Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 4 / c / Solution

Transform both fields geometrically, $g\mapsto {\varphi }^{\ast }g$ and $A\mapsto {\varphi }^{\ast }A$. Naturality givesandEach Lagrangian four-form therefore transforms by pullback. Integration of a top-degree form is unchanged under an orientation-preserving diffeomorphism, so both the Einstein-Hilbert and Maxwell terms, and hence the full action, are invariant.