Alphabeta MathArticle items · 15 — Hawking horizon generator and analytic global rigidity
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 5–6.
1 · Prerequisites
- Stationary vacuum model and asymptotic normalization
- Regular bifurcate null horizon data before rigidity
- Hawking and rotational fields near a bifurcate sphere
- Bifurcate Killing horizons, rotation and normalization
- Rotation, surface gravity and horizon degeneracy
- Staticity, axisymmetry and black-hole regularity hypotheses
- Periodic isometries and invariant normal averaging
- Affine null-generator profiles vanish exactly on the bifurcation section
- Affine generator profiles give constant signed acceleration
- For a Killing field the acceleration and surface-gravity equations agree
- An exact local Killing field is Killing to every order on a horizon
- Constant combinations preserve horizon symmetries and normalize periods
- S. Alexakis, A. D. Ionescu and S. Klainerman, Hawking’s local rigidity theorem without analyticity · §1 Theorems 1.1–1.2, pp.1–2; §2 affine null setup, pp.4–5; Proposition 3.1 and (3.1), p.7; §4 extension and §5 Propositions 5.1–5.2
- P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes · Corollary 2.4; §2.3.1 equation (2.13); Theorem 4.14, Remark 4.15 and its extension/completeness paragraph; §7.1
- P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §3.4.2 Rigidity without analyticity, PDF p.20, first paragraph
- Geometric electrovacuum equations and charge convention
- Killing fields of a Lorentzian metric
- The Axiom of Choice
- Weighted Hölder norms on a three-dimensional exterior
- Weighted exterior elliptic domains and admissible weights
- Fredholm and Banach continuation interfaces
- Exterior Dirichlet parametrix and compact decaying potentials
- Compact potential multiplication without a Dirichlet restriction
- Fredholm index is stable under compact perturbations
- Decaying Dirichlet kernel and the exact closed-component exception
- Stationary Lorentzian data and tensor conventions
- Spacelike sections, maximality and lapse–shift data
- Differentiated asymptotic bounds and parameter uniformity
- Complex self-dual forms and metric-dependent contraction
- Smooth invariant Harrison primitive near the horizon and at infinity
- Positive Harrison denominator and its parameter identity
- Lifting a normalized stationary primitive to the Harrison parameter
- Regular Lorentzian metric from the normalized rank-one map
- Smooth Maxwell tensor formula without division by the stationary norm
- Scope of smooth field equations at an exterior horizon boundary
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 GR/SBS and maximality; §3.1–3.3, Proposition 3.1, (3.5)–(3.25); Appendix A.1–A.3
- Radu small-ADM global and bifurcation package
- The ADM-source mass and angular-momentum parameters
- Maximality and smooth boundary in the ADM theorem
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1: setup, GR and (1.1), SBS and the separate maximality paragraph; quantitative-coordinate paragraphs immediately before (1.2), equations (1.2)–(1.3), and Theorem 1.1 for mass-one application.
- Harrison electromagnetic form
- Stationary maximal graphs and normalized graph heights
- Surface measure, flux densities and null normal gauge
- Null expansions, screen quotients and focal data
- Comparison spheres and their stationary cylinders
- Mass-one convention and Harrison denominator
- Compact remainder, causal cut and generator endpoint map
- Stationary section time and translated maximal sections
- Causal exterior, chronological boundaries and Cauchy data
- Every subset of ℝ̄ has a least upper bound and a greatest lower bound in ℝ̄, agreeing with the real supremum and infimum on nonempty sets bounded in ℝ
- Local and global flows generated by a vector field
- Kerr metric, parameters, exterior and extremality
- Kerr exterior tensor including the extremal parameter range
- Normal graphs and the nonlinear mean-curvature map
- An existing stationary graph supplies global projection before any estimate
- Uniform stationary return persists under arbitrary tangential drift
- Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces
- Smooth maps are continuous
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
- 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 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
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Vacuum Killing and Ernst differential identities
- Ernst sublevels and the Mars–Simon tensor interface
- Periods and primitives on a spherical exterior
- 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
- A compact spacelike surface has a normal exponential tube
- A smooth radial primitive can be returned to a weak end gauge
- Stokes theorem for smooth singular chains
- Stationary end normalization from quotient compactification
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- The pullback of a differential form
- The exterior derivative commutes with pullback
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- The smooth inverse function theorem on manifolds
- 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)
- The standard smooth step function
- The general Stokes theorem
- Uniform interior graph geometry and a bounded finite chart cover
- Multiplicative interior Harnack for geometric Schrödinger solutions
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Laplace–Beltrami operator as the trace of the Hessian
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Choice-free smooth inverse function theorem in Euclidean space
- On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂
- For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
- Positive-lapse stationary graphs give temporal level functions and closed restrictions
- A smooth spacelike reference section is a bounded rapidity graph
- A complete timelike collar flow is a reference product
- Circle-invariant normal coordinates linearize a bifurcate Killing boost
- Products of smooth manifolds have a canonical product smooth structure
- Embedded smooth submanifolds with boundary
- A continuous image of a connected space is connected, and connectedness is a topological property
- 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
- A product of finitely many compact spaces is compact in the product topology
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- The inverse metric, musical isomorphisms and index raising
- Smooth manifolds and their smooth charts
- Smooth charts, atlases, and structures with boundary
- Smooth collars of a manifold boundary
- Temporal cutoff estimate for normals to maximal hypersurfaces
- Compact sets admit relatively compact open neighborhoods and finite chart pieces
