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

Ava K. Tse, Olivia M. Markowich, Trung V. Phan

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.

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 . Our approach naturally extends to nontrivial spatial topologies.

Paper Structure

This paper contains 4 sections, 27 equations, 1 figure.

Table of Contents

  1. EPD identity in $\mathbb{R}^3$
  2. The rescaling factor $M(b)$ in $\mathbb{R}^n$
  3. Potentials possess spherical property in $\mathbb{S}^n$ and $\mathbb{H}^n$
  4. Potentials possess spherical property in $\mathbb{T}^n$

Figures (1)

  • Figure 1: Generalizing shell theorem to sphere forms in all dimensions and topologies. Here, we focus on translationally invariant pairwise interaction potentials. (A) The generalized shell theorem: outside a uniform spherical shell, the field is indistinguishable from that of a central point mass, scaled by a radius-dependent factor. (B) The field at a test point can be calculated by summing the contributions from all surface elements of the spherical shell. (C) A list of geometries we are interested in: the Euclidean flat space $\mathbb{R}^n$, the spherical space $\mathbb{S}^n$, the hyperbolic space $\mathbb{H}^n$, and the hypercubic toroidal space $\mathbb{T}^n$.