Theory like general relativity for charges According to GR, the distribution of mass curves spacetime and the curvature in spacetime then moves all masses in the spacetime in a specified fashion. Is there an analogous theory that says the distribution of charge curves spacetime in so-and-so manner and the induced curvature in spacetime is then responsible to move charges in the spacetime in so-and-so manner. Browsing the electromagnetic theories gives the impression that movement of charges is studied using theories based on fields and not spacetime. This is a hobby question borne out of leisurely study of physics rather than serious involvement with it. Please try to give as much background information as possible intuitively with as less maths as possible.
 A: All gauge theory, electrodynamics included, can be interpreted geometrically and there are "curvatures," but the picture is very likely not the same as what you might have in mind thinking about GR.
The first and clearest difference is that not everything is electrically charged, and so not everything would see the electric "curvature," whereas everything is gravitationally "charged." As a result everything sees the curvature in gravity. There are other more technical things one can say, but I think this observation kills most hope for an exact analog between the two.
A: The point why gravity is can be absorbed into a theory of curved space-time is that it is universal. In Newtonian terms, every particle at a given point will feel the absolutely same gravitational acceleration. This is known as the weak equivalence principle. This means that you can say that the gravitational acceleration is apparent, that is, non-inertial, similar to e.g. the centrifugal force. However, the issue is that you can establish the notion of inertiality only locally, there is no global inertial Cartesian frame anymore. Marry these principles with special relativity and you obtain general relativity.
Now the issue with electromagnetism is that the electromagnetic force causes different acceleration to different particles. Specifically, the acceleration will be proportional to the specific charge of the particle $q/m$, where $q$ is the charge of the particle, and $m$ its mass. Different specific charge, different acceleration - there is nothing that can be done about it really. For example, if you have zero charge, you do not "see" any of the electric and magnetic fields. So what would be the space-time geometry corresponding to electromagnetism if you can just "turn it off"? Obviously, there is no natural notion.
There are some historical theories that tried to do this nonetheless, Kaluza-Klein theory being one of them. However, there is also a way electromagnetism gets "sort of" a geometrical interpretation in field theory. The point is that classical and quantum fields couple to electromagnetism in a way that can be interpreted as coupling to some sort of geometry through a gauge covariant derivative. The electromagnetic field strengths can then be viewed as a sort of "curvature". However, this geometry is still charge-dependent and this is more of a formal analogy.
