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

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.

💡 Research Summary

The paper extends the classical shell theorem—originally formulated for Newtonian gravity in three‑dimensional Euclidean space—to arbitrary dimensions, constant curvature manifolds, and non‑trivial topologies. The authors define the “spherical property” as the condition that the gravitational field outside any uniform spherical shell is indistinguishable from that of a point mass placed at the shell’s centre, possibly multiplied by a radius‑dependent scaling factor.

Starting from flat space ℝⁿ, they introduce the spherical mean Φ(a,b) of a translationally invariant pairwise potential ϕ(r) over a shell of radius b. By invoking the Euler‑Poisson‑Darboux (EPD) identity ∂²_bΦ + (n−1)/b ∂_bΦ = Δ_a Φ, they derive a partial differential equation that any spherical mean must satisfy. Imposing the spherical property forces Φ to have the form Φ(a,b)=½