Alphabeta MathArticle items · 07 — Asymptotically flat spacelike end and derivative decay
Presents 29 items from the article and its immediate prerequisites, covering dependency level 0.
1 · Prerequisites
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Metric compatibility, torsion freedom and the metric-selected connection class
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Diffeomorphisms and local diffeomorphisms of manifolds
- Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ
- Killing fields of a Lorentzian metric
- Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor
- Causal solutions, Killing horizons, escape and metric extensions
- The Axiom of Choice
- H. S. Reall, General Relativity (Part III, October 12, 2022) · Section 5.3, printed p. 56 (PDF p. 57), equations (5.21)–(5.22) and adjacent Killing-field text.
- Local and global flows generated by a vector field
- Embedded submanifolds and slice charts
- The timelike set has exactly two convex components, described in orthonormal coordinates
- Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral
- The differential of a smooth map
- The pullback of a covariant tensor field
- Coordinate formula for the differential
- The chain rule for differentials of smooth maps
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- The Darboux and Riemann definitions agree: a bounded f on [a,b] is Darboux integrable with integral I if and only if for every real ε > 0 there is a real δ > 0 such that |S(f,P,ξ) - I| < ε for every tagged partition of mesh below δ
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality
- Continuous functions on a closed interval attain their extrema
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Oriented Lorentzian fields, observers and coordinate conventions
- Lorentzian exterior calculus and Hodge star
- The inverse metric, musical isomorphisms and index raising
- The direct-sum model V⊕ iV of a complexification
- Smooth vector bundles, rank, fibres, and trivial bundles
- Smooth sections, local sections, and support
- Vector bundle maps over a smooth base map
- Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique
- The Axiom of Countable Choice (AC_ω)
- Lecture Notes on General Relativity · §1, equations (1.87)–(1.88), PDF p. 30
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Riemannian metric and riemannian manifold
- Laplace–Beltrami operator as the trace of the Hessian
- Smooth charts, atlases, and structures with boundary
- Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025) · Section 8, opening definitions surrounding Exercise 8.1, printed p. 139 (PDF page 139): the displayed Hölder seminorm precedes the exercise; the C^k,alpha convention follows it on the same page.
- Banach space
- C k map between Banach spaces
- Fredholm operator cokernel and index
- Compact linear operator
- Implicit function theorem for Banach spaces
- Fredholm index is stable under compact perturbations
- The standard product norms on a finite product of normed spaces
- Finite products of Banach spaces are Banach
- In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle
- Fréchet derivative between Banach spaces
- Metric continuity characterisations, with countable choice for the sequential converse
- The reals are complete
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- The derivative of the metric density and the chart formula for the divergence of a vector field
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
- Smooth embeddings
- The image of a smooth embedding is an embedded submanifold
- E. Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity (2007) · Sections 2.3.1-2.3.2, printed pp. 21-22, equations (2.27)-(2.31); section 2.3.4, printed pp. 23-24, equations (2.36), (2.42); section 2.4.2, printed p. 31, equations (2.74)-(2.77).
- Null normal pairs and screen geometry
- Local null normal pairs and the positive screen quotient
- Null focal points exclude later horismos generators
- AC implies DC implies countable choice
- Causal curves, chronological and causal reachability, and inextendibility
- Geometric electrovac equations and flux normalization
- Integration on an oriented embedded submanifold
- Pullback of forms is smooth functorial and preserves wedges
- Compact support of a differential form
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- Orientation reversal changes the integral sign
- Riemannian distance on a connected manifold
- Riemannian distance is a metric
- Embedded smooth submanifolds with boundary
- Cauchy sequence in a metric space
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- 3+1 Formalism and Bases of Numerical Relativity · Section 4.2.2, equations (4.28)–(4.31), p. 56
- A specified asymptotically Euclidean ADM end
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025) · Section 8, introductory Hölder definitions, printed p. 139 (PDF page 139), before Section 8.1.
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Oriented smooth manifolds and oriented charts
- Sean M. Carroll, Lecture Notes on General Relativity (1997) · Chapter 2, printed pp. 47 and 49 (PDF pp. 54 and 56), equations (2.28), (2.34) and adjacent metric/signature definitions.
- Sean M. Carroll, Lecture Notes on General Relativity (1997) · Chapter 5, printed pp. 135 and 140 (PDF pp. 142 and 147), equations (5.17)-(5.18) and (5.41).
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- The trace tr(A) as the sum of the diagonal entries
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- The Riemannian gradient is the metric dual of the differential
- Real and complex inner product spaces, with the inner product linear in the first argument
- Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, proof of Lemma B.1, (B.8)–(B.10), printed/PDF pp.38–39
- Smooth manifolds and their smooth charts
- A smooth tensor field
- Immersions and embeddings for manifolds with boundary
- Pullback of covariant tensors is smooth and functorial
- Riemannian volume density
- Pullback of densities by local diffeomorphisms
- Orientation-free density integration and its properties
- Collar neighborhood theorem
- Smooth partitions of unity exist on manifolds with boundary
- Existence uniqueness and smooth dependence of geodesics
- The exponential domain is open and the exponential map is smooth
- The smooth inverse function theorem on manifolds
- Domain and exponential map of a connection
- Smooth collars of a manifold boundary
- The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold
- Smooth functions on relatively open half-space sets
- Levi civita connection
- Fundamental theorem of riemannian geometry
- Geodesic scaling identity
- The fundamental theorem on flows
- Inward, outward, and boundary-tangent vectors
- Chain rule for smooth half-space maps
- Time-t flow maps are diffeomorphisms between open domains
- The boundary tangent space is the boundary-tangent hyperplane
- The differential of a diffeomorphism is an isomorphism
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- The Hessian matrix and critical points of a scalar field
- Positive definite, negative definite, semidefinite, and indefinite quadratic forms
- Second-order Taylor expansion f(a+h)=f(a)+∇ f(a)· h+1/2h^TH_f(a)h+o(‖h‖²)
- Codimension and hypersurfaces
- The inclusion of an embedded submanifold is a smooth embedding
- Immersions, submersions, and constant-rank maps
- The differential sends derivations to derivations and is linear
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- Geodesic of an affine connection
- Affine connection on a smooth manifold
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- 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
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Sections 1.2–1.3, 1.5 and 1.7, pp. 6–11; section 1.11, Proposition 1.16, pp. 15–17; section 2.3, p. 28; Theorem 2.12 and Remark 2.14, pp. 32–33; Proposition 2.23 and Theorem 2.24, p. 36. PDF page numbers equal printed page numbers.
