Gravitation Inside A Uniform Hollow Sphere

The gravitational force inside a hollow sphere shell of uniform areal mass density is everywhere equal to zero, and may be proved by the following argument:

Let the sphere have a radius a. Place a point P inside the sphere at a distance r from the center where r < a; i.e., r is strictly less than a. Draw a line through P to intersect the sphere at two opposite points. Call these points and. Let the distance from P to be r₁, and the distance from P to be r₂.

Now place a differential area dA at , and project straight lines through P to acquire its image dA at . These two areas subtend a solid angle d at P. Let the sphere have areal mass density (kg/m²). Then the net differential attraction dF of dA and dA at P directed toward is just

dF = ( dA /r₁² - dA/r₂²).

But dA = r₁² d, and dA = r₂² d by definition of the solid angle. Thus,

dF = ((r₁² d)/r₁² - (r₂² d)/r₂²) = 0.

This result is true for all choices of dA and dA. The gravitational force within the sphere is everywhere equal to zero.