Sobolev Spaces, Schwartz Spaces, and a Definition of the Electromagnetic and Gravitational Coupling

Abstract

The concept of multiplicity of solutions was developed in [1] which is based on the theory of energy operators in the Schwartz space S - (R) and some subspaces called energy spaces first defined in [2] and [3]. 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 ∈Z + ) using the theory of Sobolev 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 Electro Magnetic (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 [4] 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.