Hertz Potentials and Differential Geometry
Abstract
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.
Subject
Hertz potentialHertz
EM
electromagnetism
electric
magnetic
Casimir
quantum field
quantum field theory
differential geometry
Citation
Bouas, Jeffrey David (2011). Hertz Potentials and Differential Geometry. Master's thesis, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /ETD -TAMU -2011 -05 -9409.