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.