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
- Causal exterior, chronological boundaries and Cauchy data
- Spacelike sections, maximality and lapse–shift data
- Differentiated asymptotic bounds and parameter uniformity
- Differentiated stationary end bounds imply the weighted Hölder interface
- A complete stationary flow preserves its compact tangent boundary
- Black and white exclusion is an ambient reachability condition
- Isometries preserve maximality and transported boundary
- Stationary graph translation preserves source asymptotics
- Piotr T. Chruściel and Robert M. Wald, Maximal Hypersurfaces in Asymptotically Stationary Spacetimes · Definitions 2.1 (2.1)–(2.5), pp.8–9; 2.3, p.11; (2.13)–(2.17), pp.11–12; Definition 2.4, p.13; (4.1)–(4.3) and Theorem 4.2, pp.24–25; Remark 3, p.25
- 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
- Cartan's magic formula
- Poincare's lemma on a star-shaped domain: every closed C1 field is exact
- The exterior derivative by the invariant vector-field formula
- The exterior derivative of a function is its differential
- Null second fundamental form descends to the screen quotient
- Rescaling a null generator preserves zero expansion
- The Lie derivative of a tensor field
- The Lie derivative is a derivation of the tensor algebra
- Tensor Lie derivative agrees with X on functions and bracket on vector fields
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- Lifting a normalized stationary primitive to the Harrison parameter
- Positive Harrison denominator and its parameter identity
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.2, equations (3.11), (3.12), (3.20)–(3.23) and the paragraph immediately following (3.23).
- Exact asymptotic and orbit package for the IK criterion
- Bifurcate Killing horizons, rotation and normalization
- Local null normal pairs and the positive screen quotient
- Constant-rank kernels and images of bundle maps over one base are subbundles
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §1.2, SBS and footnotes 5–6, printed p. 7
- Ernst sublevels and the Mars–Simon tensor interface
- Time-t flow maps are diffeomorphisms between open domains
- A smooth tensor field
- A type (r,s) tensor on a finite-dimensional vector space
- The pullback of a covariant tensor field
- A diffeomorphism pulls back tensor fields and forms isomorphically
- The differential of a diffeomorphism is an isomorphism
- Exact asymptotically stationary end contract
- Exact AIK global, horizon and quantitative hypotheses
- The Axiom of Choice
- Transverse embedded submanifolds intersect in the expected codimension
- For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact
- 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
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
Stationary maximal hypersurface existence with a horizon boundary
Lemma
Closed Ernst contraction and the real potential
Lemma
Metric transport along a null generator
Lemma
Regular Lorentzian metric from the normalized rank-one map
Definition
Exact bifurcate-sphere package for the IK criterion
Lemma
Stationary transport of vanishing from a section
Lemma
Coordinate cones and the separate stationary generator
Lemma
Complete stationary saturation of the end
Lemma
Horizon smallness is the induced norm of the tangential stationary field
Lemma