I work with a near-field which has no residues at the origin.

ajb, the answer about the degenerate metric just fell on my head like a brick. There is generated in my scheme a set of terms on the RHS with no m-dependence, namely the original Lorentzian flat-space statement of the E&M source tensor. There are no such terms to balance them on the LHS since in Cartesian coordinates the Christoffel symbols of the first kind are linear in m. Therefore I cannot use this presumed form of the metric tensor in the interior. This is an important change of direction for me.

Sorry Albers, been away for christmas.

 

I am no expert in building solutions. I assume that you can indeed have very different metrics in different regions, but you will have to match them in a smooth way. I beleive that this matching can be difficult.

Thanks, ajb. There is plenty I can do learning to walk, yet. My text humbles me by assigning the low-field Lense-Thirring problem as an exercise, so I'm working on this for one thing.