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
- Bifurcate Killing horizons, rotation and normalization
- Spacelike sections, maximality and lapse–shift data
- Differentiated asymptotic bounds and parameter uniformity
- Complex self-dual forms and metric-dependent contraction
- Geometric electrovacuum equations and charge convention
- Periodic isometries and invariant normal averaging
- The Axiom of Choice
- Smooth invariant Harrison primitive near the horizon and at infinity
- Positive Harrison denominator and its parameter identity
- Lifting a normalized stationary primitive to the Harrison parameter
- Regular Lorentzian metric from the normalized rank-one map
- Smooth Maxwell tensor formula without division by the stationary norm
- Scope of smooth field equations at an exterior horizon boundary
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 GR/SBS and maximality; §3.1–3.3, Proposition 3.1, (3.5)–(3.25); Appendix A.1–A.3
- Complete regular static exterior data and its normalized end
- Vacuum curvature of a globally static product
- The pointwise static end bounds imply the original weighted hypotheses
- Regular static boundary geometry and positivity of the end coefficient
- From a Schwarzschild spatial isometry to the normalized spacetime exterior
- Adopted Einstein field equation in the direct-equation presentation
- G. L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, General Relativity and Gravitation 19 (1987), 147–154 · p.148, static-data assumptions, equations (2)–(5) and Assumption 1; p.150, Theorem 2; p.153, Riemannian doubling and conclusion of the main proof
- P. Miao, A Remark on Boundary Effects in Static Vacuum Initial Data sets, arXiv:gr-qc/0504081v1 · pp.1–2, static equations (1) and paragraph beginning “A classic result of Bunting and Masood-ul-Alam”; p.3, boundary-regularity qualification for Miao’s different minimal-boundary theorem
- Exact asymptotically stationary end contract
- Coordinate cones and the separate stationary generator
- Complete stationary saturation of the end
- Chronological equivariance and the no-white-hole condition
- Causal exterior, chronological boundaries and Cauchy data
- Rácz–Wald, Global Extensions of Spacetimes Describing Asymptotic Final States of Black Holes · End of §2, printed/PDF pp. 8–9; Theorem 4.2, pp. 15–16; §5 opening paragraph, p. 17
- Staticity, axisymmetry and black-hole regularity hypotheses
- Riemannian volume of a compactly supported smooth density
- Diffeomorphisms and local diffeomorphisms of manifolds
- The pullback of a covariant tensor field
- P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §2.1 and §§2.3–2.4 (PDF pp. 5–6); §2.5.3, equations (2.20)–(2.21) (PDF p. 10); §2.6, Definition 2.1 (PDF pp. 10–11); §3.2.1 (PDF p. 13); §3.2.7, Theorem 3.2 (PDF p. 15).
- Exact AIK global, horizon and quantitative hypotheses
- Quantitative lapse and shift bounds force annular timelikeness
- The quantitative asymptotic end is uniformly timelike
- Local and global flows generated by a vector field
- S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · §1.1.3 Theorem 1.2, PDF p. 4; final Proof of Proposition 1.3, PDF p. 10
- S. Alexakis, A. D. Ionescu and S. Klainerman, Hawking's local rigidity theorem without analyticity · Abstract, PDF p. 1; Theorem 1.2 and (1.1), PDF p. 2; section 5, Proposition 5.2, PDF p. 17, and its proof, PDF pp. 17–18
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Han–Khuri–Weinstein–Xiong, Kerr Black Hole Uniqueness · §1, Weyl–Papapetrou setting (1.1)–(1.4), ordered rods (1.5), axis traces (1.6), and orientation-qualified jump formula (1.7)
- Horizon smallness is the induced norm of the tangential stationary field
- Subextremal Kerr exterior in length coordinates
- S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · Abstract and Introduction, PDF pp. 1–2; §1.1.3 Theorem 1.1 and (1.5), PDF p. 4; §1.2 (1.10), PDF p. 6
- Rácz–Wald, Global Extensions of Spacetimes Describing Asymptotic Final States of Black Holes · End of §2, printed/PDF pp. 8–9; Theorem 4.2, pp. 15–16; §5 opening paragraph, p. 17
- Lie group
- Quotient by a closed normal subgroup is a Lie group
- Quotient manifold by a closed Lie subgroup
- The constant-rank theorem for manifolds
- Products of smooth manifolds have a canonical product smooth structure
- A map into a product is smooth iff its components are smooth
- Identity maps and composites of smooth maps are smooth
- Smooth left actions of Lie groups
- AC implies DC implies countable choice
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
Theorem
Regular Harrison Einstein–Maxwell family
Physics theorem
Static vacuum Schwarzschild classification contract
Definition
Precise Rácz–Wald global collapse class
Definition
Smooth global data for the axisymmetric comparison
Lemma
The non-timelike region is confined to the local-rigidity collar
Definition
Connected rotating degenerate vacuum comparison data
Definition
Ordered horizon rods and twist-jump labels
Physics theorem
Smooth rigidity from a quantitatively small horizon field
Theorem
Nonzero constant surface gravity and global bifurcate extension
Lemma