Causal and Cauchy-surface inputs for smooth electrovacuum developments

Assume the Axioms of Countable Choice and Dependent Choice. Let $(M,g,N)$ be a time-oriented smooth globally hyperbolic Lorentzian four-manifold with the causal-curve and global-hyperbolicity conventions of Chronology, causality, strong causality, global hyperbolicity and time functions and Causal curves, chronological and causal reachability, and inextendibility. The following interfaces are available for the local and global geometry of smooth electrovacuum developments:

1. Every point has an arbitrarily small strictly convex normal neighborhood. For a future causal curve $\sigma$ in such a neighborhood from $p$, its length satisfies $L_g(\sigma )\le \sqrt{-g_p(v,v)}$, where $v={\exp }_p^{-1}(\sigma (1))$. Equality is characterized by the reparametrized radial geodesic when $v$ is future timelike and the curve does not return to $p$ after its start. The future causal and timelike sets are the exponential images of their tangent cones.
2. Each connected component admits a complete auxiliary Riemannian metric. The causal relation is closed, and the causal hull $J^{+}(K_1)\cap J^{-}(K_2)$ of two compact sets is compact.
3. Let $h$ be the complete auxiliary metric $h\mathrm{\prime }$ of The auxiliary Riemannian metric h=g+2n-flat tensor n-flat is complete after a proper-exhaustion conformal factor, and its closed bounded sets are compact clause (b), and let ${\alpha }_k:[0,b_k]\to M$, $0<b_k<\mathrm{\infty }$, be future causal curves parametrized by $h$-arclength, with ${\alpha }_k(0)\to x$ and ${\alpha }_k(b_k)\to z$ in $M$. After passage to a subsequence with $b_k\to b\in [0,\mathrm{\infty }]$, the bounded-length, zero-length and unbounded-length conclusions of Two-endpoint limit curve theorem: bounded lengths give joining limits, unbounded lengths give two inextendible tails with the closure relation hold: for $0<b<\mathrm{\infty }$ the common-domain reparametrizations converge uniformly to a joining causal curve, for $b=0$ the curves converge uniformly on their varying domains to $x=z$, and for $b=\mathrm{\infty }$ the forward and backward tails converge on compact parameter intervals to the respective inextendible curves with the supplier's sequential closure relation. Strong causality excludes the unbounded case when all images lie in one compact set.
4. Lorentzian time separation is finite on causal pairs in a globally hyperbolic spacetime, is attained by a causal geodesic, is upper semicontinuous, and is continuous at timelike-related points, with the degenerate coincident-endpoint case as specified in Time separation in a globally hyperbolic manifold: finiteness, existence of maximizing geodesics and the radial formula on causally convex normal regions.
5. Every smooth embedded spacelike hypersurface has a variable-width temporal neighborhood whose intrinsic domain-of-dependence interior is globally hyperbolic with that hypersurface Cauchy. Every globally hyperbolic spacetime also admits a smooth Cauchy temporal splitting as in Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow.

These are mathematical causal inputs. They add no field equations, empirical assumptions, completeness conclusion for causal geodesics, or uniform time-width assertion for noncompact hypersurfaces.