Generalizing the Shell Theorem to Constant Curvature Spaces in All Dimensions and Topologies

Abstract

A gravitational potential has the spherical property when the field outside any uniform spherical shell is indistinguishable from that of a point mass at the center. We present the general potentials that possess this property on constant curvature spaces, using the Euler–Poisson–Darboux identity for spherical means. Our results are consistent with known findings in flat three-dimensional space and reduce to Gurzadyan’s cosmological theorem when the rescaling factor is exactly 1. Our approach naturally extends to nontrivial spatial topologies.