Suppose you have a perfectly spherical planet all to yourself, and a lot of resources.

On this planet, you build a large number of support columns of equal height. Then, you use these as a base to build a ceiling over the entire planet. The ceiling is perfectly uniform in all places and has no flaws.

Then, using teleporters or vaporizers or something, you knock out all the columns at the exact same time.

What happens to the ceiling?

Shouldn't it remain completely unmoved? It would be unstable, but until another force acts on it, it should stay as is.

It depends upon the material. Assuming gravity is equal in all places around the planet, and the roof is perfectly flawless without crack or dent, the roof would support itself. There is no weak point on the roof.

If it were made out of a lighter material, or a material liable to crumble under pressure, it would collapse or crumble inward.

Assuming it is flawless, pressure is evenly distributed across the entire surface. So long as the material used to build said roof is built to withstand the pressure of gravity across the roof, it should theoretically stay in the same place. There force pushing it downward from the poles, for example, would be evenly distributed across the impossibly perfect roof, thus cancelling each other out.

I'm no physics expert, but this is what I see happening.

That would depend on if it was strong enough not to just crumble in on it'self due to gravity. If it is and the force from gravity is uniform all over (and there is no other mass in the universe to pull it towards other objects) it should stay perfectly still or remain in motion if the planet is moving/spinning.

Edit: It seems i was beaten to it.

I have wondered the same thing actually.

I like to think that it would stay in place

The material would need to be strong enough to not compress. Besides that, it wouldn't move.

The system would never be stable, like holding a piece of metal between two magnets and expecting it to float in the middle. Just one molecule of air (or possibly even a single photon) randomly bumping into one side of the shell would be enough to tip the balance (assuming there is no friction impeding movement). Besides, you'd never be able to position the ceiling *exactly* in the right position to start with.

But even if you ignore all these practical details and consider only pure theories of physics, Heisenberg's uncertainty principle of quantum mechanics implies that the position/speed of the shell must have some inherent uncertainty that would always tip the balance. Then chaos theory would do the rest and the shell would hit the earth on one side or the other. This is completely unavoidable, even according to pure theory.

I don't think this would work. Gravitational force decreases with distance. That means if it was pushed slightly one way, the points getting closer to the surface would feel a greater force (towards the surface) than those moving away from it, accelerating the motion. It would work if you had inverse gravity, then it would balance itself even if it was pushed out of position.
As Kelvin said, it would only work under ideal circumstances with quantum mechanics disabled. Even in a perfect vacuum there would be particles to start the sphere moving.

This is the key point but I think you have to do more that turn off quantum mechanics, you also have to run the system at absolute zero or pretend that the sphere and shell aren't made of particles with thermal energy. Assuming that, classically the system is unstable but would remain static.

It occurs to me that this might work in the "real world" if you gave the planet a magnetic field and made the shell out of a superconducting material (and cooled to the appropriate temperature). The Meissner effect should cause any wiggle from the original static state to induce a magnetic field in the shell to balance out the wiggle.

Yes, but that's cheating - you may as well keep the pillars in place!

Little known physics fun fact, but based on the Shell Theorem
Assuming the shell/ring was perfectly spherical, there would be 0 net gravitational force between the shell and the planet no matter if it was perfectly centered or offset.

OK, that IS interesting, and it does make sense, so my previous answer does not apply to this example. Thanks for posting this.

if we ignored physics for a bit and said that there was no outside influence at all and everything was in a perfect vacuum (plus you'd have to not be there) and all of the resources would have to be perfectly equal around the planet it would in theory work
however it is necessary to have a light source (sun) and you would have to orbit it unless you want to go straight into it so there would without a doubt be some influence that would push the ceiling off course and have it fall on the planet
plus with the ceiling you wouldn't get any sunlight yourself

Hey, that's pretty cool. I wasn't aware of this. This morning I considered something very similar to this. I realized that if the planet's center of gravity was directly on top of the shell's center of gravity then it shouldn't feel any gravitational effect. For a moment I thought it didn't matter if the planet moved, it still wouldn't feel anything but I rejected the idea on hand-wavy grounds. I need to wave less and Google more it seems!

If you ignored physics, there would be no means of answering this question.

Plus you completely destroyed the whole point of the question

Just judging from the title "A physics question"

Good concept

I might also like to add that nobody has yet suggested that the shell could be positioned so that it's equator lies in a geostationary orbit. Communications satellites sit in this orbit 24/7 without exerting any force upon it surroundings to stay put. Meaning if the shell is rotating such that centrifugal force counteracts gravity. However, this would still be a problem as since we are dealing with a shell, the centrifugal force would still be zero at the poles

Another problem that one would inevitably encounter is that no material known to man would be strong enough to withstand the intense tensile and/or compressive forces of gravity on a planetary level. If the shell material is acted upon by the gravitational forces of the planet, the compressive force acting on the material would be astronomically high. No material, not diamond, nor even carbon nanotubes would be strong enough to resist the forces. Even if the shell is rotating, there will be astronomical compression along the longitudal lines and astronomical tensile forces along the latitudal lines.The shearing force acting on the dome would rip it to shreds.

The only way for a practical shell to remain in suspension is for the planet to have an atmosphere with sufficient pressure to equalize the weight of the dome. You would measure the density of the dome material, and multiply the result by the thickness of the material to determine mass per surface area. Calculate the gravitational force, g, at the altitude of the dome and multiply by the result to get the force required to support the dome. Divide by surface area to determine the required atmospheric pressure under the dome.

Now here's the fun part: Because the atmospheric pressure increases at lower altitudes and decreases at higher altitudes, the dome will stay in equilibrium. Assuming that GuiltyBystander was right about the gravitational force being consistent across the dome, if a portion of the dome sags closer to the surface, the atmospheric pressure becomes greater than the weight of the dome and pushes it back up. If a portion of the dome drifts away from the surface, the atmospheric pressure supporting it becomes lesser that the weight of the dome, and it sinks back to the equilibrium point. Another interesting prospect of the domed world is the greenhouse and weather effects on the dome. Warm air expands, increasing the overall pressure and pushing the dome up. Cold air contracts and the pressure decreases, causing the dome to fall. Thus, the dome will shift slightly towards the sun, rising slightly during the day (warm side of planet) and falling slightly during the night (cold side of planet).

Would the sun heating part of the sphere mess up the balance? It would doubtlessly distort the sphere to some degree.

Funny that your question reminded me of another long forgotten physics problem I conjured up back in 2005:

What if the world was shaped like a donut? Could a donut-shaped planet actually exist? To avoid hijacking this thread, I have started a new one here:
viewtopic.php?f=7&p=243055