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

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.

Solved by gpt-5.6-sol high.