Alphabeta MathSupporting prerequisites · 03 — Komar flux equals the original stationary mass coefficient
Presents 21 supporting prerequisites, covering dependency levels 5–15.
1 · Prerequisites
- Regular non-degenerate stationary vacuum spacetime
- Differentiated asymptotic bounds and parameter uniformity
- Extrinsic curvature decay on the original stationary end
- Spacelike sections, maximality and lapse–shift data
- Riemannian volume density
- Integral of a compactly supported smooth density
- The Axiom of Choice
- A maximal stationary exterior has future access to its far tail
- Past access to the far tail by exact time reversal
- Causal curves, chronological and causal reachability, and inextendibility
- Stationary Lorentzian data and tensor conventions
- Integer-order Sobolev spaces and their norms
- Supplied CW logarithmic maximal ends have weak weighted initial data
- Interior Schauder estimate for uniformly elliptic equations
- Higher-order Sobolev embedding
- Spherical harmonic modes are orthogonal, complete for continuous data, and have polynomial evaluation bounds
- P.T. Chruściel and J.L. Costa, On uniqueness of stationary vacuum black holes · §2.1, PDF pp.5–6, especially (2.1)–(2.8) and footnote 3 on p.6
- D. Kennefick and N. Ó Murchadha, Weakly decaying asymptotically flat static and stationary solutions to the Einstein equations · §II, PDF p.6, (2.22)–(2.28); §III, p.7, Definitions 1–2; §IV, pp.10–12, (4.1)–(4.23)
- P.T. Chruściel, Asymptotic estimates in weighted Hölder spaces for a class of elliptic scale-covariant second order operators · §2 Theorem2.1, printed pp.27–29, (2.14)–(2.24), and the scaling estimates (2.12)–(2.13)
- R. Beig and P.T. Chruściel, The asymptotics of stationary electro-vacuum metrics in odd space-time dimensions · §4 equations(4.1)–(4.3), PDF pp.6–7; §2 footnote2 on PDF p.2
- Causal exterior, chronological boundaries and Cauchy data
- Exact asymptotically stationary end contract
- 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
- Mean-curvature variation and the stationary lapse equation
- Exterior Schrödinger maximum principles, Hopf growth and Harnack
- Connected components, quasicomponents, and totally disconnected spaces
- For a topological space the following agree: no separation exists, the only clopen subsets are ∅ and X, and every continuous map to the two-point discrete space is constant
- 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
- The adjugate gives an explicit inverse formula over a field
- 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
- Weighted exterior elliptic domains and admissible weights
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Normal graphs and the nonlinear mean-curvature map
- A timelike transverse field gives a fixed-boundary admissible graph neighborhood
- 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
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- The Axiom of Countable Choice (AC_ω)
- The stationary maximal vacuum divergence identity with the article sign
- Orientation induced on a hypersurface by a coorientation
- Riemannian divergence theorem
- The Cauchy-Schwarz inequality for finite sums
- Complete boundary initial-data and vanishing momentum interface
- Vacuum exterior-section constraints imply the dominant energy condition
- Zero boundary expansions without maximality
- Dan A. Lee, Martin Lesourd and Ryan Unger, Density and positive mass theorems for initial data sets with boundary · Definition 2.1 p.5; expansion convention p.6; Theorem 4.3 p.25
- Smooth positive inversion and composition in the weighted multiplier class
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Stationary parameter families have a decaying reference-slice residual
- Weighted differentiation and multiplication estimates
- The maximal equation upgrades pointwise order-four graph decay to weighted order five
- Weighted Hölder norms on a three-dimensional exterior
- Taylor's theorem with an integral remainder
- Stationary vacuum model and asymptotic normalization
- Remark 1.3: lapse bounds and stationary product
- Closed Ernst form and existence of normalized potential
- Local Hawking and axial fields, boost and pole
- Equivariant global Harrison primitive under weak original data
- Smooth transverse horizon branches are the null axes of the normal tube
- Normal boost action and invariant two-forms at a fixed surface
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Metric compatibility, torsion freedom and the metric-selected connection class
- Bifurcate Killing horizons, rotation and normalization
- Transverse embedded submanifolds intersect in the expected codimension
- Invertibility, Lorentz signature and stationary norm
- Mass-one convention and Harrison denominator
- Stationary Harrison one-form and metric
- Harrison electromagnetic form
- Single-potential construction of the Harrison one-form family
- The mass-one Killing form has electric seed flux minus four pi
- Harrison spatial two-form differs from the seed by an integrable tail
- Positive-parameter Harrison charge limits in the original end chart
- Closed electric and magnetic two-forms transfer their periods to the horizon
- The zero Harrison parameter follows from continuity on one fixed sphere
- Geometric electrovacuum equations and charge convention
- Positive stationary mass from the bifurcate vacuum boundary
- Mass-one Ernst and radial primitive asymptotics
- The exterior Cauchy component has a complete asymptotically Euclidean closure
- The flow of a vector field tangent to a closed embedded submanifold preserves it
- A. F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.1 equation (3.1) and orientation paragraph; Section 3.2 equations (3.11)–(3.24); Appendix A.2 equations (A.12)–(A.25) and the following zero-locus and stationary-gauge paragraphs.
