# The Relation Between Maxwell's Equations and the Equations of Gravitoelectromagnetism

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?

• If you have stationary and moving masses and charges, you don't get more equations than these. You have $E$ interacting with $B$ and $E_g$ with $B_g$, but there's no reason why $E$ or $B$ would interact with $E_g$ or $B_g$.
– J.G.
Mar 17, 2023 at 15:41
• The electromagnetic field has energy and therefore, can curve spacetime as shown by the Reissner–Nordström metric. Also, a curved spacetime changes the field lines of the eltromagnetic field, resulting in effects like the self-force, when a charge repell or atract itself due to the curved lines of its own eletromagnetic field. Then, if we were to approximate this using Equations like those in GEM Theory, we would need to add terms where all the four fields can interact, right? Mar 17, 2023 at 19:16
• Only if GEM is defined so as to accommodate such subtleties. It may be taken as e.g. a limit of how GR does that; the full GR version is well-known.
– J.G.
Mar 17, 2023 at 19:46