I understand that electromagnetic fields carry energy, and this energy curves spacetime gravitationally. That's not my question.
I'm asking if anyone has tried to formulate electromagnetism in such a way that EM charges impart an EM geometry onto spacetime that is only experienced by EM charges. That is, the EM geometry of spacetime would be a function of charges such that charges themselves produce EM curvature and the motion of charges produces EM torsion, and that charged objects move according to how their charge (electrical or magnetic, positive (N) or negative (S)) experiences that geometry.
For example, a positive electric charge would appear as a "hill" to other pe charges, a "valley" to ne charges, and either a left or a right hand "whirpool" to either nm or sm charges (I'm not immediately sure which would match with which) if it had some velocity.
This formulation would be directly analogous to how energy creates gravitational curvature in spacetime (and if one accepts Einstein-Cartan, gravitational torsion comes from intrinsic angular momentum), and then this resulting geometry is experienced by the energy in that region. If Einstein is correct about gravity, would it be too much of a stretch to suppose separate metric functions for each of the fundamental forces, considering that they were a single force moments after the big bang?
I'm not currently concerned with the quantum mechanical approach to electromagnetism. I understand that to be truly fundamental, EM has to be formulated quantum mechanically, but right now I want to limit my question to macroscopic charged objects.