Exact asymptotically stationary end contract

Use the smooth time-oriented Lorentzian data $(M,g,N,T)$ and causal-curve conventions supplied by the Lorentzian and causal interfaces. Here acausal means that no two distinct points are joined by a causal curve. A hypersurface, possibly with boundary, means an embedded smooth three-dimensional submanifold of the four-manifold $M$, with a supplied manifold-with-boundary smooth structure; its codimension is $4-3=1$. A Chruściel–Wald $(k,\alpha )$-asymptotically stationary end consists of such an acausal hypersurface $\mathrm{\Sigma }={\mathrm{\Sigma }}_1\bigsqcup {\mathrm{\Sigma }}^{\mathrm{\prime }}$ and the following data and requirements. The remainder ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is supplied as an embedded smooth three-dimensional submanifold with boundary of $\mathrm{\Sigma }$; its boundary means its intrinsic manifold boundary, with the induced smooth two-manifold structure, rather than its topological boundary in $M$. Its inclusion in $M$ is also required to be a smooth embedding.

(1) Supply a diffeomorphism $e:E\to {\mathrm{\Sigma }}_1$, where $E=\{z\in {\mathbb{R}}^3:\mathrm{\mid }z\mathrm{\mid }>R_1\}={\mathbb{R}}^3\setminus B(R_1)$, $R_1>0$, and $B(R_1)$ is the closed Euclidean ball. Thus ${\mathrm{\Sigma }}_1$ has the smooth structure of an open end of $\mathrm{\Sigma }$. A coordinate system $(t,x^1,x^2,x^3)$ on an open neighborhood $O_1$ of ${\mathrm{\Sigma }}_1$ obeys the following estimates for constants $C>0$, integer $k\ge 2$, $0<\alpha \le 1$ and $0<\lambda \le 1$. Put $r=({\sum }_i(x^i{)}^2{)}^{1\mathrm{/}2}$ and require $r>0$ throughout $O_1$. The model coordinate $z$ and the spacetime coordinates $(t,x)$ are separately supplied: no identity $x=e^{-1}$, level-set condition $t{\mathrm{\mid }}_{{\mathrm{\Sigma }}_1}=0$, or inequality $r>R_1$ on $O_1$ is assumed. Put $\eta =\mathrm{diag}(-1,1,1,1)$; indices $\mu ,\nu ,{\sigma }_j$ range from $0$ to $3$, and $i,j$ from $1$ to $3$:

$g_{00}<-C^{-1},\mathrm{\mid }{g}_{\mu \nu }-{\eta }_{\mu \nu }\mathrm{\mid }\le C{r}^{-\alpha },g_{ij}{Z}^i{Z}^j\ge C^{-1}{\sum }_i(Z^i{)}^2(Z\in {\mathbb{R}}^3)\mathrm{.}$

$\mathrm{\mid }{\mathrm{\partial }}_{{\sigma }_1}\cdots {\mathrm{\partial }}_{{\sigma }_m}{g}_{\mu \nu }\mathrm{\mid }\le C{r}^{-\alpha -m}(1\le m\le k)\mathrm{.}$

Writing ${\mathrm{\partial }}^k$ for any ordered $k$ coordinate derivatives, require

$\mathrm{\mid }({\mathrm{\partial }}^k{g}_{\mu \nu })(t,x)-({\mathrm{\partial }}^k{g}_{\mu \nu })(\tau ,y)\mathrm{\mid }\le Cr(x{)}^{-\alpha -k-\lambda }(\mathrm{\mid }t-\tau {\mathrm{\mid }}^{\lambda }+\mathrm{\mid }x-y{\mathrm{\mid }}^{\lambda })$

whenever both points lie in $O_1$, $\mathrm{\mid }t-\tau \mathrm{\mid }\le 1$ and $\mathrm{\mid }x-y\mathrm{\mid }\le r(x)\mathrm{/}2$. Absolute values on components are ordinary real absolute values and $\mathrm{\mid }x-y\mathrm{\mid }$ is the Euclidean norm; the estimate holds for each component and derivative choice.

