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
- Measurable domains on smooth initial surfaces
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Assuming countable choice, every second countable space is Lindelöf
- The Axiom of Choice
- Leon Simon, Introduction to Geometric Measure Theory (2014) · Chapter 2, §1, Theorem 1.8(i)–(ii) and its proof, printed pp.44–45 (PDF page 26)
- Admissible domains for outward normal graphs
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Null normal pairs and screen geometry
- Pullback of a riemannian metric as a tensor
- Musical isomorphisms
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Killing fields of a Lorentzian metric
- Local and global flows generated by a vector field
- Causal curves, chronological and causal reachability, and inextendibility
- Asymptotically flat spacelike end and derivative decay
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Tangential area Jacobian on a measurable surface domain
- Lipschitz maps control Hausdorff measure
- Leon Simon, Introduction to Geometric Measure Theory · Chapter 2, §3 equations (3.3), (3.5), and §4 equations (4.12), (4.14), printed pp.53–54, 58 (PDF pp.30–31,33)
- The Axiom of Countable Choice (AC_ω)
- Fundamental theorem of riemannian geometry
- The chain rule for differentials of smooth maps
- Metric compatibility, torsion freedom and the metric-selected connection class
- Covariant derivative along a curve
- Geodesic of an affine connection
- Coordinate geodesic equation
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes (2008), author manuscript · §2.2, (2.9)–(2.10) and footnote 4, p. 7; §4.1, opening paragraph before Proposition 4.1, p. 13 (PDF pages 7 and 13).
- Riemannian metric and riemannian manifold
- Lipschitz graph area and calculus under Countable Choice
- Transverse hitting maps have local differentiable extensions
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Chronology, causality, strong causality, global hyperbolicity and time functions
- A regular level set is an embedded submanifold
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
Counting multiplicity of an endpoint map
Lemma
Canonical outward normals and null gauge on a graph
Definition
Complete stationary flow and its exterior region
Lemma
Area formula and endpoint bound with multiplicity
Lemma
Outgoing expansion depends smoothly on the second graph data
Definition
Null generators and once-only cross-sections
Definition