Killing two-form and normalized Ernst notation

PDF p. 3: Definition 1.1(1), stationary exterior coordinates; PDF p. 4: §1.2 before Theorem 1.1; PDF p. 7: §2.1

Work in ZFC, including the axiom of choice. Let $(M,g,N,T)$ be stationary Lorentzian data: $M$ is an oriented smooth four-manifold, $g$ has signature $(-,+,+,+)$, $N$ is a time orientation, and $T$ is a smooth Killing field. Let $\mathrm{\nabla }$ be the torsion-free metric-compatible connection of $g$, and let $T_b=g_{bc}{T}^c$. Define the Killing two-form by $F_{ab}={\mathrm{\nabla }}_a{T}_b$ and put $f=-g(T,T)$.

Let $ϵ$ be the oriented metric volume form and ${\star }_g$ the Hodge star of $g$ and the chosen orientation. Define $\mathcal{F}=F+i{\star }_{g}F$ and ${\mathcal{F}}^2={\mathcal{F}}_{ab}{\mathcal{F}}^{ab}$, where both indices are raised with $g^{-1}$ and the contraction is complex bilinear, without conjugation or a factor of one half. Thus ${\star }_g\mathcal{F}=-i\mathcal{F}$ and $({\star }_{g}F{)}_{ab}=\frac{1}{2}{ϵ}_{ab\mu \nu }{F}^{\mu \nu }$.

For normalization on an open set $U\subseteq M$, a designated coordinate end consists of a supplied open subset $U_{\mathrm{e}\mathrm{n}\mathrm{d}}\subseteq U$ and a smooth coordinate diffeomorphism $(t,x):U_{\mathrm{e}\mathrm{n}\mathrm{d}}\to I\times \{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>R_0\}$, where $I\subseteq \mathbb{R}$ is a nonempty open interval, $R_0>0$, and $T={\mathrm{\partial }}_t$ in these coordinates. Put $r=\mathrm{\mid }x\mathrm{\mid }=({\sum }_{i=1}^3(x^i{)}^2{)}^{1\mathrm{/}2}$. The orientation convention is that $dt\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$ is positive on this end. The article's asymptotic chart has $I=\mathbb{R}$; metric decay is a separate hypothesis, not part of these coordinate-end data.

On such a $U$, a normalized Ernst potential, when one exists, is a smooth complex function $\sigma =f+i\psi$, with real $\psi$, satisfying ${\mathrm{\nabla }}_a\sigma =2T^b{\mathcal{F}}_{ba}$ and $\sigma \to 1$ along the designated coordinate end. Precisely, this limit means that for every $\epsilon >0$ there is $R>R_0$ such that $\mathrm{\mid }\sigma (t,x)-1\mathrm{\mid }<\epsilon$ for every $t\in I$ and every $x\in {\mathbb{R}}^3$ with $\mathrm{\mid }x\mathrm{\mid }>R$, where $\mathrm{\mid }\cdot \mathrm{\mid }$ on complex numbers denotes their modulus. Thus the limit is uniform in the time coordinate and the spatial directions. Consequently ${\mathrm{\nabla }}_a\psi =T^b{ϵ}_{ba\mu \nu }{F}^{\mu \nu }$. Here the covariant derivative of a scalar is its ordinary differential. Neither existence of $\sigma$ nor the existence of such an end is asserted by this definition.

Stationary Lorentzian data and tensor conventions

Complex self-dual forms and metric-dependent contraction

Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

Lorentzian exterior calculus and Hodge star

The Axiom of Choice

Given: The data in the Definition, and, for the assertions about $\sigma$, a function $\sigma$ satisfying the stated conditional definition.

Goal: Justify the two-form, the displayed identities and the coordinate-end normalization conventions.

1.1 By the Killing equation, $F_{ab}=-F_{ba}$, so the smooth covariant derivative of $T^{\mathrm{♭}}$ defines a real two-form. Metric compatibility gives ${\mathrm{\nabla }}_{a}f=-2T^b{\mathrm{\nabla }}_a{T}_b=2T^b{F}_{ba}$. Thus the real part of the proposed potential equation agrees with $df$; this calculation does not establish existence of a potential. [F1, F2, algebra]

2.1 The Hodge star is extended complex linearly, so ${\star }_g\mathcal{F}={\star }_{g}F+i{\star }_g^{2}F={\star }_{g}F-iF=-i\mathcal{F}$. Its component formula gives $2T^b({\star }_{g}F{)}_{ba}=T^b{ϵ}_{ba\mu \nu }{F}^{\mu \nu }$. Taking imaginary parts of the assumed potential equation therefore yields the stated formula for ${\mathrm{\nabla }}_a\psi$. The inverse metric supplies the indicated contraction. [given, F1, F3, F4, step 1.1, algebra]

3.1 The supplied end diffeomorphism defines $r$ and each region $r>R$ explicitly, so the quantified modulus bound defines the normalization without an implicit notion of approach to infinity. Moreover $T^a{\mathrm{\nabla }}_a\sigma =2T^a{T}^b{\mathcal{F}}_{ba}=0$ by alternation of $\mathcal{F}$. Since $T={\mathrm{\partial }}_t$, the potential is independent of $t$ on the interval $I$. Hence the stated uniformity in $t$ follows already from the potential equation when the spatial limit holds on any one time slice. The end chart, its orientation and the limit are supplied conditions; their existence has not been deduced. [given, F1, F3, step 1.1, step 2.1, algebra] ∎

The article introduces this notation in the cited passages. Its citation records attribution; the identities above use the stated mathematical suppliers. The coordinate-end and quantified-limit clauses are explicit local definitions, with no imported end-existence or asymptotic-flatness theorem.