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

TL;DR

This paper extends the classical shell theorem to spaces with constant curvature across all dimensions and topologies, providing a unified framework for gravitational potentials in curved spaces.

Contribution

It introduces general potentials with the spherical property on constant curvature spaces, extending known flat space results to curved geometries and nontrivial topologies.

Findings

Consistent with flat space results

Reduces to Gurzadyan's cosmological theorem under specific conditions

Extends to nontrivial 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.