Alphabeta MathArticle items · 16 — Non-rotating staticity and Schwarzschild uniqueness
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 6–8.
1 · Prerequisites
- Stationary vacuum model and asymptotic normalization
- Regular non-degenerate stationary vacuum spacetime
- Rotation, surface gravity and horizon degeneracy
- Zero stationary field gives the local bifurcate Killing-horizon and boundary data
- The selected exterior is connected and causally convex
- Closure of the exterior Cauchy section meets only the double horizon
- The exterior Cauchy component has a complete asymptotically Euclidean closure
- The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation
- A working partial-Cauchy collar satisfies the source GR section equalities
- The nonrotating bifurcation gives positive normalized mass before staticity
- Independent regular nonrotating staticity and Schwarzschild theorem
- A. F. Radu, On rigidity of stationary black holes under a nontrapping assumption, arXiv:2609.23872v2 · §1.1, GR and SBS, PDF/printed pp. 3–4; Lemma 2.13, p. 25; §6, nonrotating branch of the proof of Theorem 1.1, pp. 69–70.
- Lapse, shift, induced metric and second fundamental form
- Mean-curvature variation and the stationary lapse equation
- Exterior Schrödinger maximum principles, Hopf growth and Harnack
- Spacelike sections, maximality and lapse–shift data
- The Axiom of Choice
- Connectedness and causal convexity of a chronological exterior
- A Cauchy section cannot end on only one chronological horizon
- A smooth Cauchy hypersurface gives a connected timelike-flow product
- A complete stationary flow transverse positively to a Cauchy section is a product
- Weighted exterior elliptic domains and admissible weights
- The adjugate gives an explicit inverse formula over a field
- Exact AIK global, horizon and quantitative hypotheses
- Horizon smallness is the induced norm of the tangential stationary field
- The non-timelike region is confined to the local-rigidity collar
- Adopted Einstein field equation in the direct-equation presentation
- Subextremal Kerr exterior in length coordinates
- S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · Abstract and Introduction, PDF pp. 1–2; §1.1.3 Theorem 1.1 and (1.5), PDF p. 4; §1.2 (1.10), PDF p. 6
- Regular Harrison Einstein–Maxwell family
- Null expansions, screen quotients and focal data
- Geometric electrovacuum equations and charge convention
- Stationary Lorentzian data and tensor conventions
- Kerr metric, parameters, exterior and extremality
- Bifurcate Killing horizons, rotation and normalization
- Mean curvature vector
- Null Raychaudhuri and focal-point bound
- The pointwise energy conditions WEC, NEC, DEC and SEC
- Rigidity of stationary black holes with small ADM angular momentum · §1.2, equality discussion; §3.1, equation (3.8) and following paragraph; §4.2, Lemma 4.2
- Otis Chodosh, Stable Minimal Surfaces and Positive Scalar Curvature (Math 258, Stanford, Fall 2021) · §1, PDF pp. 2–3: two-sided hypersurface setup, variation convention before Lemma 1.2, Definition 1.4 and Proposition 1.5
- Geometric data for the analytic multiple-horizon comparison
- Causal exterior, chronological boundaries and Cauchy data
- 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
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes · §2.1, (2.1)–(2.8), the harmonic-coordinate/maximal-time-gauge paragraph and the final isotropic Schwarzschild display.
- R. Beig and P. T. Chruściel, Killing vectors in asymptotically flat space-times: I. Asymptotically translational Killing vectors and the rigid positive energy theorem · arXiv:gr-qc/9510015v3, §2, (2.2)–(2.8), (2.13)–(2.16), and Proposition 2.2, (2.26)–(2.29), printed pp.4–8, especially PDF page 8.
- P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes — stationary structure theorem · §4.2, Proposition 4.4, Theorem 4.5 and (4.15); proof through (4.18)–(4.23) and its final two paragraphs. Use s=1.