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- A compact temporal covector persists under metric perturbation
- Inversion is uniformly stable on compact chart families
- For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide
- A compact normal-flow collar has uniformly small area distortion
- A finite temporal target cannot be reached from a sufficiently small collar
- Causal curves, chronological and causal reachability, and inextendibility
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- The Axiom of Countable Choice (AC_ω)
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- 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
- A closed subset of a compact metric space is compact
- The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
- Riemannian distance on a connected manifold
- Extended riemannian distance on a disconnected manifold
- The riemannian distance topology is the manifold topology
- Distance from a point to a subset
- Distance to a nonempty subset is one lipschitz
- Smooth global data for the axisymmetric comparison
- The chain rule for differentials of smooth maps
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- Bounded boost times give a uniform Gaussian collar cone
- Spacelike normal projection is locally invertible with positive boundary jet
- Radial comparison gives proper normal projection onto a fixed target collar
- A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point
- Mass-one Ernst decay and uniform Harrison denominator bounds
- Stationary Harrison one-form and metric
- Sum, product, quotient, and chain rules from difference quotients
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- AC implies DC implies countable choice
- Exterior Schrödinger maximum principles, Hopf growth and Harnack
- Riemannian divergence theorem
- An exterior-contained null segment to a distant target cannot start inward
- Admissible domains for outward normal graphs
- Canonical outward normals and null gauge on a graph
- Small outward normal graphs have controlled inward null initial jets
- A compact chronological target supplies a causal endpoint
- A compact-target boundary point has a null geodesic segment
- The entire target segment stays on the chronological boundary
- Endpoint orthogonality without an acausal target assumption
- Boundary membership excludes interior focal points from the target
- Achronal boundaries are locally Lipschitz graphs, topological hypersurfaces and null for smooth volume densities
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- E. Minguzzi, Lorentzian causality theory · Definition 2.86 and Theorem 2.87(iii)
- Regular non-degenerate stationary vacuum spacetime
- A uniqueness theorem for degenerate Kerr-Newman black holes · Abstract
- A smooth vacuum Bondi expansion
- Subextremal Kerr exterior in length coordinates
- Asymptotically flat spacelike end and derivative decay
- Complete stationary flow and its exterior region
- P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes · Section 2.2, PDF/printed p. 7, equation (2.9) and the black-hole-region definition immediately following it.
- Killing two-form and normalized Ernst notation
- A generic timelike product for a prescribed smooth Cauchy section
- Cauchy sections do not end at a one-sided chronological horizon
- Identifying the spherical Cauchy portion without a stationary lapse
- Fixing the Ernst constant by integrable asymptotic twist
- Extending a closed one-form potential across a bifurcation sphere
- The selected exterior is connected and causally convex
- For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected
- Every path-connected space is connected, and every path component lies inside a component
- 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
- Embedded submanifolds and slice charts
- The tubular neighbourhood theorem in a smooth ambient manifold
- A smooth bump between concentric Euclidean balls
- De rham cohomology of spheres
- De rham cohomology is smooth homotopy invariant
- De rham cohomology
- Closed and exact differential forms
- A smooth function with zero differential is constant on each connected component
- AC supplies the countable and dependent choices used in Banach integration
- Every smooth vector bundle admits a smooth bundle metric
- Smooth bundle metrics
- Tubular neighbourhoods of embedded submanifolds
- Smooth vector bundles, rank, fibres, and trivial bundles
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
Physics theoremDependency level 5
Hawking horizon generator and analytic global rigidity
RemarkDependency level 5
Reported analytic and smooth electrovacuum uniqueness literature
RemarkDependency level 5
Fredholm ingredients stated in Lemma 3.1
TheoremDependency level 5
Regular Harrison Einstein–Maxwell family
DefinitionDependency level 6
The actual geometric constant in the small-ADM theorem
DefinitionDependency level 6
Horizon null frame, flux densities and stability operator notation
DefinitionDependency level 6
Truncated causal-boundary height for smooth approximation
ExampleDependency level 6
Schwarzschild as the zero-spin member of Kerr
LemmaDependency level 6
Collar comparison and an exact end projection imply properness
LemmaDependency level 6
Equivariant global Harrison primitive under weak original data
LemmaDependency level 6
Anchored lapse propagation and quantitative stationary coordinate transfer
LemmaDependency level 6
Controlled horizon cuts, compact restrictions and a uniform end anchor
LemmaDependency level 6
Shifting a temporal function permits a controlled compact restriction
LemmaDependency level 6
Uniform localization of translated past-boundary cuts
LemmaDependency level 6
Electrovac contraction in the normalized normal plane
LemmaDependency level 6
A commuting axial action preserves the stationary field and its causal character
LemmaDependency level 6
Exact assembly of the global I-plus regularity assumptions
LemmaDependency level 6
The complete stationary flow preserves the selected causal exterior
LemmaDependency level 6
A one-sheet boundary germ propagates across the common normal collar
LemmaDependency level 6
The differentiated spatial decay supplied by the Harrison lift
LemmaDependency level 6
The selected endpoints cover the compact cut and remain on its past boundary
LemmaDependency level 6
Strict mass positivity from the stationary lapse flux
LemmaDependency level 6
The surviving source-normal ray has the outgoing affine normalization
LemmaDependency level 6
Compact-target null generators and the required endpoint conditions
Open problemDependency level 6
General stationary vacuum black-hole uniqueness conjecture
Open problemDependency level 6
Smooth rotating degenerate-horizon uniqueness
PostulateDependency level 6
Physical expectation of late-time stationarity
Physics theoremDependency level 6