Curved-Space Gravity–Electromagnetism Coupling: Actions, Boundaries, and Topology

We revisit the coupling of electromagnetism and gravity in curved spacetimes from a purely 4-dimensional, action-based viewpoint. Our starting point is a smooth vector field ξµ with metric dual ξµ := gµνξν, from which we define the two-form F := dξ♭. This identification reproduces the homogeneous Maxwell equations identically and, when fed into the Einstein–Maxwell action, yields the standard coupled field equations. We also analyze a curvature-coupled variant that realizes an effective geometric source, and discuss boundary/topological refinements that adjust charges without modifying bulk equations in vacuum. Our contribution is a concise, self-contained derivation that clarifies conventions, curvature identities, and the scope and limits of curvature-coupled generalizations