Vacuum Killing and Ernst differential identities

For the stationary Lorentzian data with $\mathrm{Ric}(g)=0$, let $F_{ab}={\mathrm{\nabla }}_a{T}_b$ and $\mathcal{F}=F+i{\star }_{g}F$. Then $F=\frac{1}{2}d{T}^{\mathrm{♭}}$, ${\mathrm{\nabla }}^a{F}_{ab}=0$, $dF=d{\star }_{g}F=0$, and the complex one-form $\eta =2{\iota }_T\mathcal{F}$ is closed. It has smooth local potentials. On any domain where $\eta =d\sigma$, $\mathrm{Re}\sigma =-g(T,T)$ up to an additive real constant on each connected component. A single constant suffices on a connected domain. Global existence and normalization of a potential require a separate period or topology argument.

Independently retrieved and read Ionescu–Klainerman, §4.1 opening and §4.2 (4.11)–(4.18), including the vacuum, Killing and Hodge conventions and the separate simply-connectedness sentence. The differential identities agree with the formulation here. That passage states the Ernst identities and refers elsewhere for their verification; its full cited proof was not read. The proof above instead composes the five locally authored prerequisites, whose only explicit external differential input is the convention-matched Ricci commutator from Reall. No global black-hole hypothesis or complete flow is needed for these local differential identities. No source proof review or native mathematical acceptance is claimed.