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

A diffeomorphism $\varphi :M\to M$ is a smooth bijection with smooth inverse. It acts covariantly on a differential form by the pullback of a differential form:For vector fields $X_0,\dots ,X_p$,The coordinate definition of the exterior derivative, or its characterization as the unique natural graded derivation extending the differential of functions, gives

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The Hodge star operator is defined byfor forms of the same degree. In four dimensions, under $\hat{g}=e^{2\psi }g$, the inner product on $p$-forms scales by $e^{-2p\psi }$ and the volume form scales by $e^{4\psi }$. HenceFor $p=2$ the exponent vanishes, so the Hodge star on two-forms is conformally invariant.

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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.

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For the infinitesimal diffeomorphism generated by $X^{\nu }$,Substitute these into the given first variation and integrate by parts. Symmetry of $E^{\mu \nu }$ givesBoundary terms vanish by assumption. Since $X^{\nu }$ is arbitrary and diffeomorphism invariance says $\delta S=0$, the coefficient must vanish:When the Maxwell equations $E^{\mu }=0$ hold, their divergence also vanishes and the identity reduces to ${\nabla }_{\mu }{E}^{\mu }{}_{\nu }=0$, the off-shell origin of stress-energy conservation.

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