Alphabeta MathArticle items · 11 — ADM section time and causal spatial speed
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 2–3.
1 · Prerequisites
- The Axiom of Choice
- Causal exterior, chronological boundaries and Cauchy data
- Spacelike sections, maximality and lapse–shift data
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- Fundamental theorem of calculus for absolutely continuous functions
- C¹ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation
- Local comparison of a riemannian metric with the euclidean metric
- Products of smooth manifolds have a canonical product smooth structure
- The Riemannian gradient is the metric dual of the differential
- Pointwise norm and angle from a riemannian metric
- The action of a vector field on smooth functions
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- AC implies DC implies countable choice
- Riemannian metric and riemannian manifold
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Stationary Lorentzian data and tensor conventions
- The stationary Killing energy is constant along every affine null segment
- Time orientation, future and past timelike/null/causal vectors, and the closed cones
- Chronology, causality, strong causality, global hyperbolicity and time functions
- Diffeomorphisms and local diffeomorphisms of manifolds
- Smooth embeddings
- In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact
- Temporal functions are time functions, and time functions forbid closed causal curves
- Local and global flows generated by a vector field
- Embedded submanifolds and slice charts
- 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
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- 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
- A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A)
- A compact spacelike surface has a normal exponential tube
- Bifurcate Killing horizons, rotation and normalization
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Metric compatibility, torsion freedom and the metric-selected connection class
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- The smooth inverse function theorem on manifolds
- Stationary vacuum model and asymptotic normalization
- Differentiated asymptotic bounds and parameter uniformity
- Null expansions, screen quotients and focal data
- The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation
- A. F. Radu, On rigidity of stationary black holes under a nontrapping assumption, arXiv:2609.23872v2 · Section 1.1, GR and SBS, PDF/printed pp. 3–4; Lemma 2.13, p. 25; nonrotating branch of the proof of Theorem 1.1, pp. 69–70; Appendix A, pp. 71–74.
- P. T. Chruściel and R. M. Wald, Maximal Hypersurfaces in Asymptotically Stationary Spacetimes, arXiv:gr-qc/9304009 · Section 2, Definition 2.1 and its Remarks 1–2, PDF pp. 8–9; Definition 2.3, p. 11; Section 4, equations (4.1)–(4.3), Theorem 4.2 and Remark 3, pp. 24–25.
- Killing fields of a Lorentzian metric
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Ernst sublevels and the Mars–Simon tensor interface
- Adopted Einstein field equation in the direct-equation presentation
- Subextremal Kerr exterior in length coordinates
- Null normal pairs and screen geometry
- Local null normal pairs and the positive screen quotient
- The Axiom of Countable Choice (AC_ω)
- S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · §1.1, GR/SBS/PK and Main Theorem, PDF pp.4–5; §2.1 (2.1)–(2.5), PDF pp.6–7; normalized fields §2.3, PDF pp.8–9
- M. Dafermos and I. Rodnianski, Lectures on black holes and linear waves · Section 5.1, The Kerr metric, PDF pp.56-57 (printed pp.56-57): Boyer-Lindquist metric, parameter interpretation and DOC paragraph; p.57 footnote 49 on spherical-coordinate charts.
