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I am always looking for geometric interpretations, and as I understand it, the Einstein field equation basically says 3 things: (1) spacetime is a manifold, (2) free particles follow geodesics in the manifold, and (3) the convergence of geodesics along a given direction in the manifold is proportional to the amount of mass/energy flowing in that direction. #3 is the field equation, #2 the equation of motion.

Which is why mass/energy can be identified as the convergence of geodesics.

With EM, I guess you could say the problem is that there's no free particle. Only charged particles respond to the EM field in the first place. As far as I can tell, Maxwell's equations in curved spacetime say roughly that a charge's 4-acceleration is orthogonal to its 4-velocity and lies in the plane containing the 4-current that produced the field. Even if that's accurate, it's not simple.

But what if charge is just a more sophisticated interplay of geodesics? Something analogous to a helix, where the orientation of the helix about its time axis determines the sign of the charge, and perhaps the helix expands outward to generate the EM field, and when two helices meet they can merge if they are locally spinning the same direction, thus causing attraction, and vice versa for repulsion. That's too simplistic, but there are many structures that can exist in a 4D manifold and why couldn't they account for EM behavior?

I've read that Kaluza Klein theory combines GR & EM, but that's 5D, and it's pretty compelling to think the universe could be constructed simply from the 3D space and 1D time we know so well.

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closed as off-topic by G. Smith, GiorgioP, Yashas, Jon Custer, ZeroTheHero May 3 at 22:09

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    $\begingroup$ You may also want to look up Wheeler's "charge without charge" concept $\endgroup$ – Slereah May 1 at 18:03
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Electric charges couple to the electric field in a similar way as masses couple to the gravitational field. In general relativity charges and masses are part of the energy momentum tensor and therefore do not allow for a geometric interpretation.

In four dimensional GR, the gravitational field is understood geometrically and the electromagnetic field also contributes to the energy momentum tensor.

In five dimensional GR (Kaluza-Klein theory), the gravitational field and the electromagnetic field are understood geometrically (allthough it also contributes to the energy momentum tensor in the effective four dimensional theory).

Consequently I don't think, a geometric interpretation of charges will be consistent in the framework of general relativity.

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