Grounded Hyperspheres as Squashed Wormholes

TL;DR

This paper introduces a geometric method using squashed Riemannian wormholes to compute Green functions for grounded hyperspheres in N-dimensional electrostatics, offering a new perspective on classical image charge solutions.

Contribution

It presents a novel geometrical approach to electrostatic Green functions using wormhole squashing, connecting classical image charge methods with Riemannian geometry.

Findings

The wormhole approach yields mathematically equivalent Green functions to traditional methods.

The wormhole method offers a different physical interpretation of image charges.

Comparison shows differences in the perceived strength and location of image charges.

Abstract

We compute exterior Green functions for equipotential, grounded hyperspheres in N-dimensional electrostatics by squashing Riemannian wormholes, where an image charge is placed in the branch of the wormhole opposite the branch containing the source charge, thereby providing a vivid geometrical approach to a method first suggested in 1897 by Sommerfeld. We compare and contrast the strength and location of the image charge in the wormhole approach with that of the conventional Euclidean solution where an image charge of reduced magnitude is located inside the hypersphere. While the two approaches give mathematically equivalent Green functions, we believe they provide strikingly different physics perspectives.