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
- Complete regular static exterior data and its normalized end
- Vacuum curvature of a globally static product
- Riemannian divergence theorem
- Gradient hessian and divergence connection formulas
- Induced connection and second fundamental form
- Totally geodesic submanifold
- Collar neighborhood theorem
- Monotone convergence for the integral
- The Axiom of Countable Choice (AC_ω)
- Riemannian volume density
- Intrinsic density measure and its chart restriction
- Measurable integration extends smooth density integration
- Stationary Lorentzian data and tensor conventions
- Complex self-dual forms and metric-dependent contraction
- The Killing derivative is an exact closed two-form
- Ricci contraction of the Killing derivative
- Vanishing divergence implies closed Hodge dual
- Killing flow preserves its complex two-form
- Closed Ernst contraction and the real potential
- Regular bifurcate null horizon data before rigidity
- Rescaling a null generator preserves zero expansion
- Null second fundamental form descends to the screen quotient
- Metric transport along a null generator
- A branch-tangent Killing field restricts to the bifurcation sphere
- Periodic isometries and invariant normal averaging
- S. Alexakis, A. D. Ionescu and S. Klainerman, Hawking’s local rigidity theorem without analyticity · §1 setup and Theorems 1.1–1.2 (pp. 1–2); §2 Definition 2.1 (p. 6); §5 Propositions 5.1–5.2 (pp. 16–17)
- Causal exterior, chronological boundaries and Cauchy data
- Bifurcate Killing horizons, rotation and normalization
- Differentiated asymptotic bounds and parameter uniformity
- Adopted Einstein field equation in the direct-equation presentation
- Subextremal Kerr exterior in length coordinates
- The Axiom of Choice
- Exact asymptotic and orbit package for the IK criterion
- Exact bifurcate-sphere package for the IK criterion
- Ernst sublevels and the Mars–Simon tensor interface
- Pathwise exhaustion by anchored Ernst sublevels
- Stationary transport of vanishing from a section
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §1.2 AF, SBS, T, equations (1.4)–(1.7), printed pp. 6–7; Main Theorem, printed p. 8
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §8, Lemma 8.1 and (8.2), printed p. 41; sublevel definition and Propositions 8.2–8.3, printed p. 42; §8.2, printed pp. 48–49; completion (8.1)
- Regular Lorentzian metric from the normalized rank-one map
- Lifting a normalized stationary primitive to the Harrison parameter
- Lorentzian exterior calculus and Hodge star
- Interior product of a form by a vector field
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.1, equation (3.5); Section 3.2, equations (3.12), (3.19)–(3.25) and the paragraphs immediately before (3.24) and after (3.25).
- Complete stationary saturation of the end
- Exact AIK global, horizon and quantitative hypotheses
- Horizon smallness is the induced norm of the tangential stationary field
- Spacelike sections, maximality and lapse–shift data
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- Timelike, null, spacelike and causal vectors of a Lorentz vector space
- S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · §2.2 Lemma 2.1, (2.9)–(2.10), PDF p. 7; Lemma 2.3, (2.15), PDF pp. 8–9
- The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation
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
Regular static boundary geometry and positivity of the end coefficient
Theorem
Vacuum Killing and Ernst differential identities
Theorem
Hawking and rotational fields near a bifurcate sphere
Physics theorem
Ionescu–Klainerman local Kerr criterion with full interface
Lemma
Smooth Maxwell tensor formula without division by the stationary norm
Lemma
Chronological equivariance and the no-white-hole condition
Lemma
Quantitative lapse and shift bounds force annular timelikeness
Lemma