- A working partial-Cauchy collar satisfies the source GR section equalities
- Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · GR/SBS pp3–4; Proposition2.1 p7; Lemma2.2 statement pp9–10, exact boost coordinates (2.4) pp10–11 and cutoff construction pp11–12, quotient-escape conclusion p13; nonrotating specialization pp13–14
- The causal domain of dependence of a partial Cauchy hypersurface is an open causally convex manifold on which the hypersurface is Cauchy
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal
- Temporal functions are time functions, and time functions forbid closed causal curves
- Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed
- The fundamental theorem on flows
- Compactly supported smooth vector fields are complete
- The smooth inverse function theorem on manifolds
- Local comparison of a riemannian metric with the euclidean metric
- A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values
- Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · Section 2.2, Lemma 2.2 proof, printed pp. 10–12, equation (2.4) and logarithmic cutoff; nonrotating extension, pp. 13–14, equation (2.5); GR/SBS, pp. 3–4.
- Smooth vacuum constraint data obtained by reflection of a canonical end section
- Smooth reflection of the stationary vacuum Killing initial data
- Riemannian speed and length
- Riemannian distance on a connected manifold
- The riemannian distance topology is the manifold topology
- Cauchy sequence in a metric space
- Complete metric space: every Cauchy sequence converges in the space
- A compact metric space is complete and totally bounded, and neither implication uses any choice principle
- Killing fields of a Lorentzian metric
- Robert Beig and Piotr T. Chruściel, Killing Initial Data · §2, vacuum Case2, pp4–5, equations(2.6) and(2.13)
- Piotr T. Chruściel, On completeness of orbits of Killing vector fields · MGHD existence/uniqueness discussion p2; Theorem1.1 p2; Corollary1.2(2) p3; Definition3.1 and sufficient falloff p12; Proposition2.4 pp9–10
- All-jet reflection parity from an exact local boost
- Reflection parities of canonical section data and stationary lapse–shift
- Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form
- A compact spacelike surface has a normal exponential tube
- Rigidity of isometric immersions of a connected Lorentzian manifold, and agreement of data-fixing embeddings on overlaps
- Robert Beig and Piotr T. Chruściel, Killing Initial Data · Section 2, printed pp. 4–6 (PDF pages 4–6): (2.6)–(2.8), (2.13), vacuum Case 2 on p. 5 and (2.15) with the vacuum discussion on p. 6.
- Piotr T. Chruściel, On completeness of orbits of Killing vector fields · Introduction, printed p. 2; Proposition 2.4 and its proof, printed p. 10 (PDF page 10).
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- 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
- The pullback of a covariant tensor field
- The Lie derivative of a tensor field
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- Tensoriality of the curvature, its coordinate formula and its point-normal metric formula
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Chronology, causality, strong causality, global hyperbolicity and time functions
- The flowout theorem
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Black and white exclusion is an ambient reachability condition
- Stationary maximal hypersurface existence with a horizon boundary
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
Lemma
Komar flux equals the original stationary mass coefficient
Lemma
Two-sided far-tail access preserves the ambient causal exterior
Theorem
Independent stationary maximal-gauge asymptotic expansion and KID input
Lemma
The source maximal-gauge expansion can be used on the same CW slice
Lemma
Identification of the spherical exterior and its positive stationary lapse
Lemma
The mean-curvature graph map is a smooth finite-jet expression
Lemma
A supplied complete maximal slice has zero extrinsic curvature in the zero-shift case
Theorem
External boundary positive-energy theorem without maximality
Lemma
Leading stationary modes and a spatial normalization of the same maximal end
Lemma
Compact-core Hölder superposition gives a continuously differentiable graph map
Lemma
The first possible shift and lapse logarithms vanish by indicial projection
Lemma
Stationarity supplies the weighted end differentiability estimates
Physics theorem
Weak original data furnish the one recovery family and pole
Theorem
External local boost collar and canonical temporal splice
Lemma
Canonical bifurcation collar joined to the original exterior Cauchy section
Lemma
Complete two-ended geometry and decay of the reflected data
Theorem
External vacuum KID development and complete stationary flow
Lemma
Vacuum development realizing the exterior and local boost germs
Lemma
A complete stationary boost band with a global local-isometry certificate
Lemma
Strongly causal complete stationary ambient gluing around the exterior
Lemma