Alphabeta MathHarrison fields, horizons and uniqueness inputs
Connect the common data to Ernst potentials, Harrison transformations, invariant horizon tensors, stationary slices and established uniqueness criteria. Keep each imported result tied to its stated hypotheses and each local consequence tied to its existing argument before article applications.
1 · Prerequisites
- Stationary Lorentzian data and tensor conventions
- Ordered horizon rods and twist-jump labels
- A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant
- Han–Khuri–Weinstein–Xiong, Kerr Black Hole Uniqueness · §1, equations (1.1)–(1.4), the target-metric display immediately after (1.2), and the normalization paragraph immediately after (1.4).
- Smooth global data for the axisymmetric comparison
- Periodic isometries and invariant normal averaging
- Commuting complete flows give the global product action
- For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix
- Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose
- For same-sized finite square matrices over a commutative ring, det(AB)=det(A)det(B)
- For every square matrix over a commutative ring, det(A^T)=det(A)
- A finite square real matrix is invertible if and only if its determinant is nonzero
- Han–Khuri–Weinstein–Xiong, Kerr Black Hole Uniqueness · §1 source-class paragraph immediately before Theorem 1.1; §2.1 opening variational hypothesis and (A1)–(A3), equations (2.3)–(2.10); Proposition 2.1(i),(iii), especially (2.23).
- Staticity, axisymmetry and black-hole regularity hypotheses
- Embedded submanifolds and slice charts
- The smooth inverse function theorem on manifolds
- Clairaut--Schwarz theorem for continuous second partial derivatives
- Multivariable Taylor formula with a Lagrange remainder along a line segment
- The inverse metric, musical isomorphisms and index raising
- Smoothness and transformation of the inverse-metric components
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- 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 continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Fermat's theorem: an interior differentiable local extremum has zero gradient
- Connected rotating degenerate vacuum comparison data
- Adopted Einstein field equation in the direct-equation presentation
- The Axiom of Choice
- Amsel–Horowitz–Marolf–Roberts, Uniqueness of Extremal Kerr and Kerr-Newman Black Holes · §2, Theorem 1, printed p.2; opening proof and footnote 2, printed p.3
- Han–Khuri–Weinstein–Xiong, Kerr Black Hole Uniqueness · §1, equations (1.1)–(1.11), source-class paragraph and Theorem 1.1; §2.1 variational hypothesis and (A1)–(A3), (2.3)–(2.10); §2.3 Proposition 2.1(i)–(iv), (2.12)–(2.26).
- Chruściel and Nguyen, A uniqueness theorem for degenerate Kerr-Newman black holes · Theorem 1.1, printed p.1; introduction describing missing global-reduction inputs; §3.3 Proposition 3.1 and §3.4; §3.6, (3.35) and Theorem 3.4, printed pp.20,22
- Chruściel–Costa–Heusler, Stationary Black Holes: Uniqueness and Beyond · §3.2.3, printed p.14; §3.2.7, Theorem 3.2, printed p.15
- Tensor fields of type (r,s) on a Lorentzian four-manifold and their components
- The tensor product of multilinear tensors
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- Causal exterior, chronological boundaries and Cauchy data
- Subextremal Kerr exterior in length coordinates
- P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §2.1, (2.1)–(2.3), PDF p. 5; §§2.3–2.4, (2.11)–(2.12), pp. 5–6; §2.6, Definition 2.1, p. 11; §§3.2.1, 3.2.3–3.2.7, pp. 13–15, especially Theorem 3.2 on p. 15; §3.3.1, p. 16.
- J. Sbierski, Characterisation of the Energy of Gaussian Beams on Lorentzian Manifolds — with Applications to Black Hole Spacetimes, arXiv:1311.2477 (12 January 2016 text) · §3.2 opening, printed/PDF p. 34; §3.2.1 opening through the coordinate introduction, printed/PDF p. 35.
- S. Aretakis, Decay of Axisymmetric Solutions of the Wave Equation on Extreme Kerr Backgrounds, arXiv:1110.2006 (10 October 2011) · §2.1, metric components and equation (2), printed/PDF p. 7; §2.2, differential structure, regular Lorentzian metric and D=r>M, printed/PDF p. 8.
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
Definition
Harmonic-map and Weyl reconstruction comparison data
Lemma
Gram determinant for the two symmetry directions
Definition
Exact mixed-rod boundary class for the balance theorem
Lemma
A regular weight-one axis has zero logarithmic angle defect
Physics theorem
Analytic extremal and multiple-horizon comparison contracts
Lemma
Orthogonal projection away from the symmetry plane
Lemma
Algebraic reconstruction from orbit and horizontal metric data
Physics theorem