(2) Let $\bar{E}=\{z\in {\mathbb{R}}^3:\mathrm{\mid }z\mathrm{\mid }\ge R_1\}$ have its usual smooth manifold-with-boundary structure, and let $S$ be a connected component of the intrinsic boundary $\mathrm{\partial }{\mathrm{\Sigma }}^{\mathrm{\prime }}$. Supply a diffeomorphism $h:\mathrm{\partial }\bar{E}\to S$, meaning a bijective smooth map with smooth inverse. Its realization in the given smooth hypersurface is part of the data: supply a smooth embedding $j:\bar{E}\to \mathrm{\Sigma }$ whose restriction to $E$ is $e$ followed by the inclusion of ${\mathrm{\Sigma }}_1$, and whose restriction to $\mathrm{\partial }\bar{E}$ is $h$ followed by the inclusion of $S$. Require $j(\bar{E})\cap {\mathrm{\Sigma }}^{\mathrm{\prime }}=S$ and $j(\bar{E})\cup {\mathrm{\Sigma }}^{\mathrm{\prime }}=\mathrm{\Sigma }$. These maps specify the smooth boundary identification implicit in the source's disjoint end/remainder notation. No compactness of ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is imposed, and existence of such a gluing realization is required data, not an asserted gluing theorem.

(3) The smooth Killing field $T$ on all of $M$ has a complete flow ${\varphi }_u$, $u\in \mathbb{R}$, satisfies $g(T,T)<-C^{-1}$ on ${\mathrm{\Sigma }}_1$, and has asymptotics

$T=(A+O(r^{-\alpha })){\mathrm{\partial }}_t+{\sum }_{i}O(r^{-\alpha }){\mathrm{\partial }}_{x^i},A\mathrm{\ne }0.$

Here a complete flow means a smooth global flow of $T$: the map $\varphi :\mathbb{R}\times M\to M$, $\varphi (u,p)={\varphi }_u(p)$, is smooth, ${\varphi }_0={\mathrm{id}}_M$, $\frac{d}{du}{\varphi }_u(p)=T_{{\varphi }_u(p)}$, and ${\varphi }_{u+v}={\varphi }_u\circ {\varphi }_v$ for all real $u,v$. The component $O(r^{-\alpha })$ bounds hold uniformly in the end coordinates as $r\to \mathrm{\infty }$. Constants may be enlarged to one common $C$. These are imposed data, not consequences of the word stationary.

Stationary Lorentzian data and tensor conventions

Causal exterior, chronological boundaries and Cauchy data

Embedded smooth submanifolds with boundary

Codimension and hypersurfaces

Diffeomorphisms and local diffeomorphisms of manifolds

The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold

Local and global flows generated by a vector field

Given: The displayed smooth structures, maps, coordinate domain and conditions.

Goal: Define a precise end contract without asserting existence for arbitrary stationary data.

1.1 The Lorentzian and causal conventions give meanings to the metric, Killing condition and acausality. The supplied embeddings make $\mathrm{\Sigma }$ a codimension-one smooth submanifold, and give ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ and $j(\bar{E})$ their stated manifold-with-boundary structures. These are requirements on the data, so no smooth structure is inferred from an arbitrary set-theoretic remainder. [given, F1, F3]

1.2 In the chosen spacetime chart every derivative and norm in (1) is a derivative or norm of a specified real component. The explicit condition $r>0$ on all of $O_1$ makes each negative radial power defined, including at both points in the Hölder condition. This uses no relation between the model radius $\mathrm{\mid }z\mathrm{\mid }$ and the coordinate radius $r$, and does not infer $r>R_1$ from the topology of ${\mathrm{\Sigma }}_1$. [given, F1, F2]

2.1 The supplied boundary diffeomorphism $h$ is between smooth two-manifolds; the restrictions of $j$ identify its boundary values with that map and its interior values with the end model. Thus (2) is a definite compatibility condition on supplied smooth maps. Clause (3) separately requires the smooth global flow generated by $T$ in [F5] and its component bounds; a local Killing field alone does not supply it. The conjunction retains the cited metric estimates, Hölder condition, boundary identification and uniform timelikeness, with the necessary structural and domain conventions explicit and with no existence conclusion. [given, F1, F2, F3, F4, F5, step 1.1, step 1.2] ∎

The end estimates, boundary identification and Killing-field conditions are imported from Chruściel–Wald Definition 2.1, printed/PDF pp. 8–9. The preceding paragraph supplies its embedded codimension-one hypersurface convention. The smooth boundary realization and positive-r domain are explicit formulation data here; no coordinate or gluing existence result is imported or proved.