Exact Inverse Operator on Field Equations

Differential equations of electromagnetic and similar physical fields are generally solved via antiderivative Green’s functions involving integration over a region and its boundary. Research on the Kasner metric reveals a variable boundary deemed inappropriate for standard anti-derivatives, suggesting the need for an alternative solution technique. In this work I derive such a solution and prove its existence, based on circulation equations in which the curl of the field is induced by source current density and possibly changes in associated fields. We present an anti-curl operator that is believed novel and we prove that it solves for the field without integration required.


Introduction
Nonlinear field equations are generally impossible of solution-the few existing (Schwarzschild, Kerr) being well-known. These known solutions are static and expressed in spherical coordinates with boundaries at infinity. The Kasner solution, treated by Vishwakarma [1], is dynamic with dynamic boundaries and does not fit well into the static approach. Even physical interpretation of the exact solution has been largely nonexistent. For this reason an alternative approach was sought. The goals of this approach were: 1) A linear formulation that avoids the nonlinearity of Einstein's field equations.
3) A physical interpretation that makes sense of the new solution, and 4) An existence proof for the new solution technique. This paper primarily treats the last problem.

Relevant Mathematical Background
Physicists are familiar with Maxwell's vector-based equations describing electric and magnetic fields, E and B respectively, in terms of changes over space and time, based on differential operators ∇ and t ∂ and sources of the fields Most are familiar with the tensor formulation: A growing number of physicists are familiar with Hestenes' geometric algebra [2] derived from Clifford and Grassmann algebras, in which every entity has both an algebraic and a geometric interpretation. For classical 3D physics the entities consists of scalars, vectors, bivectors, and trivectors/pseudoscalars. In this formulation the electromagnetic field is a multivector [3], where E is the usual electric field vector and the magnetic field is a bivector, a directed 2D-planar entity representing rotation of one vector into another, and i is the pseudoscalar or dual operator that maps bivector B into its vector dual.
Magnetic field vectors do not behave like vectors under reflection, they are not really vectors; axial vectors are not encoded in vector space formalism [4].
However such behavior is encoded in bivector formalism, therefore the geometric algebra function better represents electromagnetic fields. A key geometric algebra entity is the geometric product of two vectors, u and v : where outer product ∧ u v does not exist in vector calculus but is dual [5] of the cross product If we let the first vector in Equation (4) be the vector derivative ∇ we obtain is the curl of f . Hence in geometric calculus we have the unique relation gradient = divergence + curl which has no analog in vector calculus. Relevant identities for the outer product are: -curl of gradient of scalar field is zero (7) = ⋅ + ∧ uv u v u v cannot be expressed with differential forms [6]; such limitations are addressed by extending the formalism with the theory of fiber bundles.
Forms are insufficient for most applications and must be supplemented by other algebraic systems such as matrices or tensors. In geometric calculus spinors, tensors, linear transformations and differential forms are developed in a unified mathematical system. Forms have no analog of are two distinct parts of the single fundamental quantity, coderivative f ∇ , that makes it possible to reduce Maxwell's eqns to a single equation: calculus, whose fundamental theorem on m-dimensional manifolds provides a unified mathematical tool for physics; Equation (9) subsumes Gauss's theorem, Stoke's theorem, Green's theorem, and the Cauchy integral formula, as well as generalizing vector calculus. Although the results of this paper are intended to augment geometric calculus, we will limit our proofs to vector calculus.
Arthur's geometric algebra treatment of Maxwell's equations provides a (3 + 1)D formulation of 3-space vector and 1-time scalar in detail, and then develops the 4D formulation in which time is promoted from scalar to vector.

Biot-Savart Inverse Operator and Laplacian
Much of the literature on the use of Biot-Savart operators and Green's function-based inverse operation deals with the scalar Laplacian in the Poisson equation, where Laplacian ∆ = ⋅ ∇ ∇ . The inverse operator yields depending on the distance only. In [7] the right inverse for the Laplace operator on forms is constructed and proved based on an integral kernal determined by function A which they prove to always exist. They further show that a right inverse 1 − ∆ of the Laplacian operator on differential forms provides a right inverse to the Cartan differential d and co-differential * d simultaneously.

Real Physical Field Solutions
Real physical fields are sourced by distributions or current flows. The differential equation describing field dependence on source current density has the form ∂ = f j (12) which is formally solved via the inverse differential operator The inverse operation where Green's function ( ) ( ) . This is the general solution corresponding to anti-derivative

Discrete Operator
It is the purpose of this paper to establish the discrete operator analog of the anti-derivative-based Consider two situations; the electric field E induced by a static charge density ( ) ρ x and the magnetic field B induced by moving charge density ( ) We consider each separately: ∇ is an integral (anti-derivative) operator determined by a Green's function.

of Applied Mathematics and Physics
In the absence of sources the first integral vanishes and the field within ℜ is given by the line integral of its value over the boundary ∂ℜ , which can be seen to be the Cauchy integral formula after suitable change of variables.
We next consider the coderivative of the magnetic field To derive this solution we need to show that the inverse curl operator exists: The remainder of this paper establishes the existence of the inverse curl operator.

The Uncurl Operator Theorem
Theorem I ( ) ( ) There is an anti-curl operator ( ) 1 − × ∇ with inverse curl property such that, applied to differential equation × = f j ∇ solves for field vector Our purpose in this paper is to present a 3 3 which will allow solution at specific field points r distant from the source current density. The solution is generally simpler, more intuitive, and avoids integration over boundaries of the vector manifold. It also allows mathematical solutions in situations with undefined or dynamic boundaries as seen in the Kasner metric-based exact solution to Einstein's field equations of general relativity [9].
Of course any way of finding a good solution is legitimate; typically guessing based on intuition or analogy, however contained within the beauty of mathematical physics is the fact that the physical world often suggests the answer. In the case of electrodynamic fields we know the relation of the field to the source current density. We know that = × f r j , with unit scale factor, therefore we solve × = f j ∇ for f by operating with anti-curl operator ( ) Given = × f r j we have the suggested association ( ) ( ) . We will prove that use of radial operator ( ) × r in place of inverse curl operator ( ) satisfies Equation (21) above.
In the theorem ( ) ( ) ( ) pulate that the source current density . Physically this traces to Birkhoff's theorem, a generalization of Newton's Shell theorem in which the source inside an isotropic homogeneous spherical shell can be treated as concentrated at the center of the shell and all source outside the shell can be ignored as its contributions cancel.