Sobolev Spaces, Schwartz Spaces, and a definition of the Electromagnetic and Gravitational coupling

TL;DR

This paper extends the theory of solutions to linear PDEs in Schwartz and Sobolev spaces, introducing the concept of Energy Parallax and applying it to electromagnetic and gravitational coupling phenomena.

Contribution

It generalizes previous solution theories to Sobolev spaces and defines Energy Parallax, linking energy variations to solution spaces and coupling effects.

Findings

Variation of EM energy density affects wave solutions

Energy Parallax introduces additional solutions in energy subspaces

Theoretical framework for EM and gravitational coupling

Abstract

The concept of "multiplicity of solutions" was developed in arXiv:1509.02603v2 which is based on the theory of energy operators in the Schwartz space S^-(R) and some subspaces called energy spaces first defined in arXiv:1208.3385 and arXiv:1308.0874. The main idea is to look for solutions of a given linear PDE in those subspaces. Here, this work extends previous developments in S^-(R^m) (m in Z^+) using the theory of Sobolev spaces, and in a special case the Hilbert spaces. Furthermore, we also define the concept of "Energy Parallax", which is the inclusion of additional solutions when varying the energy of a predefined system locally by taking into account additional smaller quantities. We show that it is equivalent to take into account solutions in other energy subspaces. To illustrate the theory, one of our examples is based on the variation of ElectroMagnetic (EM) energy density within the skin depth of a conductive material, leading to take into account derivatives of EM evanescent waves, particular solutions of the wave equation. The last example is the derivation of the Woodward effect with the variations of the EM energy density under strict assumptions in general relativity. It finally leads to a theoretical definition of an electromagnetic and gravitational (EMG) coupling.