Aspherical gravitational monopoles

TL;DR

This paper presents a method to construct extended, non-spherical bodies of uniform density that behave exactly like point masses gravitationally, having applications in modeling and theoretical physics.

Contribution

It introduces a novel construction technique for aspherical gravitational monopoles that replicate the gravitational effects of point masses, extending to any number of dimensions.

Findings

Bodies generate spherically symmetric potential outside

Interaction energy with external potential is equivalent to a point mass

All higher multipole moments vanish, mimicking a point mass

Abstract

We show how to construct non-spherically-symmetric extended bodies of uniform density behaving exactly as pointlike masses. These ``gravitational monopoles'' have the following equivalent properties: (i) they generate, outside them, a spherically-symmetric gravitational potential $M/|x - x_O|$; (ii) their interaction energy with an external gravitational potential $U(x)$ is $- M U(x_O)$; and (iii) all their multipole moments (of order $l \geq 1$) with respect to their center of mass $O$ vanish identically. The method applies for any number of space dimensions. The free parameters entering the construction are: (1) an arbitrary surface $\Sigma$ bounding a connected open subset $\Omega$ of $R^3$; (2) the arbitrary choice of the center of mass $O$ within $\Omega$; and (3) the total volume of the body. An extension of the method allows one to construct homogeneous bodies which are gravitationally equivalent (in the sense of having exactly the same multipole moments) to any given body.