1. 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.
2) A “boundaryless” solution, eliminating the usual anti-derivative-based approach.
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.
2. Relevant Mathematical Background
Physicists are familiar with Maxwell’s vector-based equations describing electric and magnetic fields,
and
respectively, in terms of changes over space and time, based on differential operators
and
and sources of the fields Most are familiar with the tensor formulation:
(1)
where
,
,
(2)
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],
(3)
where
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
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,
and
:
(4)
where outer product
does not exist in vector calculus but is dual [5] of the cross product
(5)
If we let the first vector in Equation (4) be the vector derivative
we obtain
(6)
where
is the gradient,
is the divergence, and
is the curl of
. 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 a curl is zero
—curl of gradient of scalar field is zero (7)
—divergence of a divergence is zero
Geometric calculus generalizes vector calculus and Cartan’s calculus of differential forms. The elementary combination of inner and outer products,
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
wherein
and
are two distinct parts of the single fundamental quantity, coderivative
, that makes it possible to reduce Maxwell’s eqns to a single equation:
. (8)
When
it can be directly solved for
, whereas
and
cannot, because each is only part of the derivative of
. Finally, geometric calculus, whose fundamental theorem on m-dimensional manifolds
(9)
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.
3. 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,
(10)
where Laplacian
. The inverse operator yields
(11)
where
has right inverse
with integral kernel
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
of the Laplacian operator on differential forms provides a right inverse to the Cartan differential d and co-differential
simultaneously. The formalism yields a Biot-Savart operator motivated by the analogy of the equations with Maxwell equations for a magnetic field
caused by stationary current J. For Laplacian
they derive
as an integral kernal determined by function A.
4. 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
(12)
which is formally solved via the inverse differential operator
. (13)
The inverse operation
is satisfied by the existence of the Green’s function anti-derivative method of solution for physical field equations. The Dirac delta distribution-based technique provides field solutions in terms of boundary conditions. If i is the unit pseudo-vector and M is an m-dimensional smooth oriented vector manifold with piecewise smooth boundary
then
(14)
where Green’s function
. This is the general solution corresponding to anti-derivative
for equation
. Explicitly, values of
at interior points of M are determined by values of
on boundary
, as well as by values of
inside M.
5. Discrete Operator
It is the purpose of this paper to establish the discrete operator analog of the anti-derivative-based
. In place of integrating over every point in the region, as well as all points in the boundary of the manifold, the preferred formulation would be
.
Consider two situations; the electric field
induced by a static charge density
and the magnetic field
induced by moving charge density
. We consider each separately:
(15)
If
is not changing,
and
, hence
where
is an integral (anti-derivative) operator determined by a Green’s function. An interesting case [8] is for
with 2D symmetry in 2D region
with boundary
:
(16)
In the absence of sources the first integral vanishes and the field within
is given by the line integral of its value over the boundary
, which can be seen to be the Cauchy integral formula after suitable change of variables.
We next consider the coderivative of the magnetic field
. (17)
With gauge field
and
and vector identity
, this reduces to
, the prototype equation for our theorem and proof. This choice is compatible with the universal equation describing the field circulation induced by source current density:
(18)
where source current density
with
, q is source charge and
is flow velocity.
“The [geometric algebra] vector derivative
actually has an inverse, in complete contrast to traditional 3D vector analysis where only the gradient has a definite inverse and the curl and divergence do not.”
Despite this definite statement, we search for a solution of the form symbolically represented as
. (19)
To derive this solution we need to show that the inverse curl operator exists:
. (20)
The remainder of this paper establishes the existence of the inverse curl operator.
6. The Uncurl Operator Theorem
Theorem I
There is an anti-curl operator
with inverse curl property such that, applied to differential equation
solves for field vector
.
Our purpose in this paper is to present a
-operator-based inverse
which will allow solution at specific field points
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
, with unit scale factor, therefore we solve
for
by operating with anti-curl operator
on both sides of
. (21)
Given
we have the suggested association
. We will prove that use of radial operator
in place of inverse curl operator
satisfies Equation (21) above.
In the theorem
where
we stipulate that the source current density
is defined using the same volume radius
as is used in the inverse curl operator
. This is not an arbitrary condition but is implied by the physics
and is seen to be compatible with the Green’s function-based anti-derivative
. 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.
Hypothesis: The inverse curl operator
is the radius operator
.
The plan of our proof is to prove that
where
. We explicitly expand the expression to obtain the terms
, where we replace the source current density
with the completely equivalent momentum density
. Here and in the following, the expression
and
.
(22)
The expansion of this equation is below:
Proposition: Terms on the left-hand side of
are organized into three expressions, the x-, y-, and z-components of the field vector
where
and
is the source current density
that replaces current density
for consistency with other problems.
lemma 1.
We regroup the terms in the x-component of the left hand side as follows:
. (23)
Similarly we regroup the remaining terms in the x-component of the left hand side as follows:
. (24)
Grouping these two terms we see that they are equal to the x-component of the right hand side:
(25)
and similarly for y- and z-components. The inverse curl operator
is radius operator
.
Theorem II
There is an anti-curl operator
with inverse curl property such that, applied to source current density
satisfies differential equation
when field vector
and
is perpendicular to
,
.
As before we substitute the momentum density
for current density
with no mathematical effect.
(26)
Proposition: The terms on the left-hand side of
can be organized into three terms,
,
, and
where
is the source current density
.
lemma 1.
By visual inspection, the first column contains terms
.
lemma 2.
After removing the
terms we find terms of the form
plus
but the
term is missing; therefore we add and subtract
to the x-component terms. Combining these three positive terms we substitute
and obtain
. The same procedure applied to the y- and z-components leads to
. (27)
lemma 3. The remaining terms in the x-component are
(28)
The y- and z-component terms similarly yield
and
. Summing these three terms we obtain
. (29)
lemma 4. The result of the above steps yields
: We have obtained equation
(30)
which is true if and only if
. By definition
since this term represents the source current density or momentum density that induces a circulation in the field in question. As source current density does not have meaning at a point, distance from this nonexistent point cannot be zero hence
. The remaining possibility is
, whence
.
From Theorem I
therefore
implies
iff
. Thus if we define
we obtain
.
7. Results
We hypothesized the existence of an exact un-curl operator,
which solves the field equation
yielding
and we have shown that this matches the known solution
. Operators require operands to operate upon; equivalence relations are summarized
(31)
(32)
We proved these operator relations in terms of the field equations:
,
. (33)
Discussion of Results
The inverse operation on field equations typically derives an anti-derivative-based solution for the field based on integrals of the field derivative for every point within the bounded region and an integral evaluating all points on the boundary of the region. Contrast the two solutions for the magnetic field at
from a source current; the Biot-Savart law [10] for a steady current
inside a region
, where
could represent a curve, surface, or volume; the magnetic field associated to
is
versus
. (34)
If we wish to integrate over a length of current path, the anti-derivative is preferable. The inverse operation on field equations derived herein is significantly simpler and derives the value of the field at a point a distance
from source current density. Much effort has been spent on generalizing Green’s function results in curved space; nevertheless; there exist gravitomagnetic problems in metric space wherein the boundary varies dynamically, and flat space is a realistic assumption. The anti-curl operator enables new approaches to the physics of fields that are effectively unobtainable using Green’s function-based anti-derivative inverses.