Stationary ADM speed bounds and Cauchy sections

Assume the Axiom of Choice. Let $\bar{\mathrm{\Sigma }}$ be a connected smooth three-manifold with compact boundary $B$ and interior ${\mathrm{\Sigma }}^{\circ }$. Let $h$ extend to a smooth positive-definite metric on $\bar{\mathrm{\Sigma }}$, complete up to its compact boundary, meaning that its intrinsic metric with the boundary included is complete. Let $N:\bar{\mathrm{\Sigma }}\to \mathbb{R}$ be a smooth real-valued function and $X$ a smooth tangent vector field on $\bar{\mathrm{\Sigma }}$, with $N>0$ in the interior. On $E={\mathbb{R}}_v\times {\mathrm{\Sigma }}^{\circ }$ put $g=-N^{2}d{v}^2+h_{ij}(d{x}^i+X^{i}dv)(d{x}^j+X^{j}dv)$, oriented by $n=N^{-1}({\mathrm{\partial }}_v-X)$; all coefficients are independent of $v$.

Supply a collar $[0,ϵ)\times B$ with coordinate $\rho$, and constants with $N\le A\rho$, $\mathrm{\mid }X(\rho )\mathrm{\mid }\le A\rho$, and $\mathrm{\mid }d\rho {\mathrm{\mid }}_h\le A$ there. Supply a smooth nonnegative proper exhaustion $R$ of $\bar{\mathrm{\Sigma }}$: each $\{R\le m\}$ is compact. Outside a compact core require $\mathrm{\mid }dR{\mathrm{\mid }}_h(N+\mathrm{\mid }X{\mathrm{\mid }}_h)\le C$. A sufficient end-coordinate realization is to provide a compact core with finitely many ends $[r_0,\mathrm{\infty })\times F_j$, compact $F_j$, let $R$ equal their unbounded radial coordinates sufficiently far out, and impose this estimate there. A bounded function of radius is insufficient. If $B$ is empty, omit the collar conditions.

Then each inextendible causal curve, with locally absolutely continuous nonzero causal velocity almost everywhere, has $v$-image all of $\mathbb{R}$. Consequently every $v$-level is a spacelike Cauchy hypersurface, and $(E,g,n)$ is globally hyperbolic. This is completeness of causal section time, not completeness of geodesic affine parameters. The smoothness and proper-exhaustion conditions explicitly specify all escape mechanisms; spatial metric completeness alone does not replace the proper-radius estimate.

math:def-axiom-of-choice

AC implies DC implies countable choice

Causal exterior, chronological boundaries and Cauchy data

Spacelike sections, maximality and lapse–shift data

ADM section time and causal spatial speed

Combining the collar and end barriers into interior compact confinement

An interior compact causal trajectory has a finite-time endpoint

The existence of a Cauchy set implies strong causality and compact causal diamonds

Proper end radius and exclusion of finite-time spatial infinity

The proper-exhaustion version is the general contract; the finite product-end version is a sufficient way to supply it, rather than a consequence of an arbitrary radial notation. The theorem does not construct a stationary product from a stationary flow, and does not assert existence of a maximal section. Applications to a transformed geometry must separately supply the global product, the smooth boundary/collar data and the actual proper end radius for each fixed parameter. No uniform estimate as a deformation parameter tends to zero is asserted. The spatial completeness hypothesis is retained from the scaffold, although the compact-exhaustion proof does not need it once the explicit escape controls are supplied.