A Shape-Preserving Shell Theorem
Abstract
We identify all radial interaction potentials that generalize Newton's exterior shell theorem, for which a uniformly distributed spherical shell is indistinguishable from a point-source up to strength renormalization and radial rescaling. We call these shape-preserving shell interactions, extending Gurzadyan's cosmological theorem which considers strength renormalization alone. We further expand the theory from monopolar to multipolar sources and establish the corresponding classification in arbitrary spatial dimensions. We find potential families beyond the spherical property subclass and show that the classifications for multipolar shape-preserving interactions form a nested hierarchy, with each lower-rank class being a subset of the next higher-rank class.
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BibTeX
@misc{messinger2026shapepreserving,
title = {{A Shape-Preserving Shell Theorem}},
author = {Penelope Messinger and Stephany M. Rodriguez and Trung V. Phan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11020},
eprint = {2610.11020},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11020},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Shape-Preserving Shell Theorem
AU - Penelope Messinger
AU - Stephany M. Rodriguez
AU - Trung V. Phan
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11020
UR - https://arxiv.org/abs/2610.11020
AB - We identify all radial interaction potentials that generalize Newton's exterior shell theorem, for which a uniformly distributed spherical shell is indistinguishable from a point-source up to strength renormalization and radial rescaling. We call these shape-preserving shell interactions, extending Gurzadyan's cosmological theorem which considers strength renormalization alone. We further expand the theory from monopolar to multipolar sources and establish the corresponding classification in arbitrary spatial dimensions. We find potential families beyond the spherical property subclass and show that the classifications for multipolar shape-preserving interactions form a nested hierarchy, with each lower-rank class being a subset of the next higher-rank class.
ER -