Alphabeta MathArticle items · 14 — Geometric data for the analytic multiple-horizon comparison
Presents 28 items from the article and its immediate prerequisites, covering dependency level 5.
1 · Prerequisites
- Stationary Lorentzian data and tensor conventions
- Staticity, axisymmetry and black-hole regularity hypotheses
- Rotation, surface gravity and horizon degeneracy
- Causal exterior, chronological boundaries and Cauchy data
- The Axiom of Choice
- Kerr exterior tensor including the extremal parameter range
- A causal barrier identifies the future Kerr event horizon
- The Komar angular-momentum flux of the Kerr end
- The Kerr end is an ADM rest frame
- A specified future ingoing Kerr extension and its time orientation
- Outgoing Kerr null curves and the selected future null infinity
- Asymptotic mass and axial angular-momentum conventions
- The Boyer–Lindquist spatial end and differentiated expansion
- The asymptotic metric flux of Kerr equals its mass parameter
- H. S. Reall, Black Holes, Part III lecture notes, 15 March 2024 · §9.4, paragraph following equation (9.20), printed p.118 (PDF p.119)
- Éric Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity (6 March 2007) · §2.3.4 (2.43), printed p.24 (PDF p.24); §7.5.1 (7.60)–(7.63), printed p.113 (PDF p.113); §7.6.3 (7.100)–(7.104), printed pp.121–122 (PDF pp.121–122)
- Stationary section time and translated maximal sections
- Comparison spheres and their stationary cylinders
- Null expansions, screen quotients and focal data
- Spacelike sections, maximality and lapse–shift data
- Riemannian distance on a connected manifold
- 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
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Null normal pairs and screen geometry
- Lorentzian exterior calculus and Hodge star
- Riemannian volume form on an oriented manifold
- Fundamental theorem of riemannian geometry
- Riemann curvature four-tensor
- Admissible domains for outward normal graphs
- Canonical outward normals and null gauge on a graph
- Outgoing expansion depends smoothly on the second graph data
- Linear and path-independent normal-graph variation of expansion
- Boost dependence of expansion variation and the MOTS gauge
- The Axiom of Countable Choice (AC_ω)
- Integral of a compactly supported top form
- Riemannian volume is the radon measure of the riemannian density
- Integrable real and complex functions, and their integrals
- Measurable integration extends smooth density integration
- Stationary maximal graphs and normalized graph heights
- Local and global flows generated by a vector field
- A smooth spacelike reference section is a bounded rapidity graph
- The stationary product extends the local boost and circle to a complete collar
- Circle-invariant normal coordinates linearize a bifurcate Killing boost
- Diffeomorphisms and local diffeomorphisms of manifolds
- Smooth collars of a manifold boundary
- 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
- 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
- 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
- Riemannian speed and length
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
- The entire target segment stays on the chronological boundary
- Endpoint orthogonality without an acausal target assumption
- Null focal points exclude later horismos generators
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- A compact chronological target supplies a causal endpoint
- A compact-target boundary point has a null geodesic segment
- Local null normal pairs and the positive screen quotient
- The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Metric compatibility, torsion freedom and the metric-selected connection class
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- AC implies DC implies countable choice
- Every tangent vector is the velocity of a smooth curve
- Constant homothety preserves the original vacuum and maximal section
- Vacuum Killing and Ernst differential identities
- Killing two-form and normalized Ernst notation
- Positive-lapse stationary graphs give temporal level functions and closed restrictions
- 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
- Riemannian metric and riemannian manifold
- The flow-box theorem
- The smooth inverse function theorem on manifolds
- Smooth dependence of ODE solutions on parameters
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- Riemannian volume density
- Intrinsic density measure and its chart restriction
- Extended riemannian distance on a disconnected manifold
- The riemannian distance topology is the manifold topology
- Leon Simon, Lectures on PDE · Lecture 18, (P), (*) and (**), printed pp.216–217, Theorem 1 p.217 (PDF sheet 111); Lecture 12, Theorem 2, printed pp.126–127 (PDF sheet 66), m=1.
