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
- Temporal normal angle and compact positive-height portions
- A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Real powers for positive bases, with the zero-base positive-exponent convention
- Positive real powers preserve measurability
- Continuity and derivatives of positive-base real powers
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
- The exponential definition of real powers agrees with the existing rational powers
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous
- Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined
- General spacelike two-surfaces, null gauges and fluxes
- Outgoing expansion variation at a general MOTS
- Weighted exterior elliptic domains and admissible weights
- Fredholm and Banach continuation interfaces
- Differentiated asymptotic bounds and parameter uniformity
- Scalar exterior inverse from exhaustion and uniform annular estimates
- Compact extra components contribute index zero
- Compact multiplication by a faster-decaying potential
- Fredholm index is stable under compact perturbations
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Causal exterior, chronological boundaries and Cauchy data
- The inverse metric, musical isomorphisms and index raising
- Compact sets admit relatively compact open neighborhoods and finite chart pieces
- Compact timelike-length bounds restrict to open subdomains
- A temporal covector has uniform compact negativity and orientation margins
- Inversion is uniformly stable on compact chart families
- A compact temporal covector persists under metric perturbation
- The Axiom of Countable Choice (AC_ω)
- A. F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Appendix B, Lemma B.2, equation (B.2)
- Compact-domain comparison with a nonnegative potential
- Strong minimum principle from the annular barrier
- 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 push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- The Axiom of Choice
- Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points
- The differential of the squared-displacement function and the position vector of the reversed geodesic
- Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic
- AC implies DC implies countable choice
- Pairing of an observer with future causal vectors
- Every tangent vector is the velocity of a smooth curve
- Fermat's interior extremum theorem: if f has a local extremum at a point c interior to its domain and is differentiable at c, then f'(c) = 0
- The differential of a smooth real-valued function
- The velocity derivation of a smooth curve
- Future null geodesic completeness and incompleteness
- Collar barrier against finite section-time boundary arrival
- Proper end radius and exclusion of finite-time spatial infinity
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- The interior is an open smooth n-manifold
- A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- Temporal angle and tangential height gradient identity
- From hyperbolic angle to logarithmic angle inequalities
- The Hessian is negative semidefinite at an interior local maximum
- Gradient hessian and divergence connection formulas
- Laplace–Beltrami operator as the trace of the Hessian
- Sum, product, quotient, and chain rules from difference quotients
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- The natural logarithm as the inverse of the exponential function
- Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
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
Compact weighted maximum and the actual boundary term
Lemma
MOTS boosts preserve the spacetime stability sign
Lemma
Exterior Dirichlet parametrix and compact decaying potentials
Theorem
Compact timelike-length bounds and local temporal functions
Lemma
Exterior exhaustion bounds for normalized Schrödinger solutions
Lemma
The entire target segment stays on the chronological boundary
Lemma
Endpoint orthogonality without an acausal target assumption
Lemma
Combining the collar and end barriers into interior compact confinement
Lemma