Stationary end normalization from quotient compactification

Put $V=g(T,T)$, $a_i=g_{0i}\mathrm{/}V$, $h_{ij}=g_{ij}-g_{0i}{g}_{0j}\mathrm{/}V$, and $\bar{h}=(-V)h$. The given bounds yield $a=O_6(r^{-2})$, $h=\delta +O_6(r^{-1})$ and $V+1=O_6(r^{-1})$. The stationary vacuum twist equation gives a normalized twist with twice-differentiated decay on this spherical tail: its derivative is a contraction of $T,\mathrm{\nabla }T$ and the volume form, hence $O(r^{-3})$; its additive limit is set to zero. Thus the local-at-infinity hypotheses of source 0 hold with any $1\mathrm{/}2<\alpha <1$. Import its arbitrary finite-order expansions and $C^m$ conformal quotient compactification, choosing $m$ sufficiently large.

In the inverted compact coordinates let $\rho =1\mathrm{/}r$, rescaled potentials ${\mathrm{\Phi }}_M,{\mathrm{\Phi }}_S$ and $Q={\mathrm{\Phi }}_M^2+{\mathrm{\Phi }}_S^2$. The source mass coefficient is nonzero, so $Q(i)>0$. Dain's potential-defined factor is ${\mathrm{\Omega }}_D={\rho }^{2}2Q\mathrm{/}[B_0^2(\sqrt{1+4{\rho }^{2}Q}+1)]$, with $B_0^2=Q(i)$. Its ratio to ${\rho }^2$ is $C^m$, positive and one at $i$. It therefore preserves $C^m$ compactified regularity and the conformal Hessian normalization. This supplies Dain's actual $C^{4,\alpha }\mathrm{/}{C}^{2,\alpha }$ hypotheses, rather than merely assuming them. Import source 1 Theorem2.1 to obtain analytic quotient fields and use its orthonormal normal coordinates $x=\xi$. In this notation $\gamma ={\mathrm{\Omega }}_D^2\bar{h}$, with $\bar{h}=(-V)h_{\mathrm{q}\mathrm{u}\mathrm{o}\mathrm{t}}$ from step 1.1. The spatial chart is obtained by inversion of precisely these normal coordinates, so the compactification conditions and the chart identity in the Statement refer to the same constructed chart.

Source 1 Lemma2.5 constructs the convergent radial connection ${\beta }_i={ϵ}_{ijk}{f}_1^j(x)x^k+{\rho }^{-1}{ϵ}_{ijk}{f}_2^j(x)x^k$. Set $C_0=f_2(0)$ and invert to $y=x\mathrm{/}{\rho }^2$. The leading covector is radial-orthogonal, so the reflection part of the inversion differential leaves it unchanged; its physical components are ${\beta }_i={ϵ}_{ijk}{C}_0^j{y}^k\mathrm{/}{R}^3+O_{\mathrm{\infty }}(R^{-3})$. The convergent radial/angular expansions justify every finite derivative bound. Converting the source $+---$ convention to $-+++$ gives $g_{0i}=-\lambda {\beta }_i$, with $\lambda =1+O_{\mathrm{\infty }}(R^{-1})$. Define $S=C_0\mathrm{/}2$ to obtain the displayed coefficient, without equating differently normalized source spin symbols. The other reconstructed components have the asserted convergent expansions. Both old and new quotient metrics are asymptotically Euclidean, so their coordinate transition has bounded first derivative and inverse; the Christoffel transformation gives $D^{2}y=O(r^{-2})$. Integrating radially and comparing angles yields $y=Q_0{x}_{\mathrm{o}\mathrm{l}\mathrm{d}}+O(\log r)$, $Dy=Q_0+O(r^{-1})$, with $Q_0$ orthogonal. Thus $R\mathrm{/}r\to 1$. The scalar limit $r(V+1)\mathrm{/}2=M$ identifies the new leading mass with the old one; convergence of the new radial expansion supplies its $O_{\mathrm{\infty }}(R^{-2})$ remainder.

Both stationary connections reconstruct the same norm/twist/quotient fields. The stationary connection–twist relation in source 0, equations (2.36) and (2.70), therefore gives equal exterior derivatives, so their difference is closed. The spherical-tail exactness of [F2] gives the stationary time gauge; it preserves $T$. The new section $\mathrm{\Sigma }=\{t^{\mathrm{\prime }}=0\}$ is a smooth embedded hypersurface because it is a coordinate slice. Hereafter $h_{ij}=g_{ij}{\mathrm{\mid }}_{\mathrm{\Sigma }}$ denotes its induced metric, rather than the quotient metric in step 1.1. Step 3.1 gives $h_{ij}={\delta }_{ij}+O_{\mathrm{\infty }}(R^{-1})$. Increase the tail radius until $\mathrm{\parallel }h-\delta {\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 1\mathrm{/}2$ in Euclidean coordinates. Then $h_{ij}{v}^i{v}^j\ge \mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}2$ for every tangent vector $v$, so $\mathrm{\Sigma }$ is spacelike. The ambient tail is smooth and time-oriented, with $T$ future timelike; hence all hypotheses of [F4] hold. It supplies the smooth future unit normal $n$ and second fundamental form $K$, and the Statement's $k$ is $-K$. Put $X_i=g_{0i}$, $X^i=h^{ij}{X}_j$ and $N=(-g_{00}+h^{ij}{X}_i{X}_j{)}^{1\mathrm{/}2}$. These fields are smooth, $N=1+O_{\mathrm{\infty }}(R^{-1})>0$, and $n=N^{-1}(T-X^i{\mathrm{\partial }}_{y^i})$: this vector is unit, orthogonal to the slice, and future directed since $g(T,n)=-N<0$.

Let $D_{U}W$ be the tangential projection of ${\mathrm{\nabla }}_{U}W$. Ambient metric compatibility and torsion freedom imply the same properties for $D$ with respect to $h$, so $D$ is the induced metric connection. Expanding ${\mathcal{L}}_{T}g=0$ by metric compatibility and torsion freedom and substituting $T=Nn+X$ gives $0=2N{K}_{ij}+D_i{X}_j+D_j{X}_i$ on tangential pairs; derivatives of $N$ multiply $n$ and pair to zero. Thus $k_{ij}=-K_{ij}=(D_i{X}_j+D_j{X}_i)\mathrm{/}(2N)$ in exactly the Statement's convention. By steps 3.1 and 4.1 its leading term is $3[(S\times y{)}_i{y}_j+(S\times y{)}_j{y}_i]\mathrm{/}{R}^5$, with $O_{\mathrm{\infty }}(R^{-4})$ remainder, and its leading trace vanishes. Pair $k-{\mathrm{tr}}_{h}kh$ with $A\times y$ and the outward $h$-unit normal to a coordinate sphere and integrate using its $h$-area measure. Writing $\omega =y\mathrm{/}R$, the identity ${\int }_{S^2}{\omega }_i{\omega }_{j}d\omega =(4\pi \mathrm{/}3){\delta }_{ij}$ gives ADM angular momentum $A\cdot S$ after multiplication by $(8\pi {)}^{-1}$. The normal and area corrections are relative $O(R^{-1})$, and the $O(R^{-4})$ tensor remainder contributes $O(R^{-1})$, so all errors vanish in the limit. This fixes the physical spin coefficient and the exact AIK factor two.