- Precise Rácz–Wald global collapse class
- Rácz–Wald, Global Extensions of Spacetimes Describing Asymptotic Final States of Black Holes · End of §2, printed/PDF pp. 8–9; Theorem 4.2, pp. 15–16; §5 opening paragraph, p. 17
- A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point
- An existing stationary graph supplies global projection before any estimate
- Collar comparison and an exact end projection imply properness
- The smooth inverse function theorem on manifolds
- Normal graphs and the nonlinear mean-curvature map
- Smooth embeddings
- Transverse embedded submanifolds
- An m-dimensional submanifold transverse to vertical fibres is locally a graph
- 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
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Complete boundary initial-data and vanishing momentum interface
- Vacuum exterior-section constraints imply the dominant energy condition
- Extrinsic curvature decay on the original stationary end
- Zero boundary expansions without maximality
- External boundary positive-energy theorem without maximality
- External ADM–Komar equality on ordinary stationary vacuum ends
- Komar flux equals the original stationary mass coefficient
- Asymptotic mass and axial angular-momentum conventions
- The maximal equation upgrades pointwise order-four graph decay to weighted order five
- The mean-curvature graph map is a smooth finite-jet expression
- Differentiated asymptotic bounds and parameter uniformity
- Weighted Hölder norms on a three-dimensional exterior
- Stationary maximal graphs have an explicit uniformly spacelike flux
- One extra pointwise derivative supplies scaled Hölder control
- Weighted differentiation and multiplication estimates
- Smooth positive inversion and composition in the weighted multiplier class
- A timelike transverse field gives a fixed-boundary admissible graph neighborhood
- Stationary Harrison one-form and metric
- Single-potential construction of the Harrison one-form family
- Smooth invariant Harrison primitive near the horizon and at infinity
- Mass-one Ernst decay and uniform Harrison denominator bounds
- Taylor's theorem with an integral remainder
- Uniform localization of translated past-boundary cuts
- Compact causal diamonds close the selected-generator endpoint domain
- The closed selected domain has smooth endpoint extensions
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed
- Stationary section time and translated maximal sections
- AC implies DC implies countable choice
- A compact normal-flow collar has uniformly small area distortion
- Achronal boundaries are locally Lipschitz graphs, topological hypersurfaces and null for smooth volume densities
- Truncated causal-boundary height for smooth approximation
- 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
- Riemannian speed and length
- The differentiated spatial decay supplied by the Harrison lift
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Killing two-form and normalized Ernst notation
- Sum, product, quotient, and chain rules from difference quotients
- Radu small-ADM global and bifurcation package
- The ADM-source mass and angular-momentum parameters
- Maximality and smooth boundary in the ADM theorem
- The actual geometric constant in the small-ADM theorem
- Mars–Simon tensor and its smooth weighted extension
- The selected endpoints cover the compact cut and remain on its past boundary
- Temporal functions are time functions, and time functions forbid closed causal curves
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Smooth charts, atlases, and structures with boundary
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- A subset of ℝⁿ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology
- 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)
- The surviving source-normal ray has the outgoing affine normalization
- The stationary Killing energy is constant along every affine null segment
- Admissible domains for outward normal graphs
- Canonical outward normals and null gauge on a graph
- Smooth maps are continuous
- First contact lies on the chronological and causal future boundary
- Compact-target null generators and the required endpoint conditions
- Positive smooth horizon area and the exact degenerate angular-momentum test
- The exhaustive analytic horizon-component case split
- Canonical stationary end expansions give the precise source slice decay
- Cofinal stationary end representatives have the same causal exterior
- Han–Khuri–Weinstein–Xiong, Kerr Black Hole Uniqueness · §1, geometric consequence immediately after Theorem 1.1: paragraph beginning “Under the standard analyticity, asymptotic flatness, and I-plus-regularity hypotheses”; Theorem 1.1 supplies the underlying negative angle defects in its map class
- Huang–Li–Zhu, Nonexistence of Multiple Black Holes in Stationary Analytic Vacuum Spacetimes · §1, Main Theorem “Nonexistence of non-rotating and degenerate black holes”, printed p.4; preceding (H1)–(H3) and Definitions 1.3–1.5, printed pp.2–3
- Huang–Li–Zhu, Nonexistence of Multiple Black Holes in Stationary Analytic Vacuum Spacetimes · §1, Lemma 1.7 and following regularity/positive-area paragraph; all-nonrotating nondegenerate paragraph immediately before the Main Theorem; final Corollary “Uniqueness of stationary, asymptotically flat, analytic vacuum spacetimes”, printed p.4
- Eric Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity · §7.2.1, (7.1)–(7.4), printed p.104; §7.3.1, (7.13)–(7.15), printed p.106 (PDF pages 104 and 106).
