Newton’s Shell Theorem in an Infinite Solid Universe.

Despite the ominous title, you will find no math in this entry. You’re welcome.

Some of you will be familiar with Newton’s shell theorem. Basically, it says this; if you are inside a sphere, or a thin shell, of homogeneous mass, then the gravitational forces of the mass around you will cancel out, and you will float freely no matter where you are inside that sphere; you will not be attracted to the center of the cavity, and you will not be attracted to the inside edge of the sphere.  See Figure 1 below.

I’ve done the math for the basic shell theorem (Fig. 1). It works. This is a fairly common undergrad physics problem. 

newton1

Since you can add additional shells of mass ad infinitum, then you can fill an infinite volume, a universe, with these shells, and the object in the spherical cavity will still be floating freely, since all the forces will balance out. See Figure 2 below.

newton2

This works great if you have a spherical universe.

A while back, while working on an SF story, I speculated on how a universe that was completely solid might work, especially if you had an Earth-sized cavity in it. My conclusion was that you would be attracted to the side of the cavity, which at first seems to contradict Newton’s shell theorem. But let’s take a look a that.

newton3

So the big rectangular gray area in Figure 3 represents the infinite, solid universe of whatever density you like. Marshmallows, quartz, whatever. We’ve carved out two Earth-sized spheres of whatever this stuff is, and discovered that if you stand where the little guy is standing, then the mass is completely symmetric around him, pulling with equal force in all directions. There is no gravitational gradient there.

Now, we fill in one of those spheres.

newton4

In Figure 4, on your LEFT, there’s a volume the size of the Earth, but it’s full of the aforementioned universal solid. On your RIGHT, there is an empty hole the size of the Earth. Since, as we’ve seen in Figure 3, all the rest of the mass in the universe balances out, not pulling you any direction at all, then the only mass pulling on you is that sphere to the left. If the average density of the universal ‘stuff’ is the same as the Earth, then you would feel a force of 1 gravity no matter where you stood on the inside of that hollow sphere. Feel free to imagine Figure 4 without the extra circle on the left. It’s not really necessary except to help with the explanation.

This is an issue when trying to solve problems involving an infinite universe. In this example, the universe would have to either be infinite and eternal, or closed in such a way that the distribution of mass was the same everywhere, such that if you went in a straight line forever, you would end up back where you started (though digging that hole would be very time consuming).

This is an interesting SF universe to play in, however. Pockets of air scattered around would create their own gravity, as described above, little bubbles in an unknowable solid universe. What if gravity in a large pocket became so great that it overcame the structural integrity of the matter that made up the universe? Would it propagate outward, consuming the universe with a growing ball of vacuum?

If you like this speculation, be sure to check out my short stories listed at my website.

 

 

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