THE ZERO-VECTOR AXIOM
This slide shows a standard mathematical statement of the zero-vector axiom, used in many texts.
As can be seen, all zero-vectors are made identical, and no substructure is considered.
Such an abstract system of vector mathematics does not fit physical reality. However, when all one’s EM foundations concepts are strongly conditioned in this fashion, the gravitational aspects of EM are discarded.
Further, EM theory, experiments, and equipments will be developed along these limiting lines. In that case, EM engineering of gravitational effects will not be developed. One will assume that ~t ~s ~impossible to have a locally curved spacetime, and hence all local frames will be Lorentz frames. Therefore the conservation laws will rigorously apply.
Further, the nonlocal quantum effects will stay firmly in the microscopic world where they belong. Physical macro-reality will stay stable and predictable in a classical manner.
General relativity (in curved spacetimes) will be restricted to the special case of the local special relativistic (uncurved) spacetime, with only distant perturbations (spacetime curvatures).
The world will stay sane and "normal", and the totally variable insanity of physical reality will remain bottled up in the minute microworld or the distant maelstrom of suns, stars, black holes, etc.
The Aharonov-Bohm heresy will stay kaput in the real world and will not be unleashed beyond mesoscopic reality -- a few thousand angstroms or so.
With such an error in the application of vector mathematics to physical science, we become "flatland physicists," so to speak. We become like skimmer bugs on the surface of a pond; we have no appreciation of the dynamics of the depths below or of the heavens above.
Worse, we condition our development of instruments according to the fixed notions in our heads. Then we use these "biased" instruments to do experiments which reaffirm our notions. Any deviation suggested to this is considered heresy and nonsense.
And as Max Planck once pointed out, you will get a new physics only when the old physicists -- who so adamantly oppose it -- die off.