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### quick ?: you are in a metal sphere in the center of earth.

Posted: Fri Jul 02, 2010 5:31 pm UTC
So, a friend proposed this question to me, and we have argued much since. You are in a magic supermaterial sphere filled with air at normal pressure. The sphere is at the center of the earth. For the purpose of the thought experiment, the sphere won't crush, melt, etc.

Will you be pulled apart or squished by gravity, or just fine?

*sorry if this has been asked, a simple search found nothing.

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 5:37 pm UTC
You'll be just fine (provided that the temperature stays nice), also, you'll float.

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 5:38 pm UTC
Are you familiar with Gauss's law?

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 5:40 pm UTC
It will be just like the Earth wasn't there, because it's gravity would be pulling all parts of you equally in all directions, canceling itself out. You will be in orbit around the sun, just like the Earth, and therefore weightless.

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 6:10 pm UTC
My friend thinks this has not answered his argument. He thinks that your extremities(and limbs), since they are off-center will be pulled unequally, and off your body or something. Is this correct?

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 6:17 pm UTC
rath358 wrote:My friend thinks this has not answered his argument. He thinks that your extremities(and limbs), since they are off-center will be pulled unequally, and off your body or something. Is this correct?

If this were true, we'd experience that sort of gravity gradient on the surface.

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 6:26 pm UTC
Those are tidal forces, and we do feel them (at least on the surface).. in the sense of to be affected physically, not in the sense of sensation, since they are very small. I'm not sure offhand how they work inside a shell with the usual reasonable assumptions about density distribution.

If he's trying to use black holes as an analogy, he can't. Black holes have all their mass concentrated in a very tiny region at the center. Earth does not.

Tidal forces are due to the divergence of the field lines (a flat, infinitely extended mass would produce none)... inside a hollow sphere, umm...

But if there's no field, there shouldn't be any field lines to diverge, I would think.

But in general, the divergence of the lines of a spherically symmetric mass increases as you get closer to the center, so that would tend to increase the strength of the tidal forces. But the mass is less, so it isn't that simple. The Wikipedia article should help.

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 6:35 pm UTC
From another forum:
KJW wrote:Assuming that the hollow sphere is the only thing in the universe, the spacetime inside the hollow sphere is **flat** (even in GR). Thus, you would be weightless. It differs from being in orbit around the sphere because, even though one is weightless, the spacetime is **not** flat when in orbit around the sphere. Thus, a person in orbit still experiences the tidal forces. There are **no** tidal forces inside the hollow sphere.

If there is a gravitationally significant mass orbiting the hollow sphere, an object at the centre-of-gravity of the hollow sphere will be weightless. However, there will be tidal forces inside the hollow sphere associated with the orbiting mass.

The flatness of the spacetime inside the hollow sphere (the only thing in the universe) depends on the spherical symmetry of the distribution of energy-momentum. It is independent of the radial distribution. However, if the hollow sphere is **rotating** (relative to an asymptotically Minkowskian metric), then the distribution of energy-momentum is no longer spherically symmetric, and there will be gravitomagnetic effects (frame dragging) inside the hollow sphere.

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 6:39 pm UTC
rath358 wrote:My friend thinks this has not answered his argument. He thinks that your extremities(and limbs), since they are off-center will be pulled unequally, and off your body or something. Is this correct?

Technically, given a long enough period of time and total isolation from outside forces, yes. Only a spherical cow would feel truly equal forces of gravity in all directions. Someone pendatic enough could even point out that the gravitational field of the earth isn't perfectly uniform - mascons would introduce instabilities and introduce unpredictable behavior for a perfect sphere.

Realistically, differences in the earth's gravitational forces should be negligible. There isn't that much of a gradient. Any gravitational forces would be utterly lost in the noise caused by biological processes (you do want to breath, right?).

Basically, you can both be right. You just need to be clear on the terms of the hypothetical. What's the timeline? Outside forces (moon, tidal forces, solar quakes, 'wobble', mascons, etc)? Is this a human who needs to breath / pass gas / move?

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 8:06 pm UTC
You wouldn't even have tidal effects in this scenario. The gravity from a spherically symmetrical shell, at all points (not just the center) within that shell is zero (And a hollow space at the center of the earth would be a (roughly) spherically symmetrical shell, just a really thick one). It's easy to see that at the exact center of it, that it would be zero, but it's also zero at every point inside of the hollow space as well.

You can think of it like this: If you move toward one side of the shell, you're closer to the mass on that side, so it pulls at you with a greater force, but at the same time, if you're on that side of the shell, that means there's more mass toward the far side of the shell then there is to the near side, so that additional mass means additional force. If you do the math, you find that these two factors will exactly counter each other at every point within a spherically symmetrical shell.

Also, a nice result of this, say that you're digging down toward the core of the Earth. You can determine what the gravity would be at a certain point by treating the Earth as two objects, a spherically symmetrical shell, of all of the Earth that's above you, from which the gravity sums to zero, and a spherically symmetrical sphere below you, which the gravity sums up to the same as a point with the same mass at the sphere's center.

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 10:23 pm UTC
Doesn't the argument simply boil down to the fact that Earth's gravity isn't powerful enough to pull someone's extremities off on its own?

I know it's already been stated, but isn't the debate over by now?

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 10:38 pm UTC
Oh, the debate was over a long time ago.

What you say is true (the Earth's gravity isn't that strong), but more importantly, in the scenario mentioned, the person inside the shell *wouldn't feel any gravity at all* (from Earth), for the reasons Soralin lays out.

### Re: quick ?: you are in a metal sphere in the center of eart

Posted: Fri Jul 02, 2010 11:50 pm UTC
Well, they'd feel a teeny weeny compression towards the equator because of earth being a spheroid, and would probably be rotating at the same rate as the earth, so they'd also feel an equally teeny centrifugal force.

But yeah, gauss' law. Spheres don't exert any gravity on people inside them, even if they're off-center.