Alphabeta MathArticle items · 12 — A smooth spacelike reference section is a bounded rapidity graph
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 3–4.
1 · Prerequisites
- Spacelike sections, maximality and lapse–shift data
- Circle-invariant normal coordinates linearize a bifurcate Killing boost
- The parametrized implicit function theorem with Cᵏ regularity
- The smooth inverse function theorem on manifolds
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral
- Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Tube lemma: if K is compact and an open N ⊆ X × Z contains K × {z₀}, then N contains K × W for some open W ∋ z₀
- 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 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)
- 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)
- Stationary Lorentzian data and tensor conventions
- Bifurcate Killing horizons, rotation and normalization
- Normal boost action and invariant two-forms at a fixed surface
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Coordinate geodesic equation
- Smooth dependence of ODE solutions on parameters
- Local and global flows generated by a vector field
- The Axiom of Choice
- Causal exterior, chronological boundaries and Cauchy data
- Smooth event-boundary branches are achronal null geodesic sheets
- Smooth transverse horizon branches are the null axes of the normal tube
- A compact spacelike surface has a normal exponential tube
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- Through each point there is a unique maximal integral curve
- Taylor's theorem with an integral remainder
- In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Differentiated asymptotic bounds and parameter uniformity
- Killing two-form and normalized Ernst notation
- Mass-one convention and Harrison denominator
- Regular non-degenerate stationary vacuum spacetime
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- Sum, product, quotient, and chain rules from difference quotients
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- The fundamental theorem of calculus and integration by parts
- The selected exterior is connected and causally convex
- Closure of the exterior Cauchy section meets only the double horizon
- A generic timelike product for a prescribed smooth Cauchy section
- For a topological space the following agree: no separation exists, the only clopen subsets are ∅ and X, and every continuous map to the two-point discrete space is constant
- The riemannian distance topology is the manifold topology
- A compact metric space is complete and totally bounded, and neither implication uses any choice principle
- A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Local comparison of a riemannian metric with the euclidean metric
- Every path-connected space is connected, and every path component lies inside a component
- Embedded smooth submanifolds with boundary
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- Relative stationary deformation and extension obstruction
- Nomizu extension and uniqueness in the analytic Lorentzian setting
- Uniqueness forces an extended axial field to commute
- An arbitrary metric isometry transports analyticity and Killing fields
- Null generators and once-only cross-sections
- Ambient closure and intrinsic boundary of a hypersurface
- Complex self-dual forms and metric-dependent contraction
- Positive Harrison denominator and its parameter identity
- Lifting a normalized stationary primitive to the Harrison parameter
- Embedded submanifolds and slice charts
- Real and imaginary parts, complex conjugation, and modulus
- Cʳ and smooth maps between smooth manifolds
- Continuity of a map of topological spaces at a point and globally
- The exterior derivative by the invariant vector-field formula
- The Lie derivative of a differential form
- Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed
- Causal curves from a common chronological boundary obey two barriers
- Causal curves, chronological and causal reachability, and inextendibility
- Periodic isometries and invariant normal averaging
- Metric compatibility, torsion freedom and the metric-selected connection class
- A circle action on a sphere has a pole with trivial normal action
- A Killing field vanishing on a spacelike surface has a normal boost derivative
- Affine generator profiles give constant signed acceleration
- For a Killing field the acceleration and surface-gravity equations agree
- Geometric electrovacuum equations and charge convention
- Unit-sphere area form and outward radial flux
- The exterior derivative commutes with pullback
- The general Stokes theorem
- Asymptotic mass and axial angular-momentum conventions
- The Boyer–Lindquist spatial end and differentiated expansion
- The Kerr Metric · §2, equations (2.1), (2.5)–(2.8), PDF pp. 2–3
- Kerr exterior tensor including the extremal parameter range
- Kerr-Newman metric and potential on the exact Boyer-Lindquist domain
- The Kerr Metric · §2, equations (2.1), (2.5)–(2.8), PDF pp. 2–3
- A specified future ingoing Kerr extension and its time orientation
- The Kerr Metric · §2, equations (2.1), (2.5)–(2.9), PDF pp. 2–3
- Normalized Komar data on a nonrotating horizon section
- The normal binormal is invariant under oriented boosts
- Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor
- Cauchy sections do not end at a one-sided chronological horizon
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- AC supplies the countable and dependent choices used in Banach integration
- For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected
- Diffeomorphisms and local diffeomorphisms of manifolds
- The n-sphere with its standard smooth atlas
- Products of smooth manifolds have a canonical product smooth structure
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Connected components, quasicomponents, and totally disconnected spaces
- Paths, path-connected spaces and path components
- A continuous image of a connected space is connected, and connectedness is a topological property
- ADM section time and causal spatial speed
- Fundamental theorem of calculus for absolutely continuous functions
- Gronwall's integral inequality with variable and constant coefficients
- Interior and boundary of a manifold with boundary
