Ernst and self-dual curvature data in the AIK convention

For the AIK perturbative GR/SBS package, let $U=\tilde{M}$ be the open neighborhood of ${\mathrm{\Sigma }}^0\cap \bar{E}$ constructed in AIK §2.1. Use the normalized smooth Ernst potential $\sigma =f+i\psi$ on $U$, with $f=-g(T,T)$, $d\sigma =2{\iota }_T\mathcal{F}$ and $\sigma \to 1$ at infinity. Here $F_{ab}={\mathrm{\nabla }}_a{T}_b=\frac{1}{2}(d{T}^{\mathrm{♭}}{)}_{ab}$, $\mathcal{F}=F+i{\star }_{g}F$, ${\mathcal{F}}^2={\mathcal{F}}_{ab}{\mathcal{F}}^{ab}$ uses complex bilinear contraction, and the spacetime orientation fixes $ϵ$ and ${\star }_g$. Existence on $U$ is imported explicitly in External input; the definition of a potential alone asserts no existence.

For normalization, designate the product end $U_{\mathrm{e}\mathrm{n}\mathrm{d}}={\mathrm{\Phi }}_0(I\times E_{2R_1})\subset U$, where $I=(-{\epsilon }_0,{\epsilon }_0)$ and $R_1,{\epsilon }_0$ are from the AIK §2.1 construction. In its source coordinates $(t,x)$, put $y=x$ when $dt\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$ is positive for the given spacetime orientation; otherwise put $y=(-x^1,x^2,x^3)$. Use $(t,y)$ as the designated normalization chart, so $dt\wedge d{y}^1\wedge d{y}^2\wedge d{y}^3$ is positive, $\mathrm{\mid }y\mathrm{\mid }=\mathrm{\mid }x\mathrm{\mid }$ and $T={\mathrm{\partial }}_t$. The GR/SBS source chart and its component hypotheses remain the original $(t,x)$ chart.

Write $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ for the stationary-data curvature operator. Define the source covariant array by $R^{\mathrm{A}\mathrm{I}\mathrm{K}}(X,Y,Z,W)=g(X,R(Z,W)Y),R_{abcd}^{\mathrm{A}\mathrm{I}\mathrm{K}}=g_{ae}{R}^e{}_{bcd}\mathrm{.}$ This fixes all four slots and the sign explicitly; its identification with the AIK convention uses the curvature formula in §5.1 after (5.10), rather than (2.16) or a contraction with $T$ alone. Define ${\mathcal{R}}_{abcd}=R_{abcd}^{\mathrm{A}\mathrm{I}\mathrm{K}}+\frac{i}{2}{ϵ}_{cd}{}^{ef}{R}_{abef}^{\mathrm{A}\mathrm{I}\mathrm{K}},{\mathcal{I}}_{abcd}=\frac{1}{4}(g_{ac}{g}_{bd}-g_{ad}{g}_{bc}+i{ϵ}_{abcd})\mathrm{.}$ Thus the Hodge operation on curvature is in its last alternating pair, all raised indices use this $g$, and $\mathcal{I}$ is the displayed covariant four-tensor. The source identity (2.20), ${\mathrm{\nabla }}_{\mu }{\mathcal{F}}_{ab}=T^{\nu }{\mathcal{R}}_{\nu \mu ab}$, is an imported identity in this convention, not a definition of the full curvature array or a locally proved differential identity. No conversion to a different curvature convention is asserted.

Given: The GR/SBS data, the declared mathematical conventions and the two explicit source inputs.

Goal: Establish the displayed constructions and their compatibility with the source convention.

1.1 Since $F$ is alternating and the connection is torsion-free, $(d{T}^{\mathrm{♭}}{)}_{ab}={\mathrm{\nabla }}_a{T}_b-{\mathrm{\nabla }}_b{T}_a=2F_{ab}$. The GR/SBS data meet the hypotheses of the imported neighborhood-potential input [F3], so $\sigma$ is a supplied smooth function on $U$, with the stated real part, differential and slice normalization. The same source construction supplies $U_{\mathrm{e}\mathrm{n}\mathrm{d}}={\mathrm{\Phi }}_0(I\times E_{2R_1})\subset U$ with $T={\mathrm{\partial }}_t$. This end is connected, so the nowhere-zero form $dt\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$ has constant orientation sign there. In the negative case the fixed reflection in the Definition multiplies that form by $-1$; in the positive case the chart is unchanged. Both choices are smooth coordinate diffeomorphisms onto $I\times E_{2R_1}$, preserve the radius and $t$, and give the positive coordinate volume form required by [F2]. This changes only the designated coordinates, not the spacetime orientation, Hodge star or potential. By alternation, $T^a{\mathrm{\nabla }}_a\sigma =2T^a{T}^b{\mathcal{F}}_{ba}=0$. Since $T={\mathrm{\partial }}_t$ on this product end, $\sigma (t,y)=\sigma (0,y)$ for every $t\in I$. For any $\delta >0$, the imported slice limit gives $R>2R_1$ such that $\mathrm{\mid }\sigma (0,x)-1\mathrm{\mid }<\delta$ whenever $\mathrm{\mid }x\mathrm{\mid }>R$. The equality of radii and the independence of $t$ therefore give $\mathrm{\mid }\sigma (t,y)-1\mathrm{\mid }<\delta$ for every $t\in I$ and $\mathrm{\mid }y\mathrm{\mid }>R$, exactly the uniform end normalization of [F2]. This uses the imported existence result, without claiming a local topological or potential-existence proof. [F1, F2, F3, algebra]

1.2 For any smooth local frame $(v_a)$, substitution of the curvature operator of [F1] into the covariant array gives $g(v_a,({\mathrm{\nabla }}_{v_c}{\mathrm{\nabla }}_{v_d}-{\mathrm{\nabla }}_{v_d}{\mathrm{\nabla }}_{v_c}-{\mathrm{\nabla }}_{[v_c,v_d]})v_b)$, exactly the source convention in [F4]. Thus this identification fixes every slot without inferring the full array from a contraction with $T$. The array is a smooth covariant tensor and is alternating in its last two slots because $R(Z,W)=-R(W,Z)$ directly from its definition. Applying the smooth Hodge operation to those two slots is therefore intrinsic and well defined. [F1, F2, F4, algebra]

2.1 Metric tensor products, the oriented metric volume form, contractions and constant complex linear combinations preserve tensor type and smoothness. Hence the displayed $\mathcal{R}$ and $\mathcal{I}$ are smooth covariant four-tensors independent of the frame, and the displayed scalar contraction has precisely the complex bilinear meaning of [F2]. Together with the supplied smooth potential this proves well-definedness of the package. Identity (2.20) retains its imported status [F4]; it is not used to infer its own curvature convention or to prove potential existence. [F1, F2, F4, step 1.1, step 1.2] ∎

The first statement-bearing AIK source record supplies the neighborhood potential, its normalization and the stationary product end in the neighborhood construction under GR/SBS. Its source argument is accepted as an external input; no local potential-existence proof is claimed.

The second statement-bearing AIK source record supplies the explicit curvature convention and identity (2.20). The identification with the stationary-data operator and the well-definedness of the displayed constructions are justified locally above. This definition imports no global rigidity theorem.