Comparison of covariant form of Maxwell equations with Einstein's GR

If you want to compare Maxwell's EM with GR, note they're respectively obtainable from extending a global $$U(1)$$ variance to a local one and Lorentz invariance to invariance under general coordinate transformations. Thus the equivalent of introducing $$A_\mu$$ in the gauge covariant derivative is introducing Christoffel symbols in the connection, while the commutator $$F_{\mu\nu}$$ of gauge covariant derivatives is analogous to the commutator $$R_{\mu\nu\rho\sigma}$$ of connections acting on a vector field. So the equivalent of (1) is Einstein's usual equations, while the equivalent of (2) is $$R_{\mu[\nu\rho\sigma]}=0$$. In neither case do we need the second equation; a Lagrangian formulation obtains the first, but the second is just a tautology.

1. The counterpart to $$(g,R)$$ in GR is $$(A,F)$$ in E&M.

2. If you write $$F=F(A)$$ in E&M, then the Bianchi relation (2) is not needed/a tautology, cf. e.g. this Phys.SE post.

3. For how GEM appears as a limit of GR, see e.g. this Phys.SE post.