- Induced metric and graph normal inherit the required weighted regularity
- Smooth positive inversion and composition in the weighted multiplier class
- Weighted Hölder spaces and their Dirichlet kernels are Banach
- Weighted Hölder norms on a three-dimensional exterior
- One extra pointwise derivative supplies scaled Hölder control
- Weighted differentiation and multiplication estimates
- Pairing of an observer with future causal vectors
- Smooth partitions of unity exist on manifolds with boundary
- Normal graphs and the nonlinear mean-curvature map
- The fundamental theorem on flows
- Weighted exterior elliptic domains and admissible weights
- 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)
- The maximal equation upgrades pointwise order-four graph decay to weighted order five
- Geometric electrovacuum equations and charge convention
- Stationary maximal graphs have an explicit uniformly spacelike flux
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- Lipschitz height functions, cut measure and Jacobians
- Multivariable Taylor formula with a Lagrange remainder along a line segment
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Clairaut--Schwarz theorem for continuous second partial derivatives
- Continuous functions have an oriented Riemann integral
- Killing fields of a Lorentzian metric
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Bounded boost times give a uniform Gaussian collar cone
- Spacelike normal projection is locally invertible with positive boundary jet
- A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point
- 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
- A closed subset of a compact metric space is compact
- A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
- 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
- Proper maps between Euclidean open sets
- Smooth charts, atlases, and structures with boundary
- Distance from a point to a subset
- Distance to a nonempty subset is one lipschitz
- Local comparison of a riemannian metric with the euclidean metric
- Absolute continuity on a compact interval
- Temporal functions are time functions, and time functions forbid closed causal curves
- Fundamental theorem of calculus for absolutely continuous functions
- Gronwall's integral inequality with variable and constant coefficients
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- The inverse metric, musical isomorphisms and index raising
- The observer rest-space decomposition and the spatial projection
- The exterior Cauchy component has a complete asymptotically Euclidean closure
- Stationary end access and future orientation need no maximal slice
- A working partial-Cauchy collar satisfies the source GR section equalities
- Zero stationary field gives the local bifurcate Killing-horizon and boundary data
- Differentiated asymptotic bounds and parameter uniformity
- Bifurcate Killing horizons, rotation and normalization
- Regular non-degenerate stationary vacuum spacetime
- Embedded submanifolds admit local defining submersions
- D. Sudarsky and R. M. Wald, Mass Formulas for Stationary Einstein–Yang–Mills Black Holes and a Simple Proof of Two Staticity Theorems · PDF p.5, equation (23) and following paragraph; p.2, horizon generator normalization (4)
- H. S. Reall, Black Holes, Part III lecture notes, 15 March 2024 · §8.1, printed pp.105–106 (PDF pp.106–107), (8.2) and the orientation example (8.11)–(8.13); §8.2, printed pp.107–108 (PDF pp.108–109), (8.14) sphere convention and (8.24) Komar definition; §9.3, printed p.117 (PDF p.118), zeroth-law theorem and (9.16)–(9.19); §8.4, printed p.113 (PDF p.114), stationary ADM/Komar equality and (8.43)–(8.45)
- The riemannian volume density is coordinate independent
- Riemannian volume of a compactly supported smooth density
- Orientation-free density integration and its properties
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Integration on an oriented embedded submanifold
- H. S. Reall, Black Holes, Part III lecture notes, 15 March 2024 · Conventions, printed p.1; section 8.2, equation (8.25), printed p.109 (PDF page 110).