- Relative stationary deformation and extension obstruction
- Analytic Kerr coordinates across the chosen nondegenerate horizon
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Ionescu–Klainerman, On the local extension of Killing vector-fields in Ricci flat manifolds · §1 Theorem 1.3, printed/PDF p. 4; §3.2 pp. 19–20; proof of Theorem 1.3 pp. 23–27, especially (3.39) and the horizon coordinates on p. 26
- Complex self-dual forms and metric-dependent contraction
- Periodic isometries and invariant normal averaging
- Stokes theorem for smooth singular chains
- Relative homotopy primitive on a normal bundle
- Equivariant homotopy and forced contraction constant
- Radial primitive and five differentiated decay bounds
- Stationary lifting of a spatial primitive
- Stationary exact corrections preserve two prescribed gauges
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 GR/SBS and maximality; Lemma 2.1; §3.1 (3.2) and the following global-existence and extension paragraph; §3.2 (3.9), (3.13)–(3.15); Appendix A.1 (A.2)–(A.11), PDF pp. 29–30; Appendix A.3 Lemma A.1 at q=1
- Exact asymptotic and orbit package for the IK criterion
- Constant-rank kernels and images of bundle maps over one base are subbundles
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §1.2, SBS and footnotes 5–6, printed p. 7
- Killing two-form and normalized Ernst notation
- AIK perturbative global regularity and bifurcation package
- S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · Section 1.1 setup and GR/SBS; section 2.1 neighborhood construction preceding (2.3); section 2.3, (2.13)–(2.15) and the potential-existence paragraph immediately before (2.16). PDF v1 pp. 4–7 and 8–9.
- 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, (2.13), (2.16)–(2.17), (2.20), PDF pp. 8–9; section 5.1, (5.9)–(5.10) and the immediately following unnumbered curvature formula, PDF p. 27.
- Radu nontrapping global regularity package
- Transverse embedded submanifolds
- Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · Version 1, section 1.1 SBS and its two following explanatory paragraphs; Theorem 1.1; section 1.2 opening paragraph.
- S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · Version 1, section 1.1, SBS paragraph, especially its final sentence (after equation (1.3) and before PK).
- S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · Version 2, section 1.1.2, SBS paragraph, final sentence immediately before section 1.1.3.
- S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · Section 1.1 setup and GR/SBS; section 2.1, (2.1)–(2.5), especially the three coordinate-construction paragraphs between (2.2) and (2.3).
- Equivalence relation, equivalence class, and the quotient set A/∼
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- Continuity of a map of topological spaces at a point and globally
- Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · Version 1, §1.1, printed pp. 4–5: quotient-space definition after SBS and the compact-set clarification immediately after NT.
- Mass-one convention and Harrison denominator
- Through each point there is a unique maximal integral curve
- Stationary Harrison one-form and metric
- Geometric electrovacuum equations and charge convention
- Lipschitz graph area and calculus under Countable Choice
- Measurable domains on smooth initial surfaces
- Tangential area Jacobian on a measurable surface domain
- Counting multiplicity of an endpoint map
- Transverse hitting maps have local differentiable extensions
- Area formula and endpoint bound with multiplicity
- A regular level set is an embedded submanifold
- The image of a smooth embedding is an embedded submanifold
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- The reals form a totally ordered field
- Future timelike cylinder translations make their chronological pasts increase
- Sufficiently far stationary end points can be routed to a chosen cylinder
- An open chronological exterior is causally convex without a timelike-end assumption
- Causal curves, chronological and causal reachability, and inextendibility
- Maximal stationary shift has zero divergence and tangent boundary
- Complete volume preserving shift flow on a spherical exterior
- Inaccessible set is the closure of its relative interior
- Finite volume and interior density force exact flow invariance
- The Borel sigma-algebra of a topological space
- Sigma-algebras
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Gaussian maximal graph equation on a compact spacelike slope cone
- Uniform elliptic extension and flattening before the boundary gradient estimate
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- 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)
- Leon Simon, Lectures on PDE · Lecture 12, Theorem 2-prime (local boundary non-divergence Schauder estimate), printed pp.134–135, PDF sheet70, specialized to m=1 and homogeneous Dirichlet data
- The fundamental theorem on flows
- The parametrized implicit function theorem with Cᵏ regularity
- Stationary coefficients in a transverse collar retain the tangential motion
- The fundamental theorem of calculus and integration by parts
- Riemannian speed and length
- Local compactness of full weighted Hölder inputs
- Weighted product and uniform potential tails on the full input space
- The closure Hölder spaces are Banach spaces
- Fredholm and Banach continuation interfaces
- Weighted Hölder norms on a three-dimensional exterior
- Weighted exterior elliptic domains and admissible weights
- 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
- Equivalence of the absolutely-continuous and continuous causal conventions, and one-sided inextendible extensions
- Connectedness and causal convexity of a chronological exterior
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed
- Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal
- A generic timelike product for a prescribed smooth Cauchy section
- The differential of a smooth real-valued function
- The chain rule for differentials of smooth maps
- The differential of a smooth map
- Existence of a positive smooth volume density of total mass one on a smooth Lorentzian four-manifold
- The Geroch volume function: a continuous strictly increasing time function whose levels are Cauchy sets
- Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow
- Stationary maximal graphs have an explicit uniformly spacelike flux
- Uniform coefficient regularity before the final graph estimate
- Linear fourth derivative bound with controlled realized coefficients
- Interior Schauder estimate for uniformly elliptic equations
- Trace-free three-dimensional coercivity for the logarithmic gradient
- Uniform scalar angle inequalities from local background tensors
- Boost-frame bounds for a smooth observer and arbitrary spacelike normals
- Laplace–Beltrami operator as the trace of the Hessian
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1 hypotheses, printed pp.35–36; proof (B.8)–(B.11), printed pp.38–39 (PDF pages 35–36 and 38–39).