- Eric Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity — kinematic identities · §4.2.3, (4.43)–(4.48), printed pp.57–58; §§4.3.1–4.3.2, (4.59)–(4.63), (4.70), (4.74), printed pp.59–60 (PDF pages 57–60, one-based).
- Smooth global data for the axisymmetric comparison
- Staticity, axisymmetry and black-hole regularity hypotheses
- The complete stationary flow preserves the selected causal exterior
- A commuting axial action preserves the stationary field and its causal character
- Weight-one regular axis gives elementary flatness of small axial orbits
- Exact assembly of the global I-plus regularity assumptions
- Vacuum metrics embed as stationary electrovacuum data with zero field
- P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §3.2.7 Theorem 3.2, p.15; vacuum Kerr specialization in §3.2.4, p.15; no-analyticity distinction in §3.3.1, p.16; definitions §2.1 and §2.3, p.5 and Definition2.1, pp.10–11; global axisymmetric framework §3.2.1, p.13
- Rotating Kerr admits local stationary vacuum deformations with an axial obstruction
- A commuting-extension obstruction excludes arbitrary analytic-model isometries
- Hawking and rotational fields near a bifurcate sphere
- Complex self-dual forms and metric-dependent contraction
- Closed Ernst form and existence of normalized potential
- Vacuum Killing and Ernst differential identities
- Radial primitive and five differentiated decay bounds
- Lorentzian exterior calculus and Hodge star
- Interior product on forms is a graded antiderivation
- Mass-one convention and Harrison denominator
- Harrison electromagnetic form
- Transverse embedded submanifolds intersect in the expected codimension
- Ionescu–Klainerman local Kerr criterion with full interface
- Ernst sublevels and the Mars–Simon tensor interface
- Strict mass positivity from the stationary lapse flux
- Stationary end normalization from quotient compactification
- Extending normalized end coordinates while preserving the section
- A working bifurcation collar supplies the exact GR section
- The IK continuation also fixes one global reciprocal invariant
- Stationary product time is a Cauchy temporal coordinate
- Bounded spacelike stationary graphs remain Cauchy
- A decaying stationary time gauge gives bounded section displacement
- The reciprocal-potential invariant determines the end spin bound
- Normalized Ernst data and zero Mars–Simon survive stationary gauges
- AIK global Kerr identification under the full GR/SBS/PK contract
- Periods and primitives on a spherical exterior
- Smooth Schoenflies for an embedded two-sphere in the three-sphere
- The isotopy extension theorem
- Exact asymptotic and orbit package for the IK criterion
- Exact bifurcate-sphere package for the IK criterion
- 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, pp.4–5; §2.1 (2.1)–(2.5), pp.6–7; Appendix A (A.9)–(A.12), p.37
- A. D. Ionescu and S. Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §1.2 AF/SBS/T and Main Theorem, PDF pp.6–8; §7 (7.1)–(7.3), p.35; Lemma 7.1 (7.17)–(7.18), p.37; §8 (8.1) and Lemma 8.1, pp.41–42; §8.2 completion, pp.48–49
- Allen Hatcher, Notes on Basic 3-Manifold Topology · Chapter 1 §1.1, smooth-category convention and Theorem 1.1, printed p.1/PDF p.2; boundary sphere-map extension paragraph, printed p.4/PDF p.5.
- The Isotopy Extension Theorem, University of California Riverside differential topology handout · PDF p.11, closed-collar uniqueness; PDF pp.11–12, Cerf–Palais Disk Theorem; PDF p.2, compact-source isotopy extension.
