An Analogy with Maxwell's Equations

Let J = electric current (flux of electric charge), a (n-1)-form. The Fredholm equation implies that there exists a form A of the same class (in standard texts, *A is used, a 1-form) such that:

    ÐA = J          (Or, substituting the definition of Ð)
    dðA + ðdA = J

It is known separately that electric charge is conserved, or dJ = 0, and hence the ðdA part is suppressed in standard texts, and without that restriction A is not unique, but can be decomposed per Kodaira's theorem into A1 + A2 where A2 = ðG, the "gauge field", which can be chosen arbitrarily without changing the inferred current density J.

d*A = E	Electric and magnetic fields
*E = uH	Electric and magnetic induction, scaled by the vacuum permittivity and permeability
dH = -J	The electric current, which is conserved.

These are Maxwell's equations, and A is the magnetic vector potential (extended to 4D with the electric potential). Any conserved scalar current obeys Maxwell's equations, a fact not known when they were developed empirically. However, the fields E also appear in the formula for the force on a current, and there is no mathematical necessity that the fields actually have a physical significance of that kind.

(A pet peeve: dE gives the density and flux of magnetic monopoles. What a joke! E = d*A, so dE = d² *A = 0 identically.)

One can formulate an analogous set of equations involving the connection. Let C = Christoffel symbols (referring to the middle index when differentiating). Let R = Riemann-Christoffel tensor. Let T = a source term, as yet uninterpreted.

    Ð (*C) = T
    dð(*C) + ðd(*C) = T
    d((-*d*)(*C)) + ðd(*C) = T

    R = dC, so d*(*C) = -R
    d*R + ðd(*C) = +T

It's important to assess the ðd(*C) term. What is d(*C), the "divergence" of the Christoffel symbols? Using our generalized Kodaira's "theorem" we can decompose C as

    C = C1 + C2 + C3
        where C1 = ðZ1, C2 = dZ2 and C3 is harmonic.

As R = dC, the C2 portion has no physical effect; this term contains the Coriolis force and similar synthetic effects. Thus, while ðd(*C) is in general nonzero, we are justified in suppressing it, similar to neglecting the "gauge field" in electromagnetism. Here's an example: in a rotating coordinate system, we have a centrifugal force. Thinking Newtonianly, this force would be the gravitational field of a cylinder of uniform negative mass density and infinite length, coaxial with the spin axis. The negative mass is as real as the Coriolis and centrifugal forces are.

How arbitrary is Z2? Suppose we have a coordinate patch and a value for Z2 on it, having n² components. Suppose Z2' is not too different from Z2 (i.e. the metric square of the difference is less than some bound). Then it should be possible to find a coordinate transformation on the patch that transforms C so its C2' component is dZ2'.

Since T = d*R, dT = d²*R, which we have seen is zero due to a special condition of the Riemann-Christoffel tensor. Therefore the unidentified source term T is conserved by mathematical necessity. Thus it can be interpreted as the motion of a "source substance". The force law on that substance includes the Christoffel symbols (or their derivatives: *R) and there is nothing else for it to include. Thus, unlike the electromagnetic case, the field induced by the source term has physical significance and is solo, by mathematical necessity.

When the structure of the space (Christoffel symbols) is given and the source field is computed from it, it is not necessary to solve Fredholm's equation or to apply Kodaira's theorem. But when inferring the structure of the space from a prespecified source field T, the task is very difficult, because neither the Kodaira nor the Fredholm theory are proven for composite-valued forms, and Green's function depends on the structure of the space, which is to be computed. Consider a scaled ensemble of source terms eT. It is likely that Christoffel symbols can be found such that ÐC = eT for any conserved T when e is small enough, but it is entirely conceivable that no solution exists for certain T when e is above a bound.