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
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Killing fields of a Lorentzian metric
- Metric compatibility, torsion freedom and the metric-selected connection class
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- The inverse metric, musical isomorphisms and index raising
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Oriented smooth manifolds and oriented charts
- The Axiom of Choice
- Sean M. Carroll, Lecture Notes on General Relativity (1997) · Chapter 2, printed pp. 47 and 49 (PDF pp. 54 and 56), equations (2.28), (2.34) and adjacent metric/signature definitions.
- Sean M. Carroll, Lecture Notes on General Relativity (1997) · Chapter 5, printed pp. 135 and 140 (PDF pp. 142 and 147), equations (5.17)-(5.18) and (5.41).
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Riemannian metric and riemannian manifold
- Riemannian distance on a connected manifold
- Riemannian distance is a metric
- Embedded smooth submanifolds with boundary
- Cauchy sequence in a metric space
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- 3+1 Formalism and Bases of Numerical Relativity · Section 4.2.2, equations (4.28)–(4.31), p. 56
- Null normal pairs and screen geometry
- Local null normal pairs and the positive screen quotient
- Null focal points exclude later horismos generators
- AC implies DC implies countable choice
- Causal curves, chronological and causal reachability, and inextendibility
- Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor
- Causal solutions, Killing horizons, escape and metric extensions
- H. S. Reall, General Relativity (Part III, October 12, 2022) · Section 5.3, printed p. 56 (PDF p. 57), equations (5.21)–(5.22) and adjacent Killing-field text.
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- A specified asymptotically Euclidean ADM end
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025) · Section 8, introductory Hölder definitions, printed p. 139 (PDF page 139), before Section 8.1.
- Oriented Lorentzian fields, observers and coordinate conventions
- Lorentzian exterior calculus and Hodge star
- The direct-sum model V⊕ iV of a complexification
- Smooth vector bundles, rank, fibres, and trivial bundles
- Smooth sections, local sections, and support
- Vector bundle maps over a smooth base map
- Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique
- The Axiom of Countable Choice (AC_ω)
- Lecture Notes on General Relativity · §1, equations (1.87)–(1.88), PDF p. 30
- Geometric electrovac equations and flux normalization
- Integration on an oriented embedded submanifold
- Pullback of forms is smooth functorial and preserves wedges
- Compact support of a differential form
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- Orientation reversal changes the integral sign
- Laplace–Beltrami operator as the trace of the Hessian
- Smooth charts, atlases, and structures with boundary
- Diffeomorphisms and local diffeomorphisms of manifolds
- Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025) · Section 8, opening definitions surrounding Exercise 8.1, printed p. 139 (PDF page 139): the displayed Hölder seminorm precedes the exercise; the C^k,alpha convention follows it on the same page.
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
Stationary Lorentzian data and tensor conventions
Definition
Spacelike sections, maximality and lapse–shift data
Definition
Null expansions, screen quotients and focal data
Definition
Bifurcate Killing horizons, rotation and normalization
Definition
Differentiated asymptotic bounds and parameter uniformity
Definition
Complex self-dual forms and metric-dependent contraction
Definition
Geometric electrovacuum equations and charge convention
Definition