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An Unsuccessful Simplification

We have seen that on any Hausdorff space the Christoffel symbols obey a Fredholm-type equation

     (*C) = T

where the source term T is conserved (dT = 0). But thinking Newtonianly, C contains fields, not potentials, and T as computed here is the flux of an unfamiliar derivative of mass density. Is it possible to find an equivalent set of equations at one higher level of integration?

We've seen that:

    C[i][j][k] = g(x2,x1)[i][k][2/j] = 0.5 r^2[2/i][1/k][2/j]

Keeping the ends separate in the distance function:

    C = dg

where g is interpreted as a matrix-valued 0-form. There are two differentiations on argument 2. Suppose we start with *g itself as the potential, so the source term will come out as a flux of a 1-form (rather than a matrix 0-form). We'll follow the same analogy with Maxwell's equations.

    C = -d*(*g)
    d*C = something having to do with Coriolis forces

So that approach is a dead end. If we want the Christoffel symbols involved, they come into the equations at the level of a potential. The alternatives are:

  1. Squeeze into the equations an unnatural combination of metric derivatives, giving up a relation with the Christoffel symbols.
  2. Accept a source term which is not the flux of something (a (n-1)-form).
  3. Accept a source term which is the flux of something bizarre.

The last choice is the one which will be followed from here on.


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