Alphabeta MathArticle items · 10 — Regular non-degenerate stationary vacuum spacetime
Presents 28 items from the article and its immediate prerequisites, covering dependency level 2.
1 · Prerequisites
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Killing fields of a Lorentzian metric
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Metric compatibility, torsion freedom and the metric-selected connection class
- Null normal pairs and screen geometry
- Local null normal pairs and the positive screen quotient
- Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)
- Embedded submanifolds and slice charts
- Causal exterior, chronological boundaries and Cauchy data
- Spacelike sections, maximality and lapse–shift data
- Bifurcate Killing horizons, rotation and normalization
- Differentiated asymptotic bounds and parameter uniformity
- Diffeomorphisms and local diffeomorphisms of manifolds
- The Axiom of Choice
- Annular scaling and fixed interior covers
- Ellipticity and zero-slope forcing of the stationary maximal flux
- Gradient Hölder control by differentiated flux and one-dimensional lifting
- Uniform coefficient regularity before the final graph estimate
- Linear fourth derivative bound with controlled realized coefficients
- Normal graphs and the nonlinear mean-curvature map
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Interior Schauder estimate for uniformly elliptic equations
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Fundamental theorem of calculus for Banach-valued continuous curves
- The notation Hᵏ and the reserved zero-boundary symbol
- Local weak solutions of a divergence-form operator
- 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
- 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)
- Divergence on a bounded C1 Euclidean domain
- Xavier Fernández-Real and Xavier Ros-Oton, Regularity Theory for Elliptic PDE · Theorem 3.17, printed p. 93, PDF page 99 (zero-based page 98 in the currently retrieved 236-page PDF)
- Causal curves, chronological and causal reachability, and inextendibility
- 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
- Pointwise local causality: closed cones, openness of the local chronological future, no local closed causal curves, corner smoothing and closedness of the local causal relation
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points
- Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic
- 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
- Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- AC implies DC implies countable choice
- Local and global flows generated by a vector field
- 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
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Boost-frame bounds for a smooth observer and arbitrary spacelike normals
- Laplace–Beltrami operator as the trace of the Hessian
- Fundamental theorem of riemannian geometry
- The gradient is characterized by inner products
- Gaussian maximal graph equation on a compact spacelike slope cone
- A Euclidean bump for a compact set inside an open set
- 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
- 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
- Stationary Lorentzian data and tensor conventions
- The fundamental theorem on flows
- The parametrized implicit function theorem with Cᵏ regularity
- Riemannian speed and length
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Weighted exterior elliptic domains and admissible weights
- Weighted Hölder norms on a three-dimensional exterior
- Weak compact-domain comparison for continuous nonnegative potentials
- Coordinate and weak forms of the geometric Schrödinger equation
- The riemannian volume density is coordinate independent
- Smooth partitions of unity exist on manifolds
- Positive smooth densities give Radon volume
- Measurable integration extends smooth density integration
- Lorentzian exterior calculus and Hodge star
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Regular bifurcate null horizon data before rigidity
- Complete vector fields
- The Axiom of Countable Choice (AC_ω)
- Topological manifolds are locally compact and locally path connected
- A connected, locally path-connected space is path-connected, because its path components are open
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Banach space
- A sequence of real-valued functions converges uniformly if and only if it is uniformly Cauchy
- The uniform limit of continuous real-valued functions on a metric space is continuous
- If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit
- Continuous mixed partials of order k are invariant under permutations
- A subset of ℝ is connected if and only if it is order-convex, that is, an interval
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- A closed subspace of a Banach space is Banach
- Pointwise Gauss–Codazzi identities for the normal angle
- A product of finitely many compact spaces is compact in the product topology
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1, pp.35–36 (hypotheses); proof p.38, (B.8), the following unnumbered scalar inequality, and the transition to (B.9).
