Ernst sublevels and the Mars–Simon tensor interface

Use the stationary Lorentzian data, metric connection and index conventions, together with the complex self-dual-form conventions. Let $F_{ab}={\mathrm{\nabla }}_a{T}_b$, $\mathcal{F}=F+i\star F$, and ${\mathcal{F}}^2={\mathcal{F}}_{ab}{\mathcal{F}}^{ab}$ without conjugation. For a complex two-form $U+iV$ with real $U,V$, use the interior product extended complex linearly: ${\iota }_T(U+iV)={\iota }_{T}U+i{\iota }_{T}V$, so $({\iota }_T\mathcal{F}{)}_b=T^a{\mathcal{F}}_{ab}$.

As normalization data, choose $R_0>0$ and a smooth embedding $e:\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>R_0\}\to M$. Its image is the chosen parametrized spatial end, with radial coordinate $r(e(x))=\mathrm{\mid }x\mathrm{\mid }$. This end datum specifies where the limit is taken and imposes no metric decay conditions. For this interface an Ernst potential is a supplied smooth complex scalar $\sigma$ satisfying $d\sigma =2{\iota }_T\mathcal{F}$, normalized by $\sigma \to 1$ along this end. Precisely, for every $\epsilon >0$ there is $R>R_0$ such that $\mathrm{\mid }\sigma (e(x))-1\mathrm{\mid }<\epsilon$ for all $\mathrm{\mid }x\mathrm{\mid }>R$, uniformly in angular direction. Existence of the end datum and of such a potential is supplied, not proved here.

On the open set $N=\{1-\sigma \mathrm{\ne }0\}$ define $P=(1-\sigma {)}^{-1}$ and $y=\mathrm{Re}P$. Write $ϵ={\mathrm{v}\mathrm{o}\mathrm{l}}_g$ for the oriented metric volume tensor: in every positively oriented coordinate chart, ${\mathrm{v}\mathrm{o}\mathrm{l}}_g=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$. Use the same orientation for this tensor and the Hodge star. Write $R_{abcd}$ for the curvature convention used in the cited source; let ${\mathcal{R}}_{abcd}=R_{abcd}+\frac{i}{2}{ϵ}_{cd}{}^{ef}{R}_{abef}$ and ${\mathcal{I}}_{abcd}=(g_{ac}{g}_{bd}-g_{ad}{g}_{bc}+i{ϵ}_{abcd})\mathrm{/}4$. The Mars–Simon tensor in the IK convention is ${\mathcal{S}}_{abcd}={\mathcal{R}}_{abcd}+6(1-\sigma {)}^{-1}({\mathcal{F}}_{ab}{\mathcal{F}}_{cd}-{\mathcal{F}}^2{\mathcal{I}}_{abcd}\mathrm{/}3)$. Curvature and orientation conventions must be transported together when applying this formula.

Stationary Lorentzian data and tensor conventions

Complex self-dual forms and metric-dependent contraction

Interior product of a form by a vector field

The metric volume form, current flux through 3-faces and finite-chain charges

Smooth embeddings

The Killing equation makes $F$ alternating, using Stationary Lorentzian data and tensor conventions. The supplied self-dual-form interface Complex self-dual forms and metric-dependent contraction defines the complex contractions. The real interior product in Interior product of a form by a vector field is smooth and inserts $T$ into the first slot; its complex-linear extension is well defined because the real and imaginary parts are unique. The end embedding, in the sense of Smooth embeddings, makes $\sigma \circ e$ a specified function on a Euclidean exterior; the quantified limit is therefore a predicate on the supplied data, rather than an assertion that stationary geometry provides an end or a normalized potential.

Clause (a) of The metric volume form, current flux through 3-faces and finite-chain charges supplies the smooth normalized metric volume form on an oriented Lorentzian four-manifold. Raising its indices uses the given inverse metric. On $N$ the smooth nonzero denominator permits the reciprocal and tensor formula, so they define smooth fields there. This construction does not assert a wave equation, vanishing or a Kerr characterization.

Appendix A.3 of Ionescu–Klainerman, printed p. 54, fixes $R_{abcd}=g(e_a,{\mathrm{\nabla }}_{e_c}{\mathrm{\nabla }}_{e_d}{e}_b-{\mathrm{\nabla }}_{e_d}{\mathrm{\nabla }}_{e_c}{e}_b-{\mathrm{\nabla }}_{[e_c,e_d]}{e}_b)$. This is the curvature operator of Stationary Lorentzian data and tensor conventions with its output index lowered: in a coordinate frame $R_{abcd}=g_{ae}{R}^e{}_{bcd}$. Thus the imported convention agrees with that supplier without a sign reversal or slot permutation.

Ionescu–Klainerman supplies the displayed definition formulas in §4.2–§4.3, printed pp. 22–24, and the reciprocal potential in §7, equation (7.3), printed p. 35. Appendix A.3, printed p. 54, supplies the curvature convention compared above. Its normalization is at infinity along the chosen section. The embedding and quantified angularly uniform limit above specify that convention locally. Only definitions are adopted from the source's vacuum setting; no potential-existence or analytic theorem is inferred.