Shell Theorem - Converses and Generalisations

Converses and Generalisations

It is natural to ask whether the converse of the shell theorem is true, namely whether the result of the theorem implies the law of universal gravitation, or if there is some more general force law for which the theorem holds. More specifically one may ask the question:

Suppose there is a force between masses M and m, separated by a distance r of the form such that any spherically symmetric body affects external bodies as if its mass were concentrated at its centre. Then what form can the function take?

In fact, this allows exactly one more class of force than the (Newtonian) inverse square. The most general force is:

where G and can be constants taking any value. The first term is the familiar law of universal gravitation; the second is an additional force, analogous to the cosmological constant term in general relativity.

If we further constrain the force by requiring that the second part of the theorem also hold, namely that there is no force inside a hollow ball, we exclude the possibility of the additional term, and the inverse square law is indeed the unique force law satisfying the theorem.

On the other hand, if we relax the conditions, and require only that the field everywhere outside a spherically symmetric body is the same as the field from some point mass at the centre (of any mass), we allow a new class of solutions given by the Yukawa potential, of which the inverse square law is a special case.