Under certain conditions, it is possible to approximate the effects of the theory of relativity through equations very similar to those of Maxwell, but for gravity. In these equations, our "standard" gravity is like the charge, and moving masses create a field like the magnetic field. In these equations, the gravitational field and this "gravitational magnetic field" affect each other like charge and the standard magnetism do. This is the GEM Theory.
However, this is valid for a chargeless system. Once the body has charge, the electric field, magnetic field, gravitational field and "gravitacional magnetic field" will interact with each other.
My question is what the GEM equations would look like in this situation where there's mass and charge? How many "Maxwell Equations"-like there will be?