Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 4

For any differential form $\omega$, Cartan's magic formula givesUsing $d^2=0$,Thus the Lie derivative commutes with the exterior derivative on every differential form.

Because the Levi-Civita connection preserves the spacetime volume form, only derivatives of $X$ remain in ${\mathcal{L}}_{X}T$. Lower the raised indices in $T_{ab}{}^{cd}={ϵ}_{ab}{}^{cd}$, contract successively with the supplied products of two Levi-Civita tensors, and separate ${\nabla }_a{X}_b$ into antisymmetric, trace and symmetric trace-free parts. The antisymmetric part cancels automatically, while ${\mathcal{L}}_{X}T=0$ is equivalent to vanishing of the symmetric trace-free part:Thereforeso $\boxed{\alpha =1/2}$. This is the four-dimensional Conformal Killing equation, and $X$ is a Conformal Killing vector field.

The first vacuum Maxwell equations equation is preserved for every vector field because part a givesOn two-forms in four dimensions, the mixed volume tensor $T_{ab}{}^{cd}$ represents twice the Hodge star operator. Part b therefore says that a conformal Killing field commutes with the Hodge star:It follows thatHence $\tilde{F}={\mathcal{L}}_{X}F$ satisfies both vacuum Maxwell equations whenever $F$ does.