differential geometry - Has a metric formulation of electromagnetism ever been attempted?

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.

Answer

The formulation you seek is gauge theory. It is not completely analogous to changing the metric of spacetime, but many similarities can be seen.

In this, we take as our starting point a certain gauge group $G$ (In the case of EM, $\mathrm{U}(1)$), which will induce symmetries of our theory, just as the Lorentz group of special relativity is the symmetry of that theory (but the Lorentz group is emphatically not a gauge group for special relativity!). Then, we construct a so called $G$-principal bundle over our spacetime $\mathcal{M}$, and take so-called associated vector bundles as the spaces where our fields now take value (in addition to any spacetime indices they may possess) (just like vectors in GR take values in the (co)tangent spaces).

Now, we must learn with shock that the naive derivative of our fields do not transform as proper values in the associated bundles, just like naive derivatives of Lorentz vectors are not vector either. And so we construct a covariant derivative, which yields the gauge field (the vector potential) as its connection form, just like we get the Levi-Civita connection in GR. (The analogous thing in GR to the connection form coefficients would be the Christoffel symbols, but remember that the analogy is not complete)

And from the connection that is the gauge field, we may calculate its curvature - which is the field strength tensor and will appear in the action of the theory, just like the curvature in GR appears in the Einstein-Hilbert action.

The analogy goes quite a way, but I want to stress that it breaks down as soon as the metric is concerned. Gauge theories do not change the metric, they take a spacetime with a given metric and build upon it. This is essentially why understanding gravity is so hard - because it is not obviously a gauge theory like all the other forces.