldn’t form a black hole at large radius. As the interior gets bigger and bigger, eventually you get a cosmological event horizon forming so the interior forms a white hole—so light can’t leave the shell to the interior. Since it’s balanced, for symmetry reason’s I’d expect the same to apply on the exterior. So you have this shell black hole between an interior and exterior both ‘outside’ the black hole.
Of course, this shell would actually be crushed by the stress in the radial direction, it’s only not under stress circumferentially. But, now that we’ve got this example, we can extend to an ultralight aerogel (space-o-gel?) that balances the dark energy everywhere. I’d expect this to look externally the same as the shell example, so it should also eventually form a black hole. These are just guesses though—not actually calculated.
Edit: I’m now very suspicious of this analogy between the sphere and space-o-gel and will have to think about it more.
I agree with what you’ve said. I’ll have to think about it for a while.
I did look at the case of a spinning ring and it seems stable. But I’m only using the Newtonian approximation, and you have to spin it at a speed that increases like sqrt(log(r)) so eventually that’s invalid.
I changed my mind on the space-o-gel though, at least for now.
Nice idea with the spinning ring. With relativity it should be fine as long as light itself isn’t pulled in when going in parallel with the ring.