Complete regular static exterior data and its normalized end

Regular static exterior data consist of a connected oriented three-manifold $\bar{X}$ with nonempty compact smooth boundary $B$, a positive-definite metric $h$ and real function $V$ which are smooth on $X=\bar{X}\setminus B$ and at least $C^2$ on $\bar{X}$. Require $V>0$ on $X$, $V=0$ and $dV\mathrm{\ne }0$ on $B$. Completeness up to boundary means completeness of the intrinsic $h$-distance on $\bar{X}$, with boundary included. A single asymptotically Euclidean end means that the complement of a compact subset of $\bar{X}$ is diffeomorphic to $\{x\in {\mathbb{R}}^3:r=\mathrm{\mid }x\mathrm{\mid }>R_0\}$, with no additional ends or deleted interior boundary. In that chart require $h_{ij}-{\delta }_{ij}=O_2(r^{-1})$ and $V=1-m\mathrm{/}r+O_2(r^{-2})$ for a real number $m$ not initially assumed positive.

The associated globally static spacetime exterior is ${\mathbb{R}}_t\times X$ with $g=-V^{2}d{t}^2+h$, future orientation ${\mathrm{\partial }}_t$ and stationary field $T={\mathrm{\partial }}_t$. The boundary $B$ is a boundary of the spatial data; this product formula does not define a nondegenerate Lorentzian metric at $V=0$. Write $D$ for the Levi–Civita connection of $h$, $D^{2}V(A,C)=A(CV)-(D_{A}C)V$ for its Hessian and ${\mathrm{\Delta }}_{h}V={\mathrm{tr}}_h{D}^{2}V$ for its Laplacian.

Stationary Lorentzian data and tensor conventions

Differentiated asymptotic bounds and parameter uniformity

Spacelike sections, maximality and lapse–shift data

Laplace–Beltrami operator as the trace of the Hessian

Smooth charts, atlases, and structures with boundary

Riemannian distance on a connected manifold

Given: The data in the Definition and the interfaces in [F1]–[F10].

Goal: Justify the object and completeness conventions without asserting an extension across a horizon.

1.1 Use the smooth boundary charts of [F4] and their interior counterparts. For a piecewise $C^1$ curve $\gamma :[a,b]\to \bar{X}$, set $L_h(\gamma )={\int }_a^b\sqrt{h_{\gamma (s)}(\dot{\gamma }(s),\dot{\gamma }(s))}ds$, summed over its finitely many $C^1$ pieces, and define $d_h(p,q)$ as the infimum of these lengths for curves joining $p$ to $q$, as in [F5]. Small coordinate balls or half-balls are convex, so any two points in one such chart neighborhood can be joined by a smooth coordinate segment. For fixed $p$, the set reachable from $p$ by finitely many such segments is open; its complement is open by the same local segment property. Connectedness therefore makes every point reachable. Each finite piece has finite length by continuity of $h$ on its compact trace. Thus each infimum is finite and nonnegative, using only continuity of $h$ up to $B$. [given, F4, F5]

1.2 The compact-core/end-chart requirement is a global topological condition, not a consequence of pointwise estimates. The differentiated $O_2$ notation fixes all Cartesian derivatives up to second order and is uniform in angular directions for each fixed datum. On the smooth interior the trace-of-Hessian definition fixes the sign of ${\mathrm{\Delta }}_h$ and agrees with ${\mathrm{div}}_h{\mathrm{grad}}_h$. [given, F2, F3]

2.1 Constant curves give $d_h(p,p)=0$, and reversal preserves length, so $d_h$ is symmetric. Length is additive under concatenation by splitting the defining integrals. Given $\epsilon >0$, curves from $p$ to $q$ and from $q$ to $r$ with lengths less than the respective infima plus $\epsilon$ yield $d_h(p,r)\le d_h(p,q)+d_h(q,r)+2\epsilon$. Letting $\epsilon$ decrease to zero gives the triangle inequality. [given, step 1.1]

2.2 On $X$ positivity of $V$ gives a smooth nondegenerate product tensor of signature $(-,+,+,+)$; $g(T,T)=-V^2<0$ and its coefficients are independent of $t$. Thus $T$ is Killing, complete on this product, and hypersurface orthogonal since its metric dual is $-V^{2}dt$. The construction asserts neither regular Lorentzian coordinates at $B$ nor a maximal analytic extension. [given, F1, step 1.2]

3.1 For distinct $p,q$, take a coordinate ball or boundary half-ball centered at $p$, of radius $\rho >0$, whose compact closure lies in the chart and excludes $q$. Continuity and positive definiteness give a positive minimum $c$ of $h_x(v,v)$ over this compact coordinate closure and the Euclidean unit sphere, hence $h_x(v,v)\ge c\mathrm{\mid }v{\mathrm{\mid }}^2$. Every connecting curve from $p$ to $q$ first exits this relative ball through its spherical boundary; at a boundary chart the flat face lies in the manifold and is not an exit. The coordinate displacement of the initial curve is $\rho$, so integration of its derivative gives $L_h(\gamma )\ge \sqrt{c}\int \mathrm{\mid }\dot{\gamma }\mathrm{\mid }ds\ge \sqrt{c}\rho$. Taking the infimum proves $d_h(p,q)>0$. Consequently $d_h$ is a metric also at boundary points for the stated $C^2$ data, and completeness means that every $d_h$-Cauchy sequence converges in $\bar{X}$, with $B$ included, consistently with [F6]. Completeness itself remains a requirement on the data. [given, F4, F6, step 1.1, step 2.1]

4.1 On the smooth interior product take $N=T$. Direct contraction with $g=-V^{2}d{t}^2+h$ gives $\mathrm{\nabla }t=-V^{-2}T$, hence $g(\mathrm{\nabla }t,\mathrm{\nabla }t)=-V^{-2}<0$ and $dt(N)=1>0$. Thus $t$ is temporal by [F8]. The standing ZFC convention in [F1] supplies the choice hypotheses of [F9] by [F10]. For every future causal curve $\gamma :I\to {\mathbb{R}}_t\times X$ in [F7], the function $t\circ \gamma$ is therefore locally absolutely continuous and has positive derivative almost everywhere by [F9]. Such a curve is nonconstant on every nondegenerate subinterval by [F7], so [F9] gives $t(\gamma (a))<t(\gamma (b))$ whenever $a<b$ in $I$. This proves the claimed strict increase along the declared causal-curve class. [given, F1, F7, F8, F9, F10, step 2.2, algebra] ∎