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- AC implies DC implies countable choice
- Smooth charts, atlases, and structures with boundary
- Smooth maps between manifolds with boundary
- For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
- Riemannian metric and riemannian manifold
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A product of finitely many compact spaces is compact in the product topology
- Compactness gives an earliest translated-cylinder causal contact
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact
- A bifurcate future horizon has an oriented defining pair near its section
- Admissible domains for outward normal graphs
- Canonical outward normals and null gauge on a graph
- Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure
- Stationary Harrison one-form and metric
- Stationary vacuum model and asymptotic normalization
- Local frame fields, coordinate frames and tetrad fields
- The tensor product of multilinear tensors
- Single-potential construction of the Harrison one-form family
- Rank-one tangent changes, signature and Hodge transfer
- The Lie derivative of a tensor field
- Differentiated stationary end bounds imply the weighted Hölder interface
- A complete stationary flow preserves its compact tangent boundary
- Black and white exclusion is an ambient reachability condition
- Isometries preserve maximality and transported boundary
- Stationary graph translation preserves source asymptotics
- Piotr T. Chruściel and Robert M. Wald, Maximal Hypersurfaces in Asymptotically Stationary Spacetimes · Definitions 2.1 (2.1)–(2.5), pp.8–9; 2.3, p.11; (2.13)–(2.17), pp.11–12; Definition 2.4, p.13; (4.1)–(4.3) and Theorem 4.2, pp.24–25; Remark 3, p.25
- Null expansions, screen quotients and focal data
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Event-horizon bifurcation data for the nontrapping theorem
- A smooth map transverse to an embedded submanifold
- Local null normal pairs and the positive screen quotient
- Transverse embedded submanifolds
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 opening setup; GR, equation (1.1) and the immediately following charge/decay conventions; SBS through its nonexpansion definition (before the separate maximality assumption).
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Definition 2.19, printed/PDF p. 34; Definition 2.95, printed/PDF p. 63; §2.18, Definition 2.130, printed/PDF p. 76; §3.2, Definition 3.35, printed/PDF p. 95.
- Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · Version 1, §1.1 setup, GR and SBS; §1.2 opening paragraph.
- Radu nontrapping global regularity package
- Stationary orbit space with its quotient topology
- Zero stationary energy along null geodesics
- Geodesic of an affine connection
- Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · Version 1, section 1.1: quotient definition after SBS; NT and its following maximality/compactness paragraph; energy discussion immediately after Theorem 1.1.
- Ernst and self-dual curvature data in the AIK convention
- Tensor products and contractions of smooth tensor fields are smooth
- S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · Section 2.3, equations (2.18)–(2.19), with (2.13), (2.16)–(2.17) and the weighted-smoothness sentence immediately after (2.19).
- 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
- Asymptotically flat spacelike end and derivative decay
- Complete stationary flow and its exterior region
- Acausal hypersurface with compact core and finitely many ends
- The inverse metric, musical isomorphisms and index raising
- 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
- Stationary maximal graphs and normalized graph heights
- Lapse, shift, induced metric and second fundamental form
- Natural tensors under Killing isometries
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
LemmaDependency level 3
A smooth spacelike reference section is a bounded rapidity graph
LemmaDependency level 3
Bounded boost times give a uniform Gaussian collar cone
LemmaDependency level 3
Mass-one Ernst decay and uniform Harrison denominator bounds
LemmaDependency level 3
The exterior Cauchy component has a complete asymptotically Euclidean closure
LemmaDependency level 3
A commuting-extension obstruction excludes arbitrary analytic-model isometries
LemmaDependency level 3
Cofinal stationary end representatives have the same causal exterior
LemmaDependency level 3
A common positive Harrison domain across the compact sphere
LemmaDependency level 3
The normal future and past quadrants are the local causal sets of the sphere
LemmaDependency level 3
The coefficient-one horizon generator inherits the positive lapse derivative
LemmaDependency level 3
The mass-one Killing form has electric seed flux minus four pi
LemmaDependency level 3
The asymptotic metric flux of Kerr equals its mass parameter
LemmaDependency level 3
The Komar angular-momentum flux of the Kerr end
LemmaDependency level 3
Outgoing Kerr null curves and the selected future null infinity
LemmaDependency level 3
Killing-horizon derivative contracted with the binormal
LemmaDependency level 3
Identifying the spherical Cauchy portion without a stationary lapse
LemmaDependency level 3
Collar barrier against finite section-time boundary arrival
LemmaDependency level 3
An interior compact causal trajectory has a finite-time endpoint
LemmaDependency level 3
Proper end radius and exclusion of finite-time spatial infinity
LemmaDependency level 3
First contact lies on the chronological and causal future boundary
LemmaDependency level 3
Small outward normal graphs have controlled inward null initial jets
Physics theoremDependency level 3
Coordinate components of the transformed metric
Physics theoremDependency level 3
Invertibility, Lorentz signature and stationary norm
TheoremDependency level 3
Stationary maximal hypersurface existence with a horizon boundary
DefinitionDependency level 4
Radu small-ADM global and bifurcation package
DefinitionDependency level 4
The precise maximal zero-energy nontrapping predicate
DefinitionDependency level 4
Mars–Simon tensor and its smooth weighted extension
DefinitionDependency level 4
Staticity, axisymmetry and black-hole regularity hypotheses
DefinitionDependency level 4