Alphabeta MathArticle items · 09 — Compact causal diamonds close the selected-generator endpoint domain
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 1–2.
1 · Prerequisites
- Temporal normalization makes the null geodesic flow compact over compact bases
- The Axiom of Choice
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Topological manifolds are metrizable and paracompact
- For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
- A point lies in the closure of A iff some sequence in A converges to it, and a set is closed iff it is sequentially closed
- Embedded submanifolds and slice charts
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 — dimension-independent causal conventions · Sections 1.2–1.3, PDF pp. 6–7; sections 1.5 and 1.7, pp. 9–11; sections 1.10–1.11, pp. 15–16; section 2.3, p. 28; Definition 4.74, p. 131; Definition 4.117 and following traditional-definition paragraph, pp. 146–147.
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 — closed compact-set cones and compact causal hulls · Theorem 4.12 (i)–(iv), PDF pp. 109–110; Definition 4.117, its final compact-set assertion and proof, p. 146; traditional-definition paragraph, pp. 146–147.
- Admissible domains for outward normal graphs
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Null normal pairs and screen geometry
- Pullback of a riemannian metric as a tensor
- Musical isomorphisms
- Stationary Lorentzian data and tensor conventions
- Geometric electrovacuum equations and charge convention
- Lorentzian exterior calculus and Hodge star
- The inverse metric, musical isomorphisms and index raising
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Two continuous maps into a Hausdorff space that agree on a dense subset are equal
- A smooth differential k-form
- The Lie derivative of a function
- The Lie derivative of a tensor field
- The flow-box theorem
- The coordinate formula for the Lie derivative of a covariant tensor
- Tensor Lie derivative agrees with X on functions and bracket on vector fields
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Clairaut--Schwarz theorem for continuous second partial derivatives
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- The smooth inverse function theorem on manifolds
- Smooth manifolds and their smooth charts
- Slice-chart restrictions form a smooth atlas
- eta-orthogonality and orthogonal complements in a Lorentz vector space
- Embedded smooth submanifolds with boundary
- Smooth charts, atlases, and structures with boundary
- Tangent and cotangent bundles extend over a boundary
- Inward, outward, and boundary-tangent vectors
- 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
- 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
- Bifurcate Killing horizons, rotation and normalization
- Metric compatibility, torsion freedom and the metric-selected connection class
- Future sets, past sets, achronal boundaries and their elementary properties
- Integral curves of a vector field
- Geodesic of an affine connection
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- Periodic isometries and invariant normal averaging
- A sphere has a nowhere zero tangent vector field iff its dimension is odd
- Local and global flows generated by a vector field
- The chain rule for differentials of smooth maps
- Coordinate formula for the differential
- Continuous functions on a closed interval attain their extrema
- 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)
- Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor
- Covariant derivative along a curve is independent of frame and extension
- Stokes theorem for smooth singular chains
- Integration on an oriented embedded submanifold
- Top de Rham integration equals fundamental-class evaluation
- Fundamental class of a compact oriented manifold
- Smooth singular chains compute singular homology
- Smooth singular chain and cochain complexes
- Pullback of forms is smooth functorial and preserves wedges
- The general Stokes theorem
- The exterior derivative commutes with pullback
- Differentiated asymptotic bounds and parameter uniformity
- Complex self-dual forms and metric-dependent contraction
- Real and imaginary parts, complex conjugation, and modulus
- The wedge product of differential forms
- The pullback of a differential form
- Subextremal Kerr exterior in length coordinates
- Kerr-Newman metric and potential on the exact Boyer-Lindquist domain
- The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation
- The Kerr Metric · §2, equations (2.1), (2.5)–(2.8), PDF pp. 2–3
- Null expansions, screen quotients and focal data
- Affine connection on a smooth manifold
- Covariant derivative along a curve
- Coordinate formula for the Lie bracket
- Sum, product, quotient, and chain rules from difference quotients
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- A function continuous on an interval I whose derivative vanishes at every interior point of I is constant on I; consequently two such functions with the same derivative differ by a constant
- Picard-Lindelöf local existence and uniqueness for first-order systems
- Stationary product time is a Cauchy temporal coordinate
- Smooth embeddings
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Sections 1.2–1.7 and 1.11; Definitions 1.21–1.22 and following assertion; Theorem 2.12 and Remark 2.14; Definition 2.16
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Theorem 4.119, both the Cauchy/global-hyperbolicity equivalence and the timelike-fiber topological splitting
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Definitions 3.29 and 3.35, including the observation immediately after Definition 3.35
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Killing fields of a Lorentzian metric
- Periods and primitives on a spherical exterior
- Normal neighborhood and normal coordinate chart
- R. Beig and B. Schmidt, Time-Independent Gravitational Fields · §2.1 (2.33)–(2.38), p.7; §2.3 (2.70)–(2.72), pp.10–11; §3.1 (3.5), (3.15)–(3.25), pp.17–20; §3.2 (3.51)–(3.52), p.23
- S. Dain, Initial data for stationary space-times near space-like infinity · §2 (17)–(19) and Theorem2.1, p.4; (20), pp.5–6; Lemma2.5 and (36)–(43), pp.7–8
- Observers, rest spaces and the spatial projection at an event
- Pairing of an observer with future causal vectors
- Smooth formal matter, EM tensors and observer quantities
- Pointwise Maxwell and matter energy-condition vocabulary
- EM, dust and perfect-fluid energy inequalities
- Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique
- Through each point there is a unique maximal integral curve
- Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral
- Null Raychaudhuri and focal-point bound
- Null focal points exclude later horismos generators
- 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 chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- 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)
- For A∈ Mₙ(R) and columns u,v over a commutative ring, det(A+uv^T)=det(A)+v^Tadj(A)u
- Conformal Hodge scaling in every degree
- Orientation of a finite-dimensional real vector space
- A real linear isomorphism preserves or reverses orientation according to the sign of its determinant
- Linear pullback respects tensor products and permutations
- The wedge product of alternating covectors
- Alexandru D. Ionescu and Sergiu Klainerman, On the uniqueness of smooth, stationary black holes in vacuum (arXiv:0711.0040v2) · Section 1.1, printed/PDF p. 5, equation (1.1) and adjoining Ricci-flat condition; Section 1.2, printed/PDF p. 6, opening assumptions, AF, equation (1.4), T = partialₜ, and footnote 4.
