# Has a metric formulation of electromagnetism ever been attempted? [duplicate]

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.

## marked as duplicate by Kyle Kanos, Danu, Brandon Enright, Qmechanic♦Jul 9 '14 at 20:11

• There exists a geometric formulation of gauge theory (of which EM is the simplest example). Naturally it uses many concepts that are also useful in general relativity. For example in both cases the main physical observable is a curvature tensor. However in gauge theory the curvature does not come from a metric. If you want to learn about this, I like the book Gauge Fields, Knots and Gravity by Baez and Muniain. – Robin Ekman Jul 9 '14 at 19:35
• possible duplicate of Geometric interpretation of Electromagnetism – Kyle Kanos Jul 9 '14 at 19:35
• possible duplicate of Is there any relationship between gauge field and spin connection?. Although the question is not the same, the answers may very well provide a satisfactory answer to this question as well. – Danu Jul 9 '14 at 19:37
• The core problem to doing this consistently is that electromagnetism doesn't respect the equivalence principle -- electric "mass" is not equal to inertial "mass", so it's not reasonable to expect geodesic motion. – Jerry Schirmer Jul 9 '14 at 20:31

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).