Green Functions, Sommerfeld Images, and Wormholes

Abstract: Electrostatic Green functions for grounded equipotential circular and elliptical rings, and grounded hyperspheres in n-dimension electrostatics, are constructed using Sommerfeld's method. These electrostatic systems are treated geometrically as different radial p-norm wormhole metrics that are deformed to be the Manhattan norm, namely "squashed wormholes". Differential geometry techniques are discussed to show how Riemannian geometry plays a rule in Sommerfeld's method. A comparison is made in terms of strength and position of the image charges for Sommerfeld's method with those for the more conventional Kelvin's method. Both methods are shown to be mathematically equivalent in terms of the corresponding Green functions. However, the two methods provide different physics perspectives, especially when studying different limits of those electrostatic systems. Further studies of ellipsoidal cases are suggested.