Exact asymptotic and orbit package for the IK criterion

Fix a smooth connected orientable paracompact boundaryless, time-oriented Lorentzian four-manifold $(M,g,N)$ and a smooth Killing field $T$, with the stationary Lorentzian conventions. The IK AF package consists of an open end $M^{\mathrm{e}\mathrm{n}\mathrm{d}}\cong \mathbb{R}\times \{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>R\}$, with coordinates $(t,x)$, $r=\mathrm{\mid }x\mathrm{\mid }$, $T={\mathrm{\partial }}_t$, and $g_{00}=-1+2M\mathrm{/}r+O(r^{-2})$, $g_{ij}={\delta }_{ij}+O(r^{-1})$, $g_{0i}=O(r^{-2})$, where the constant $M>0$ is the mass parameter (distinct from the manifold denoted by the same letter).

Here $u=O(r^{-b})$ means $u(0,\cdot )=O_0(r^{-b})$ in the fixed spatial chart: some $R_1>R$ and finite $C\ge 0$ satisfy $\mathrm{\mid }u(0,x)\mathrm{\mid }\le C{r}^{-b}$ for all $r\ge R_1$, componentwise for an array. The metric remainders are independent of $t$ because $T={\mathrm{\partial }}_t$ is Killing, so the same bounds hold for every $t\in \mathbb{R}$, uniformly over spatial angular directions and time. No differentiated decay is asserted by this unindexed notation.

Set $E=I_M^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})\cap I_M^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$, using chronological reachability in $M$. Require an embedded spacelike section ${\mathrm{\Sigma }}_0\cong \{\mathrm{\mid }x\mathrm{\mid }>1\mathrm{/}2\}$ agreeing with $t=0$ on the end. Denote its unique smooth future unit normal by $T_0$, so $g(T_0,T_0)=-1$, $g(T_0,N)<0$ and $g(T_0,v)=0$ for $v\in T{\mathrm{\Sigma }}_0$.

For each $p\in E$, let ${\gamma }_p:I_p\to M$ be the unique maximal integral curve of $T$ with ${\gamma }_p(0)=p$. Require every such orbit to be complete, meaning $I_p=\mathbb{R}$, and to meet ${\mathrm{\Sigma }}_0$. Require also $\mathrm{\mid }g(T,T_0)\mathrm{\mid }>0$ on ${\mathrm{\Sigma }}_0\cap E$. These requirements are assumptions, not consequences of mere stationarity or leading asymptotic decay.

Stationary Lorentzian data and tensor conventions

Causal exterior, chronological boundaries and Cauchy data

Differentiated asymptotic bounds and parameter uniformity

The future unit normal, induced metric and second fundamental form of a spacelike hypersurface

Through each point there is a unique maximal integral curve

Given: The manifold, field, end chart and spacelike section specified in the Definition.

1.1 The chronological operations determine $E$ in the ambient manifold. The normal lemma applies to the time-oriented Lorentzian manifold and embedded spacelike section, giving the stated unique smooth $T_0$. The maximal-integral-curve theorem applies to the smooth $T$ on $M$, so $I_p=\mathbb{R}$ and intersection with ${\mathrm{\Sigma }}_0$ are unambiguous additional predicates for each $p\in E$. Nonvanishing of $g(T,T_0)$ is the separately imposed transversality condition; no existence of an AF package or completeness theorem is asserted. [given, F1, F2, F4, F5]

2.1 In the end chart the local flow of $T={\mathrm{\partial }}_t$ sends $(t,x)$ to $(t+s,x)$. Differentiating its metric pullback at $s=0$ by the Lie-derivative definition gives $0=({\mathcal{L}}_{T}g{)}_{\mu \nu }={\mathrm{\partial }}_t{g}_{\mu \nu }$. Thus each metric remainder has the same value at $(t,x)$ as at $(0,x)$. Applying the $m=0$ decay convention to the time-zero remainder gives the same radius and constants for all $t$, exactly the ordinary bounds defined above. The package therefore specifies definite data and predicates without deriving any of its existence or orbit assumptions from the asymptotic estimates. [given, F1, F3, step 1.1] ∎

The independent Ionescu–Klainerman source supplies the AF predicates in section 1.2, equations (1.4)–(1.5), printed/PDF pp. 6–7. Only that definition package is imported; no vacuum equation, existence theorem or analytic continuation result is inferred here.