Would there be any gravity inside a hollow planet made of a spherical shell of basalt crust? I am writing a science-fiction story, which contains a 'crazy' planet. I would like some input into the consequences of this physics according to standard physics.
The planet in question is Earth-sized and it orbits a Sun-like star. Within the story, this planet has been taken over by a malignant 'force' and has been hollowed out, so that there is one mile of basalt crust forming a spherical shell, and the inside is a void filled with nitrogen. This force keeps the planet intact, but does not affect the planet in any other way (i.e. any standard-physics problems regarding the integrity of this spherical shell can be ignored, as the story explains them in other ways). The force has no mass.
Given this setup, what would the gravity be like at the center of the planet? In the story, the protagonist is able to fly around in a large bubble of zero-gravity at the middle. Would this be possible? (Assume that the protagonist can survive the atmosphere.) Could he, assuming he has a suitable raft, get around by blowing?
 A: To add to @GSmith comment, if you take into account the nitrogen inside, you would only feel the gravitational pull of the nitrogen between you and the center of the planet (as explained). It is easy to see that anyone who falls inside will end up right at the centre of the planet:

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*You begin to fall towards the centre of the planet due to the pull of the nitrogen inside.

*Since you can go through nitrogen (its not solid but gas!) you will go past the centre out towards the other end of the planet.

*This oscillatory motion would continue for ever would it not be for the air resistance of the nitrogen which keeps substracting speed/energy from you. This means that with every swing about the centre you reach a lower altitude each time (just like in a kid's swing) until eventually you end up in centre. Think of this motion like a real pendulum which ends up stopping at the middle but in a 3D kind of way. What a way to die!

A: If you neglect the gravity of the nitrogen gas, there is no net gravitational force anywhere inside. This is a result of the famous “Shell Theorem” for inverse-square forces. At any point, you are attracted by all the atoms in the shell, but the vector sum of all these forces — in different directions, and having different magnitudes because they are caused by atoms at various distances from you — turns out to be zero. This assumes a perfectly spherical shell of uniform mass per unit area.
If you want to take the nitrogen gas into account, at radius $r$ from the center you are attracted by the mass of all the nitrogen in a sphere of radius $r$ around the center, as if this mass were concentrated in a point at the center. The nitrogen at larger radii doesn’t attract you for the same reason that the shell doesn’t.
A: Ok let's go:

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*It is not physically possible to have a situation like you described: since your planet has the size of the earth but it's hollow it would surely collapse into itself, crushed by its own gravity, and it would form a ball with much smaller radius. This is why you don't see large hollow bodies in nature. There isn't a material strong enough to form a shell like you described and not crush under gravity, I am pretty sure about that. You would have to invent some sort of implausible sci-fi material to form the shell, really implausible material.

*You mention a force, but it's not clear how this force is produced or how it acts. You also state "the force has no mass".. this is ridiculus: a force is not an object, talking about the mass of a force is meaningless.

*But let's ignore this force you mention for now, and let's suppose also that in some way your shell planet does not collapse: then we can apply a famous theorem of calculus, usually used in the context of electrostatics but it will also work here: the shell theorem; using it we can prove that the gravitational force inside your shell planet must be zero everywhere not only in the center of it. (Of course this is true with some possible corrections, in the sense that if you have a massive mountain on the shell then the symmetry is partially broken and you will have some light gravitational pull inside your planet).

*In a context where the force acting on the protagonist is zero and there is little drag yes, I think it would be possible to move by blowing. But this would be a very inconvenient method of locomotion to say the least.

*Sci-fi is not an easy topic. You really should study some physics before attempting to write a book on it, otherwise you will likely make a joke of yourself, especially when it's time to publish it.

