Question on gravity at the center of the earth

[Austin0]

If you imagine a magically strong , insulated sphere situated at the center of the earth
could GR calculate the conditions there? Would an observer be pulled apart,,,compacted,,
or would the surrounding mass cancel itself in opposition leaving you in free fall around the sun or dragged around by the moon?
Does anyone have any idea how either Newtonian gravity or GR would approach this?

 

[Fredrik]

In Newton's theory of gravity, the gravitational field inside a spherical distribution of mass is exactly zero. This can be proved using Gauss' law. I think a similar result holds in GR (but one wouldn't use the term "gravitational field").

Edit: I meant of course that the field produced by the spherical distribution is zero on the inside, not that the surrounding mass shields you from the fields produced by external sources like the sun.

 

[Austin0]

In Newton's theory of gravity, the gravitational field inside a spherical distribution of mass is exactly zero. This can be proved using Gauss' law. I think a similar result holds in GR (but one wouldn't use the term "gravitational field").

Edit: I meant of course that the field produced by the spherical distribution is zero on the inside, not that the surrounding mass shields you from the fields produced by external sources like the sun.

Thanks
SO this would seem to mean being in free fall orbit around the sun ,,,or would the moon then become the predominant factor because there would be no angular momentum keeping you in freefall in relation to it?

 

[mgb_phys]

Gravity isn't zero at the centre of the Earth - it's zero at the centre of the Earth-Moon barycentre. If you imagine a line form the centre of the Earth to the centre of the moon then zero gravity is at some point along this where the forces balance (the Earth pulling you one way and the moon the other).
Because the mass of the Earth is rather larger this is inside the Earth's volume.

So in practive you would have a torus/donut rather than a sphere runnign around the Earth about 1800km below the surface. The moons gravity would pull you around the torus doing one lap every lunar month.

The whole thing would also be in orbit around the sun but you wouldn't notice that because you would also be in orbit and so effectively in free fall.

 

[DaveC426913]

Gravity isn't zero at the centre of the Earth - it's zero at the centre of the Earth-Moon barycentre. If you imagine a line form the centre of the Earth to the centre of the moon then zero gravity is at some point along this where the forces balance (the Earth pulling you one way and the moon the other).

So in practive you would have a torus/donut rather than a sphere runnign around the Earth about 1800km below the surface. The moons gravity would pull you around the torus doing one lap every lunar month.

I am not convinced of this. You too would be pulled by the Moon. It seems to me that you would be stable at the centre of the Earth.

Consider: If your claim were true, that would mean that, without the air chamber, the rock that makes up the core of the Earth itself is currently undergoing great tidal stresses, being pulled first 400km in one direction, then 400 km in the other direction, every month.

 

[DaveC426913]

Gravity isn't zero at the centre of the Earth - it's zero at the centre of the Earth-Moon barycentre. If you imagine a line form the centre of the Earth to the centre of the moon then zero gravity is at some point along this where the forces balance (the Earth pulling you one way and the moon the other).

So in practive you would have a torus/donut rather than a sphere runnign around the Earth about 1800km below the surface. The moons gravity would pull you around the torus doing one lap every lunar month.

I am not convinced of this. You too would be pulled by the Moon just like any other rock. You would be stable the the centre of the Earth.

If your claim were true, that would mean that, without your air chamber, the rock that makes up the core of the Earth itself is currently undergoing great tidal stresses, being pulled first x00km in one direction, then x00 km in the other direction, every month. I don't mean the whole core as a unit would be pulled, I mean each discrete rock it in it would be pulled in a torus through the solid bulk of the Earth.

Try simplifying the air-chamber-with-human-in-it to rock-with-other-rocks-around-it (or pool-of-molten-rock-with-other-pools-of-molten-rocks-around-it) and see the paradoxical results you get.

In truth, you're right that there is a toroidal path, but both you in your chamber AND the core of the Earth around you are following it. This means that you are at rest with respect to the walls of your chamber. You are just another rock.