Sources of symmetric potentials

TL;DR

This paper investigates the geometric shapes of objects with identical Newtonian potentials to optimize gravitational experiments, revealing that such shapes must include a hole, which impacts experimental design and interpretation.

Contribution

It proves that for objects with the same Newtonian potential, the shape must include a hole, advancing understanding of symmetric potentials in gravitational experiments.

Findings

Objects with identical Newtonian potentials can differ significantly in shape.

The second shape must contain a hole, as established by Theorem 1.

Numerical simulations supported the theoretical result.

Abstract

Laboratory experiments on gravitation are usually performed with objects of constant density, so that the analysis of the forces concerns only the geometry of their shape. In an ideal experiment, the shapes of the constituent parts will be optimised to meet certain mathematical criteria, which ensure that the experiment has maximal sensitivity. Using this idea, the author suggested an experiment to determine the departure of the gravitational force from Newton's force law [1]. The geometrical problem which has to be solved is to find two shapes which differ significantly, but have the same Newtonian potential. Essentially, the experiment determines whether the two objects are distinguishable by their gravitational force. Here, we consider the case when one of them is a round ball. The result, Theorem 1, establishes a fact which appeared in numerical simulations, that the second object has to have a hole in it.