- Spacelike sections, maximality and lapse–shift data
- Normal graphs and the nonlinear mean-curvature map
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- Laplace–Beltrami operator as the trace of the Hessian
- Gradient hessian and divergence connection formulas
- The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- E. Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity (2007) · Section 9.2.2, equations (9.4) and (9.15), printed pp. 153–155
- Causal exterior, chronological boundaries and Cauchy data
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum (arXiv:0711.0040v2) · Section 1.2, AF, equations (1.4)–(1.5), printed/PDF pp. 6–7, including footnote 4.
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Diffeomorphisms and local diffeomorphisms of manifolds
- Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · Version 1, section 1.1, opening setup and GR, equation (1.1), through the sentence that Sigma⁰ is t=0 on the end, immediately before SBS.
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Definitions 2.19 (achronal), 2.95 (acausal), 2.130 (edge), and 3.35 (partial Cauchy hypersurface).
- Regular bifurcate null horizon data before rigidity
- S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · Section 1.1, opening setup and footnote 3; GR, equations (1.1)–(1.3) and footnote 5; SBS and footnote 6.
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Section 2.18, Definition 2.130, printed/PDF p. 76; section 3.2, Definition 3.35, printed/PDF p. 95.
- E. Gourgoulhon and J. L. Jaramillo, A 3+1 perspective on null hypersurfaces and isolated horizons, arXiv:gr-qc/0503113v2 · Section 2.2, (2.10)–(2.11), pp. 18–19; sections 2.6–2.7, (2.46)–(2.57) and Remark 2.8, pp. 28–31; section 5.6, (5.65), (5.71) and Remark 5.2, pp. 72–74; section 5.8, (5.114), p. 81 (printed and one-based PDF pages).
- The n-sphere with its standard smooth atlas
- The pullback of a covariant tensor field
- The Kerr Metric · §2, equations (2.1), (2.5)–(2.8), PDF pp. 2–3
- Killing two-form and normalized Ernst notation
- Surface integration on compact C1 hypersurfaces
- Chart and partition independence of surface measure
- For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact
- A regular level set is an embedded submanifold
- The exterior derivative by the invariant vector-field formula
- The Lie derivative of a differential form
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
LemmaDependency level 1
Compact causal diamonds close the selected-generator endpoint domain
LemmaDependency level 1
Canonical outward normals and null gauge on a graph
LemmaDependency level 1
Extending smooth equation residuals from a dense nonzero-field locus
LemmaDependency level 1
Stationary flow-box data for a nonzero Killing field
LemmaDependency level 1
A compact spacelike surface has a normal exponential tube
LemmaDependency level 1
Smooth event-boundary branches are achronal null geodesic sheets
LemmaDependency level 1
A circle action on a sphere has a pole with trivial normal action
LemmaDependency level 1
A Killing field vanishing on a spacelike surface has a normal boost derivative
LemmaDependency level 1
Closed electric and magnetic two-forms transfer their periods to the horizon
LemmaDependency level 1
Harrison spatial two-form differs from the seed by an integrable tail
LemmaDependency level 1
Kerr exterior tensor including the extremal parameter range
LemmaDependency level 1
A nonzero degenerate generator cannot vanish at a regular bifurcation section
LemmaDependency level 1
Bounded spacelike stationary graphs remain Cauchy
LemmaDependency level 1
Stationary end normalization from quotient compactification
LemmaDependency level 1
The stationary Killing energy is constant along every affine null segment
LemmaDependency level 1
Maxwell algebra supplies timelike and null convergence
LemmaDependency level 1
Normal boost action and invariant two-forms at a fixed surface
LemmaDependency level 1
Null optical comparison with affine and screen conventions
LemmaDependency level 1
Rank-one tangent changes, signature and Hodge transfer
PostulateDependency level 1
Stationary vacuum model and asymptotic normalization
TheoremDependency level 1
Mean-curvature variation and the stationary lapse equation
DefinitionDependency level 2
Exact asymptotic and orbit package for the IK criterion
DefinitionDependency level 2
Radu nontrapping global regularity package
DefinitionDependency level 2
AIK perturbative global regularity and bifurcation package
DefinitionDependency level 2
A specified future ingoing Kerr extension and its time orientation
DefinitionDependency level 2
Mass-one convention and Harrison denominator
DefinitionDependency level 2
Stationary Harrison one-form and metric
DefinitionDependency level 2