EM 4-potential vs gravity 4-potential?

Question

In classical field theory, the electrostatic and gravitational fields have very similar differential forms:

$$\vec \nabla\cdot \vec{E}=\frac{\rho}{\varepsilon_0}$$

$$\vec \nabla\cdot \vec{g}=-4\pi G\rho$$

And their respective curls are null. This means we can express these fields as the gradient of some potential. Nonetheless, if there are any moving charges affecting our system, then we also have a magnetic vector potential $\vec A_M$ (excuse the notation) and the electric field is now given by:

$$\vec E = -\vec \nabla V-\frac{\partial \vec A_M}{\partial t}$$

Therefore, if we had moving mass, is it possible that the gravitational field could be written in a similar fashion?

$$\vec g = -\vec\nabla \Phi-\frac{\partial \vec A_G}{\partial t}$$

Answer 1

This analogy to electromagnetism is called gravitoelectromagnetism, and it is a valid approximation to the Einstein field equations under certain conditions. The most prominent of those conditions is that the weak field limit applies (see here). There are also considerations to be made when choosing a frame, as the GEM equations are not Lorentz covariant.