- The inward normal coordinate is intrinsic distance from the boundary
- Coordinate formula for riemannian divergence
- Gradient hessian and divergence connection formulas
- Riemannian volume density
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- A compact smooth boundary has a uniform inward normal exponential collar
- Riemannian volume is the radon measure of the riemannian density
- Pullback of densities by local diffeomorphisms
- Intrinsic density measure and its chart restriction
- Borel sigma-algebra of a product of second-countable spaces
- On Borel subsets of ℝᵐ⁺ⁿ, the product λₘ × λₙ agrees with λₘ₊ₙ
- The completion domain and proposed completed set function of a measure space
- The completed product measure
- Second-order contact on a closed half-domain has nonnegative Hessian
- Mean curvature at identical spacelike graph jets has positive Hessian symbol
- Equal fixed-surface flow heights imply boundary tangency
- Ordered timelike flow graphs lie on the corresponding geometric side
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Periodic isometries and invariant normal averaging
- Normal boost action and invariant two-forms at a fixed surface
- A compact spacelike surface has a normal exponential tube
- eta-orthogonality and orthogonal complements in a Lorentz vector space
- The chain rule for differentials of smooth maps
- Immersions and embeddings for manifolds with boundary
- Affine null-generator profiles vanish exactly on the bifurcation section
- The action of a vector field on smooth functions
- Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- The connected subspaces of ℝ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in ℝ"
- A continuous image of a connected space is connected, and connectedness is a topological property
- A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member
- Rotation, surface gravity and horizon degeneracy
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- The flowout theorem
- Embedded submanifolds admit local defining submersions
- Admissible domains for outward normal graphs
- Canonical outward normals and null gauge on a graph
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Future sets, past sets, achronal boundaries and their elementary properties
- The flow-box theorem
- Smooth event-boundary branches are achronal null geodesic sheets
- Embedded smooth submanifolds with boundary
- Smooth collars of a manifold boundary
- Kerr exterior tensor including the extremal parameter range
- The Kerr Metric · §2, equations (2.1), (2.5)–(2.8), PDF pp. 2–3
- Normalized Komar data on a nonrotating horizon section
- The six hyperbolic functions and their natural domains
- Addition formulas, identities, parity, and derivatives of the hyperbolic functions
- Stationary end normalization from quotient compactification
- A decaying stationary time gauge gives bounded section displacement
- Bounded spacelike stationary graphs remain Cauchy
- The smooth inverse function theorem on manifolds
- Stationary product time is a Cauchy temporal coordinate
- Christoffel formula for the levi civita connection
- Christoffel symbol transformation law
- A smooth radial profile with a prescribed inner plateau
- Extreme value theorem: a continuous real function on a nonempty compact subset of ℝ attains a greatest and a least value
- The fundamental theorem of calculus and integration by parts
- If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
- Cauchy criterion for improper integrals
- Every path-connected space is connected, and every path component lies inside a component
- A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point
- The inverse metric, musical isomorphisms and index raising
- Smooth manifolds and their smooth charts
- Smooth embeddings
- Riemannian metric and riemannian manifold
- Pullback of a riemannian metric as a tensor
- The pullback of a covariant tensor field
- Cᵏ Euclidean maps and diffeomorphisms
- Affine connection on a smooth manifold
- Christoffel symbols of an affine connection
- Levi civita connection
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- Euclidean spheres and closed balls as subspaces of ℝⁿ
- The cross product in ℝ³
- Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
- Proper maps between Euclidean open sets
- Ernst sublevels and the Mars–Simon tensor interface
- 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
- Products of smooth manifolds have a canonical product smooth structure
- Equivalence of the absolutely-continuous and continuous causal conventions, and one-sided inextendible extensions
- Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed
- AC supplies the countable and dependent choices used in Banach integration
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 2
Regular non-degenerate stationary vacuum spacetime
LemmaDependency level 2
Uniform annular Schauder bootstrap for spacelike graph equations
LemmaDependency level 2
An open chronological exterior is causally convex without a timelike-end assumption
LemmaDependency level 2
Future timelike cylinder translations make their chronological pasts increase
LemmaDependency level 2
Projection estimates for a Gaussian collar cutoff
LemmaDependency level 2
Uniform elliptic extension and flattening before the boundary gradient estimate
LemmaDependency level 2
Stationary coefficients in a transverse collar retain the tangential motion
LemmaDependency level 2
Decaying Dirichlet kernel and the exact closed-component exception
LemmaDependency level 2
Constant homothety preserves the original vacuum and maximal section
LemmaDependency level 2
Connectedness and causal convexity of a chronological exterior
LemmaDependency level 2
Weighted Hölder spaces and their Dirichlet kernels are Banach
LemmaDependency level 2
Uniform scalar angle inequalities from local background tensors
LemmaDependency level 2
Distance Laplacian and normal-collar volume density are uniformly bounded
LemmaDependency level 2
Strict mean-curvature barriers cannot have ordered boundary tangency
LemmaDependency level 2
Circle-invariant normal coordinates linearize a bifurcate Killing boost
LemmaDependency level 2
A variable-time flow through a fixed boundary has the boosted unit normal
LemmaDependency level 2
Affine generator profiles give constant signed acceleration
LemmaDependency level 2
Closure of the exterior Cauchy section meets only the double horizon
LemmaDependency level 2
The selected exterior is connected and causally convex
LemmaDependency level 2
Zero stationary field gives the local bifurcate Killing-horizon and boundary data
LemmaDependency level 2
Outgoing expansion depends smoothly on the second graph data
LemmaDependency level 2
Smooth transverse horizon branches are the null axes of the normal tube
LemmaDependency level 2
The Boyer–Lindquist spatial end and differentiated expansion
LemmaDependency level 2
The normal binormal is invariant under oriented boosts
LemmaDependency level 2
Extending normalized end coordinates while preserving the section
LemmaDependency level 2
Normalized Ernst data and zero Mars–Simon survive stationary gauges
LemmaDependency level 2
A generic timelike product for a prescribed smooth Cauchy section
LemmaDependency level 2