- The inward normal coordinate is intrinsic distance from the boundary
- Distance Laplacian and normal-collar volume density are uniformly bounded
- Divergence comparison in a smooth compact boundary collar
- Riemannian volume density
- Riemannian divergence theorem
- Smooth partitions of unity exist on manifolds with boundary
- Smooth partition of unity on a manifold with boundary
- Orientation-free density integration and its properties
- Coordinate formula for riemannian divergence
- Riemannian volume form on an oriented manifold
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 2
ADM section time and causal spatial speed
LemmaDependency level 2
An asymptotically flat stationary end bounds null speed by Killing energy
LemmaDependency level 2
Compactness gives an earliest translated-cylinder causal contact
LemmaDependency level 2
A bifurcate future horizon has an oriented defining pair near its section
Physics theoremDependency level 2
Independent regular nonrotating staticity and Schwarzschild theorem
Physics theoremDependency level 2
AIK global Kerr identification under the full GR/SBS/PK contract
TheoremDependency level 2
Rotating Kerr admits local stationary vacuum deformations with an axial obstruction
TheoremDependency level 2
Smooth invariant Harrison primitive near the horizon and at infinity
DefinitionDependency level 3
Exact bifurcate-sphere package for the IK criterion
DefinitionDependency level 3
Ernst and self-dual curvature data in the AIK convention
DefinitionDependency level 3
Event-horizon bifurcation data for the nontrapping theorem
DefinitionDependency level 3
AIK perturbative effective constant and its coordinate data
DefinitionDependency level 3
Stationary orbit space with its quotient topology
DefinitionDependency level 3
Comparison spheres and their stationary cylinders
DefinitionDependency level 3
Harrison electromagnetic form
DefinitionDependency level 3
Lipschitz height functions, cut measure and Jacobians
DefinitionDependency level 3
Stationary maximal graphs and normalized graph heights
LemmaDependency level 3
Pointwise eventual chronological entry is uniform on compact subsets
LemmaDependency level 3
Piecewise timelike transfer yields arrival on the cylinder under far-tail access
LemmaDependency level 3
A maximal stationary exterior has future access to its far tail
LemmaDependency level 3
Uniform boundary Schauder bootstrap after a C1-Holder seed
LemmaDependency level 3
Uniform stationary return persists under arbitrary tangential drift
LemmaDependency level 3
Compact potential multiplication without a Dirichlet restriction
LemmaDependency level 3
A Cauchy section cannot end on only one chronological horizon
LemmaDependency level 3
A smooth Cauchy hypersurface gives a connected timelike-flow product
LemmaDependency level 3
The maximal equation upgrades pointwise order-four graph decay to weighted order five
LemmaDependency level 3
Logarithmic angle inequality through the full range v at least one
LemmaDependency level 3