Alphabeta MathStationary geometry and analytic data
Define the common Lorentzian, causal, horizon, hypersurface and field conventions, together with weighted elliptic, graph, continuation and area data. These fifteen mathematical interfaces supply both later prerequisite pages without importing article-specific conclusions.
1 · Prerequisites
- Admissible domains for outward normal graphs
- Outgoing expansion depends smoothly on the second graph data
- The chain rule for differentials of smooth maps
- Clairaut--Schwarz theorem for continuous second partial derivatives
- The Axiom of Countable Choice (AC_ω)
- Canonical outward normals and null gauge on a graph
- Null normal pairs and screen geometry
- Asymptotically flat spacelike end and derivative decay
- Ambient closure and intrinsic boundary of a hypersurface
- Null generators and once-only cross-sections
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Killing fields of a Lorentzian metric
- Chronology, causality, strong causality, global hyperbolicity and time functions
- Metric compatibility, torsion freedom and the metric-selected connection class
- Complete stationary flow and its exterior region
- The inverse metric, musical isomorphisms and index raising
- The exterior derivative by the invariant vector-field formula
- The wedge product of differential forms
- Smooth left actions of Lie groups
- The invariant exterior-derivative formula is C^∞-multilinear
- Causal solutions, Killing horizons, escape and metric extensions
- P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes · Definition 1.1 and (1.1), PDF p. 2; §2.1, (2.1)–(2.2), PDF pp. 5–6; §2.2, (2.9)–(2.10), PDF p. 7
- P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes · §4.1, opening paragraphs, PDF p. 13; §2.3, (2.11), PDF p. 8
- P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §3.1, opening paragraph, PDF p. 12; §3.2 opening paragraphs, PDF p. 13; §2.5 surface-gravity discussion, PDF pp. 8–9
- Affine connection on a smooth manifold
- Diffeomorphisms and local diffeomorphisms of manifolds
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Lorentzian exterior calculus and Hodge star
- Riemannian volume form on an oriented manifold
- Fundamental theorem of riemannian geometry
- Riemann curvature four-tensor
- Integral of a compactly supported top form
- Riemannian volume is the radon measure of the riemannian density
- Integrable real and complex functions, and their integrals
- Measurable integration extends smooth density integration
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
Linear and path-independent normal-graph variation of expansion
Definition
Acausal hypersurface with compact core and finitely many ends
Definition
Staticity, axisymmetry and black-hole regularity hypotheses
Lemma
Boost dependence of expansion variation and the MOTS gauge
Definition