- Complex self-dual forms and metric-dependent contraction
- Harrison equation residuals are continuous functions of stationary two-jets
- Stationary flow-box data for a nonzero Killing field
- Metric continuity characterisations, with countable choice for the sequential converse
- Normalized Komar data on a nonrotating horizon section
- Constant surface gravity gives the normalized horizon integral
- A bifurcate future horizon has an oriented defining pair near its section
- The inward null family reaches the future local horizon in short affine time
- Small outward normal graphs have controlled inward null initial jets
- Exterior exhaustion bounds for normalized Schrödinger solutions
- Hopf positivity and uniform linear growth on a fixed compact collar
- Multiplicative interior Harnack for geometric Schrödinger solutions
- Finite Harnack chains on fixed compact connected interior sets
- Positive anchor values and the limits of uniform Harnack propagation
- Fredholm and Banach continuation interfaces
- Exterior Dirichlet parametrix and compact decaying potentials
- Decaying Dirichlet kernel and the exact closed-component exception
- Bounded inverse theorem
- Non-existence of multiple-black-hole solutions close to Kerr-Newman · Abstract, §1.2 and §2.2, Main Theorem
- Stationary vacuum model and asymptotic normalization
- Jörg Hennig, The balance problem for n aligned black holes (arXiv:2604.12134v1) · PDF p. 1, abstract and section 1 Introduction, especially the opening paragraph and the paragraph beginning "We investigate this problem".
- Ionescu–Klainerman local Kerr criterion with full interface
- Ernst sublevels and the Mars–Simon tensor interface
- Exact asymptotic and orbit package for the IK criterion
- Normalized Ernst data and zero Mars–Simon survive stationary gauges
- Killing flow preserves its complex two-form
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- Stationary transport of vanishing from a section
- Exact bifurcate-sphere package for the IK criterion
- Time-t flow maps are diffeomorphisms between open domains
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- A. D. Ionescu and S. Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · Section 1.2, AF and (1.5), printed/PDF pp. 6–7; Section 4.2 (4.17), p. 22, Section 4.3 (4.29), p. 24, and Section 4.4 (4.33), p. 25; Section 7 (7.1)–(7.3), p. 35, Lemma 7.1 (7.17)–(7.18), p. 37; Section 8 (8.1), Lemma 8.1 and its proof, p. 41; Section 8.2 completion, pp. 48–49.
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
DefinitionDependency level 5
Geometric data for the analytic multiple-horizon comparison
DefinitionDependency level 5
Kerr metric, parameters, exterior and extremality
DefinitionDependency level 5
Compact remainder, causal cut and generator endpoint map
DefinitionDependency level 5
Surface measure, flux densities and null normal gauge
LemmaDependency level 5
An existing stationary graph supplies global projection before any estimate
LemmaDependency level 5
Boundary membership excludes interior focal points from the target
LemmaDependency level 5
The mass factor in the recovered Ernst identity
LemmaDependency level 5
Uniform interior graph geometry and a bounded finite chart cover
LemmaDependency level 5
The transverse lapse is a positive weighted Dirichlet multiplier
LemmaDependency level 5
A timelike transverse field gives a fixed-boundary admissible graph neighborhood
LemmaDependency level 5
The graph second form and electrovacuum Jacobi potential have weighted order four
LemmaDependency level 5
A compact normal-flow collar has uniformly small area distortion
LemmaDependency level 5
Weight-one regular axis gives elementary flatness of small axial orbits
LemmaDependency level 5
A complete timelike collar flow is a reference product
LemmaDependency level 5
Radial comparison gives proper normal projection onto a fixed target collar
LemmaDependency level 5
A finite temporal target cannot be reached from a sufficiently small collar
LemmaDependency level 5
The nonrotating bifurcation gives positive normalized mass before staticity
LemmaDependency level 5
The exhaustive analytic horizon-component case split
LemmaDependency level 5
Positive smooth horizon area and the exact degenerate angular-momentum test
LemmaDependency level 5
Passing a local Harrison identity through compatible stationary jets
LemmaDependency level 5
Degeneracy forces zero mass when global Komar transport applies
LemmaDependency level 5
Fixing the Ernst constant by integrable asymptotic twist
LemmaDependency level 5
An exterior-contained null segment to a distant target cannot start inward
LemmaDependency level 5
Exterior Schrödinger maximum principles, Hopf growth and Harnack
LemmaDependency level 5
Lemma 3.1: weighted Dirichlet Laplacian isomorphism
Open problemDependency level 5
Smooth stationary multiple-black-hole nonexistence
PostulateDependency level 5
Physical motivation against stationary multiple black holes
Physics theoremDependency level 5