Thread by @gregeganSF

[1/2] Newton famously proved that, throughout the interior of a perfectly spherical shell of uniform density, you would feel no gravitational force at all.

But what would you experience inside a hollow *prolate ellipsoid* (one with two short equal axes and one longer axis)?

[2/2] There is no net force at the exact centre — but the equilibrium there is unstable, so given a small disturbance you will fall “upwards” towards the smaller, circular “equator” of the shell.

The images show the potential for two slices through the interior.

It’s well known that a circular ring of matter has a potential hill at the centre.

So maybe a more surprising result is that for an *oblate* ellipsoid — where the 3rd axis is shorter than the other 2 — you end up with a valley at the centre, and you fall inwards from any point.

But then, a flat *disk* of matter has a potential *valley* at its centre.

So:

• An oblate ellipsoid is sufficiently “disk-like” to have an attractive central point.

• A prolate ellipsoid is sufficiently “ring-like” to have a repelling central point.

CORRECTION!

For the oblate ellipsoid, although the central point is at the bottom of a potential valley *within the plane of symmetry*, that plane itself lies at the top of a hill.

So an object in the interior will actually “fall” towards the *pole* of an oblate ellipsoid.

This should have been obvious to me, but I missed it:

You can’t have a point in the vacuum where the 2nd derivative of the potential in *every* direction is positive (i.e. a true valley). The potential U obeys:

∇^2 U = 4πGρ

So in a vacuum, the second derivatives sum to 0.