Hertz Potentials and Differential Geometry

• Main Author: Bouas, Jeffrey David
• Other Authors: Fulling, Stephen
• Format: Others
• Language: en_US
• Published: 2011
• Subjects: Hertz potential; Hertz; EM; electromagnetism; electric; magnetic; Casimir; quantum field; quantum field theory; differential geometry
• Online Access: http://hdl.handle.net/1969.1/ETD-TAMU-2011-05-9409

I review the construction of Hertz potentials in vector calculus starting from Maxwell's equations. From here, I lay the minimal foundations of differential geometry to construct Hertz potentials for a general (spatially compact) Lorentzian manifold with or without boundary. In this general framework, I discuss "scalar" Hertz potentials as they apply to the vector calculus situation, and I consider their possible generalization, showing which procedures used by previous authors fail to generalize and which succeed, if any. I give specific examples, including the standard at coordinate systems and an example of a non-flat metric, specifically a spherically symmetric black hole. Additionally, I generalize the introduction of gauge terms, and I present techniques for introducing gauge terms of arbitrary order. Finally, I give a treatment of one application of Hertz potentials, namely calculating electromagnetic Casimir interactions for a couple of systems.