- Topological manifolds with boundary
- Interior and boundary of a manifold with boundary
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- |d(x,A) - d(y,A)| ≤ d(x,y), so the distance to a fixed nonempty set is 1-Lipschitz
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- 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 subset of a metric space is closed and bounded
- A closed subset of a compact metric space is compact
- Continuity of a map of topological spaces at a point and globally
- 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
- Every nonempty finite set of reals has a maximum and a minimum
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Through each point there is a unique maximal integral curve
- Covariant derivative along a curve
- The Lie bracket of smooth vector fields
- Leibniz rules for the Lie bracket with function multiples
- Disk bundle, sphere bundle, and Thom space: the differential topology interface
- Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure
- The induced tangent bundle chart
- A smooth vector field is a smooth section of the tangent bundle
- Integral curves of a vector field
- Slice-chart restrictions form a smooth atlas
- Products of smooth manifolds have a canonical product smooth structure
- Future sets, past sets, achronal boundaries and their elementary properties
- 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
- Complete vector fields
- An open subset of a smooth manifold has a canonical restricted smooth structure
- Cʳ and smooth maps between smooth manifolds
- Derivations at a point and the tangent space
- The velocity derivation of a smooth curve
- Will J. Merry, Differential Geometry (2021) · Lecture 8, Proposition 8.2(iii), equations (8.1)-(8.3); Lecture 9, Definition 9.3, equation (9.2).
- The flowout theorem
- Cauchy sets are closed, split the manifold into past, set and future, and are locally Lipschitz graphs
- 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)
- Cᵏ Euclidean maps and diffeomorphisms
- Pullback of a riemannian metric as a tensor
- 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
- Continuity and derivatives of positive-base real powers
- 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
- 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
- Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)
- The graph of a smooth map is an embedded submanifold
- Sum, product, quotient, and chain rules from difference quotients
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Fundamental theorem of calculus for absolutely continuous functions
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
- A smooth differential k-form
- Local coordinate expression for a differential form
- Riemannian volume form on an oriented manifold
- The Lebesgue integral is linear on L¹(μ)
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The modulus of an integral is bounded by the integral of the modulus
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on Y^X and on C(X,Y)
- Coordinate differentials form the dual cotangent basis
- Wedge monomials in a dual basis form a basis
- 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 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 Borel sigma-algebra of a topological space
- A continuous map has Borel preimages of Borel sets
- Integrable real and complex functions, and their integrals
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- Integrating against a density agrees with integrating the product
- Pointwise Borel nonnegative densities
- Intrinsic density measure and its chart restriction
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Finite and countable subadditivity of measures
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 0
Asymptotically flat spacelike end and derivative decay
DefinitionDependency level 0
Bifurcate Killing horizons, rotation and normalization
DefinitionDependency level 0
Periodic isometries and invariant normal averaging
DefinitionDependency level 0
Complex self-dual forms and metric-dependent contraction
DefinitionDependency level 0
Weighted exterior elliptic domains and admissible weights
DefinitionDependency level 0
Fredholm and Banach continuation interfaces
DefinitionDependency level 0
Normal graphs and the nonlinear mean-curvature map
DefinitionDependency level 0
Null expansions, screen quotients and focal data
DefinitionDependency level 0
Geometric electrovacuum equations and charge convention
DefinitionDependency level 0
Spacelike sections, maximality and lapse–shift data
DefinitionDependency level 0
Differentiated asymptotic bounds and parameter uniformity
DefinitionDependency level 0
Stationary Lorentzian data and tensor conventions
LemmaDependency level 0
Trace-free three-dimensional coercivity for the logarithmic gradient
LemmaDependency level 0
Isometric target areas remain fixed while collar errors vanish
LemmaDependency level 0
A compact smooth boundary has a uniform inward normal exponential collar
LemmaDependency level 0
Equal fixed-surface flow heights imply boundary tangency
LemmaDependency level 0
Second-order contact on a closed half-domain has nonnegative Hessian
LemmaDependency level 0
An exact local Killing field is Killing to every order on a horizon
LemmaDependency level 0
Intersecting a chronological past boundary with a spacelike section
LemmaDependency level 0
Eventual access is uniform on the trimmed compact inside core
LemmaDependency level 0
Temporal normalization makes the null geodesic flow compact over compact bases
LemmaDependency level 0
Admissible domains for outward normal graphs
LemmaDependency level 0
Causal curves from a common chronological boundary obey two barriers
LemmaDependency level 0
A stationary product rules out a zero Hawking coefficient
LemmaDependency level 0
A working bifurcation collar supplies the exact GR section
LemmaDependency level 0
A decaying stationary time gauge gives bounded section displacement
LemmaDependency level 0
The reciprocal-potential invariant determines the end spin bound
LemmaDependency level 0
Stationary product time is a Cauchy temporal coordinate
LemmaDependency level 0