Alphabeta MathMaximal slices, causal cuts and stability
Assemble the analytic and geometric inputs for maximal-slice continuation, causal completeness, null-boundary cuts, area comparison and horizon flux rigidity. Use the common data and the preceding field inputs while retaining every local argument and imported-premise qualification.
1 · Prerequisites
- Scalar metric Jacobian and weak-gradient identification
- Riemannian metric and riemannian manifold
- Lipschitz height functions, cut measure and Jacobians
- Riemannian speed and length
- Riemannian distance on a connected manifold
- Extended riemannian distance on a disconnected manifold
- The riemannian distance topology is the manifold topology
- Local comparison of a riemannian metric with the euclidean metric
- Riemannian volume density
- Intrinsic density measure and its chart restriction
- Smooth charts, atlases, and structures with boundary
- Unnormalised Hausdorff measure
- Hausdorff content at a prescribed scale
- Hausdorff measure is metric and measures every Borel set
- Positive-codimension immersed submanifolds are null
- Beppo Levi's theorem for nonnegative series
- The Axiom of Choice
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- Liviu I. Nicolaescu, The co-area formula (revision January 22, 2021) · Theorem 5.3, p. 14, equation (5.1); Section 3, p. 8, definition and Example 3.1(a).
- Causal exterior, chronological boundaries and Cauchy data
- Null expansions, screen quotients and focal data
- A compact chronological target supplies a causal endpoint
- A compact-target boundary point has a null geodesic segment
- The entire target segment stays on the chronological boundary
- Endpoint orthogonality without an acausal target assumption
- Boundary membership excludes interior focal points from the target
- Achronal boundaries are locally Lipschitz graphs, topological hypersurfaces and null for smooth volume densities
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- E. Minguzzi, Lorentzian causality theory · Definition 2.86 and Theorem 2.87(iii)
- Surface measure, flux densities and null normal gauge
- Geometric electrovacuum equations and charge convention
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- The co-area formula · Liviu I. Nicolaescu, Theorem 5.3 and Corollary 5.4, pp. 14–15; Lemma 1.1, p. 3; Section 3, p. 8
- Outgoing expansion depends smoothly on the second graph data
- Linear and path-independent normal-graph variation of expansion
- Canonical outward normals and null gauge on a graph
- The chain rule for differentials of smooth maps
- Elliptic, hyperbolic, and parabolic principal symbols
- General spacelike two-surfaces, null gauges and fluxes
- Outgoing expansion variation at a general MOTS
- MOTS boosts preserve the spacetime stability sign
- The Axiom of Countable Choice (AC_ω)
- Formal adjoint of a real drift Laplacian on a closed manifold
- Positive right and adjoint eigenfunctions have the same eigenvalue
- Ground-state identity, spectral ordering and eigenspace uniqueness
- Nonnegative supersolutions detect the principal spectral sign
- Imported positive eigenpairs for a smooth closed elliptic operator
- Lars Andersson, Marc Mars and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes · §2 null-frame conventions, printed p.5; §3 variation definition, printed p.7, equation (2), Definition 3.1 equations (5) and Lemma 3.2 equation (7), printed p.8; §5 Definition 5.1 and Proposition 5.1 with proof, printed pp.14–15
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Removing the boundary before applying scalar coarea
Lemma
Compact-target null generators and the required endpoint conditions
Lemma
Outward unit variation in the normalized null basis
Lemma
Electrovac contraction in the normalized normal plane
Lemma
Riemannian Lipschitz coarea on compact interior bands
Lemma
Outward graph stability has the negative-Laplacian principal symbol
Definition
Spacetime stably outermost condition and the slice bridge
Theorem