Note 108 — Chapter XXXIV · THE FINITENESS OF THE UNIVERSE

The reason why, under Newton’s law, a uniform distribution of matter to infinity is impossible can be understood as follows: The difference between the values of the Newtonian potential at two points and is equal to the work which must be expended against the gravitational force for a unit mass to be moved from to . But if matter were distributed uniformly to infinity, no work would be necessary, since wherever the body stood, it would be at the centre of the infinite universe, hence would be subjected to equal forces in every direction. But then would have the same constant value everywhere. This would entail the vanishing of . And so Poisson’s equation would become ; an impossibility, since , density of matter, must have a non-vanishing value. In short, it would be necessary to modify the law of gravitation and consider, for example, (where is some constant) in place of Poisson’s equation. With this latter law instead of Newton’s, an infinitely extended universe of matter would be possible.