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Comment by hydrogen7800 | Hacker News Reader

This reminds me of a paradox in my understanding of gravity. I often read about the pressure inside the earth resulting from the mass above, but the gravitational force inside a spherical shell is zero [0], and the only force acting at a given radius from the center is the mass of the sphere inside that radius.

[0]http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/sphshel...

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Think about sinking to the bottom of the Marianas Trench and then inflating a balloon just enough to make you neutrally buoyant. The net force on you will be zero, but you'd still be crushed by the pressure, which is the effect of many kilometers of seawater above you.

Another way to think of this is to imagine a small sphere (like the size of a basketball) around the very center of the Earth. The matter in that sphere feels no net gravitational force, but it has to push back against the entire mass of the rest of the Earth, which is trying to fall into it.

If all the shells above you were self supporting you'd be right, you shouldn't feel any more pressure as you go down, if anything you should feel less force because gravity decreases.

However a shell of liquid is not self supporting, and at the size of a planet pretty much all material behaves like a liquid more or less. This means something needs to be pushing back for the layers of material to stay where they are. That is the pressure you feel, and you can easily show it will increase as you go down (with each layer supporting all the layers above it).

If anything the shell theorem makes this worse because it means the top layer can't pull back on any of the layers below it, it can only be pushed back through sheer pressure.

Same effect is happening inside a black hole, thus each black hole must have a tiny hole inside, with it own event horizon, a bit similar to Arago Spot. :-/

I believe the apparent paradox is that the 'net force' is not the same thing as pressure; it's basically the vector sum of pressure. Pressure in the Earth is an isotropic distribution of stresses, so that in a north-east-up coordinate system, the upward-pushing stress is matched by the downward-pushing stress, the eastward-pushing stress is matched by the westward-pushing stress, etc. So while these sum to zero in some sense, their magnitude is the density of the rock between the point and the surface times the depth times gravity. This is like going deeper in the ocean-the forces are equal in all directions, meaning that they sum to zero in some sense and there is no directional flattening or translation, but they still increase with depth.

> I believe the apparent paradox is that the 'net force' is not the same thing as pressure

I don't think that's what OP means. He's saying that the gravitational force that acts on a given test particle that's located inside Earth at a radius r=R from the center is determined only by the amount of mass "below it", i.e. the mass inside the ball of radius R, not the matter making up the spherical shell R < r < R_E, where R_E is Earth's radius. Put differently, that (hollow) spherical shell does not cause any gravitational force on test particles inside it.

This is absolutely correct and a consequence of Gauß's law, see https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity

Meanwhile, you are talking about a different thing entirely: The pressure. The matter above (as well as below and next to) our test particle will obviously also experience gravity and get pulled down. So our test particle will experience an isotropic force from all sides, which is generally quantified as pressure (force per area). This pressure is obviously not zero (but no one ever claimed that) and does depend on the radius of the outer spherical shell above it, in the same way as the pressure under water depends on how deep you are.

The net force is 0 in the same way that if two people are pushing you from opposite sides, their net force on you is also zero. You're still being crushed from both sides.

This is a confusion between the gravitational force that the core feels from the layers above, and the pressure that the layers above push down onto the core.

That is, the core doesn't feel gravitational pull towards the outside, because it's balanced on that other side, but the outside is being pulling in towards the core, which pushes on the core.

It might be easier to simplify the mental image by picturing just a slice of the whole thing.

Also note that that is only true for (being inside) a hollow shell, not for a solid sphere.

Indeed. The acceleration due to gravity actually increases slightly with depth, reaching a peak at the surface of the core. This is because they Earth's density is not uniform.

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