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
- General spacelike two-surfaces, null gauges and fluxes
- Normal deformation families and an outgoing null gauge
- Outgoing expansion variation at a general MOTS
- MOTS boosts preserve the spacetime stability sign
- Spacetime stably outermost condition and the slice bridge
- Pointwise Maxwell and matter energy-condition vocabulary
- The Axiom of Countable Choice (AC_ω)
- Metric compatibility, torsion freedom and the metric-selected connection class
- S. Dain, J. L. Jaramillo and M. Reiris, Area-charge inequality for black holes, arXiv:1109.5602v1 · Theorem 2.1 and equations (1)–(4), PDF pp. 2–3; §3, equations (17)–(20), Definitions 3.1–3.2, PDF p. 7; Lemma 3.3 and equation (21), PDF p. 8; Lemma 3.4, equation (24), PDF p. 9; proof of Theorem 2.1, equations (30)–(34), PDF p. 10
- Surface measure, flux densities and null normal gauge
- Null expansions, screen quotients and focal data
- Outward unit variation in the normalized null basis
- Boundary-constant-preserving smooth approximation
- Riemannian Lipschitz coarea on compact interior bands
- The Axiom of Choice
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality
- Morse-Sard for smooth manifolds
- A regular level set is an embedded submanifold
- Riemannian metric and riemannian manifold
- Smooth charts, atlases, and structures with boundary
- Embedded smooth submanifolds with boundary
- Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)
- Local normal form for submersions
- AC implies DC implies countable choice
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- A nonnegative integral over a null set vanishes
- Regular and critical points and values
- Smooth maps between manifolds with boundary
- Unnormalised Hausdorff measure
- Lipschitz height functions, cut measure and Jacobians
- Liviu I. Nicolaescu, The co-area formula (2021 revision) · Section 3, printed p. 8, opening definition of Hausdorff measures; equation (2.5), printed p. 7
- Smooth partition of unity on a manifold with boundary
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Geometric electrovacuum equations and charge convention
- Electrovac contraction in the normalized normal plane
- Zero null optics supplies the missing null Ricci condition
- Scalar curvature is twice the sum of sectional curvatures of orthonormal coordinate planes
- Andersson–Mars–Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes · §2, printed pp. 5–6 (null normals, second form and expansion); §3, printed p. 7 (immersed-surface variation and varying null normal, before Lemma 3.1); Lemma 3.1, printed p. 7; equations (2)–(7), Definition 3.1 and Lemma 3.2, printed p. 8
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
Physics theorem
Stable MOTS area–charge inequality
Lemma
Agreement of expansion and normal-connection conventions
Lemma
Transferring area-controlled band energy through smooth approximation
Lemma
Choosing a smooth separating regular level from an average bound
Lemma
Coarea smoothing with fixed boundary values and area control
Lemma
General curvature coefficient in the outward hypersurface gauge
Theorem