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
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Null normal pairs and screen geometry
- Metric compatibility, torsion freedom and the metric-selected connection class
- Riemannian volume form on an oriented manifold
- Integration on an oriented embedded submanifold
- Lorentzian exterior calculus and Hodge star
- Affine connection on a smooth manifold
- Pullback connection
- Pullback connection is well defined and functorial
- Integral of a compactly supported smooth density
- Orientation-free density integration and its properties
- The Axiom of Countable Choice (AC_ω)
- Spacelike sections, maximality and lapse–shift data
- Normal graphs and the nonlinear mean-curvature map
- Stationary Lorentzian data and tensor conventions
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- Laplace–Beltrami operator as the trace of the Hessian
- Gradient hessian and divergence connection formulas
- The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- E. Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity (2007) · Section 9.2.2, equations (9.4) and (9.15), printed pp. 153–155
- Null expansions, screen quotients and focal data
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Null Raychaudhuri and focal-point bound
- Null focal points exclude later horismos generators
- The Axiom of Choice
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Differentiated asymptotic bounds and parameter uniformity
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Sum, product, quotient, and chain rules from difference quotients
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries
- If f is continuous on an interval I and |f'| ≤ M at every interior point, then |f(x) - f(y)| ≤ M|x-y| for all x,y ∈ I, so f is Lipschitz with constant M and uniformly continuous on I
- If det(A) is a unit, then A⁻¹=det(A)⁻¹adj(A)
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
Definition
General spacelike two-surfaces, null gauges and fluxes
Theorem
Mean-curvature variation and the stationary lapse equation
Lemma
Null optical comparison with affine and screen conventions
Lemma
Annular scaling and fixed interior covers
Lemma