Noticing that Newtonian gravity and electrostatics are equivalent, is there also a relationship between the general relativity and electrodynamics?

In classical mechanics, we had Newton's law of gravity $F \propto \frac{Mm}{r^2}$. Because of this, all laws of classical electrostatics applied to classical gravity if we assumed that all charges attracted each other due to Coulomb's law being analogous. We can "tweak" classical electrostatics to fit gravity.

In modern physics, does the reverse work? Can we "tweak" General Relativity to accurately describe electrostatics or even electromagnetism?