- Regular bifurcate null horizon data before rigidity
- Periodic isometries and invariant normal averaging
- Normal boost action and invariant two-forms at a fixed surface
- Smooth event-boundary branches are achronal null geodesic sheets
- A stationary product rules out a zero Hawking coefficient
- 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
- The coefficient-one horizon generator inherits the positive lapse derivative
- A compact spacelike surface has a normal exponential tube
- Smooth transverse horizon branches are the null axes of the normal tube
- Causal curves from a common chronological boundary obey two barriers
- The normal future and past quadrants are the local causal sets of the sphere
- Regular stationary event branches supply the local AIK horizon pair
- 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, (2.1)–(2.4); Proposition 3.1, p.7, (3.1); §4 extension, pp.10–11 and conclusion p.15; §5 Propositions 5.1–5.2, pp.16–17
- Invertibility, Lorentz signature and stationary norm
- The chain rule for differentials of smooth maps
- Fundamental theorem of calculus for Banach-valued continuous curves
- Formal adjoint of a real drift Laplacian on a closed manifold
- Positive right and adjoint eigenfunctions have the same eigenvalue
- Ground-state identity, spectral ordering and eigenspace uniqueness
- Nonnegative supersolutions detect the principal spectral sign
- Outward graph stability has the negative-Laplacian principal symbol
- Imported positive eigenpairs for a smooth closed elliptic operator
- Surface measure, flux densities and null normal gauge
- Lars Andersson, Marc Mars and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes · §2 null-frame conventions, printed p.5; §3 variation definition, printed p.7, equation (2), Definition 3.1 equations (5) and Lemma 3.2 equation (7), printed p.8; §5 Definition 5.1 and Proposition 5.1 with proof, printed pp.14–15
- Lipschitz height functions, cut measure and Jacobians
- Area formula and endpoint bound with multiplicity
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 6
Non-rotating staticity and Schwarzschild uniqueness
Physics theoremDependency level 6
Remark 1.3: lapse bounds and stationary product
Physics theoremDependency level 6
Smooth rigidity from a quantitatively small horizon field
RemarkDependency level 6
Reported comparison and Bertotti–Robinson strategy
TheoremDependency level 6
Stationary vacuum normal form and compatible regularity section
TheoremDependency level 6
Nonzero constant surface gravity and global bifurcate extension
LemmaDependency level 7
A proper stationary projection with the correct end has one sheet
LemmaDependency level 7
Positive stationary mass from the bifurcate vacuum boundary
LemmaDependency level 7
Stationary parameter families have a decaying reference-slice residual
LemmaDependency level 7
Every negative-time cut has a closed selected domain and smooth endpoint extensions
LemmaDependency level 7
The truncated past height is Lipschitz and has exactly the pulled-back cuts as levels
LemmaDependency level 7
Uniform spatial components and the cylinder scale
LemmaDependency level 7
Quantifiers and parameter gap in the ADM smallness hypothesis
LemmaDependency level 7
Distinct boundary generators cannot cross strictly before a spacelike cut
LemmaDependency level 7
Flattening a smooth fixed boundary preserves the elliptic graph equation
LemmaDependency level 7
Compact normalized source data give a target-independent energy bound
LemmaDependency level 7
Time reversal supplies a source-normal generator without interior source focal points
Physics theoremDependency level 7
Imported geometric analytic connectedness and degenerate-component exclusions
Physics theoremDependency level 7
Axisymmetric vacuum uniqueness with complete global hypotheses
Physics theoremDependency level 7
Bifurcation extension, local non-Kerr deformations and smooth local rigidity
Physics theoremDependency level 7
Mass-one Ernst and radial primitive asymptotics
Physics theoremDependency level 7
Positivity of the Harrison denominator and horizon bound
Physics theoremDependency level 7
Local Kerr rigidity and conditional global identification
Physics theoremDependency level 7
Local Hawking and axial fields, boost and pole
Physics theoremDependency level 7
Stationary transfer of timelike curves
TheoremDependency level 7
Principal eigenfunctions and MOTS spectral stability
LemmaDependency level 8
Level-set area transfer retains generator multiplicity
LemmaDependency level 8