Shell theorem

Remarks

Proof idea: $U^\mu(x)$ is $M$ times the average of the function $y \mapsto 1/\abs{x - y}$ over the sphere $S_R$. If $\abs{x} > R$, this function is harmonic on a neighbourhood of the closed ball of radius $R$, and the mean value property gives the value $1/\abs{x}$ at the centre. If $\abs{x} < R$, fix $y_0 \in S_R$. By rotation symmetry, the average of $1/\abs{x - y}$ over $y \in S_R$ equals the average of $1/\abs{x' - y_0}$ over $x'$ in the sphere of radius $\abs{x}$. That function of $x'$ is harmonic on the ball of radius $R$, so the average is its value $1/\abs{y_0} = 1/R$ at the centre. The value $\abs{x} = R$ follows by continuity of both formulas, or by Gauss's law applied to spheres about the origin. Since a ball with radial density is a union of concentric uniform shells, the potential of such a ball of mass $M$ is $M / \abs{x}$ outside the ball as well.

Going deeper

For a student

The theorem is the first example of how potential theory turns symmetry into exact formulas. It says that the potential of the shell is constant inside, so it is harmonic there, and that outside it is a multiple of the Newtonian kernel. In $\R^n$ with $n \ge 3$ the same argument gives $M \abs{x}^{2-n}$ outside and $M R^{2-n}$ inside, and in the plane the logarithmic potential of a uniform circle behaves in the same way.