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
- Ernst sublevels and the Mars–Simon tensor interface
- Stationary Lorentzian data and tensor conventions
- Smooth manifolds and their smooth charts
- Connected components, quasicomponents, and totally disconnected spaces
- Paths, path-connected spaces and path components
- Continuity of a map of topological spaces at a point and globally
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- The type (r,s) tensor bundle
- The direct-sum model V⊕ iV of a complexification
- Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous
- Closed real intervals are compact by completeness
- Extreme value theorem: a continuous real function on a nonempty compact subset of ℝ attains a greatest and a least value
- A subset of ℝ is connected if and only if it is order-convex, that is, an interval
- The connected subspaces of ℝ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in ℝ"
- A continuous image of a connected space is connected, and connectedness is a topological property
- The absolute value makes ℝ a metric space: d(x,y) = |x-y| is a metric, its open balls are the intervals (x-r, x+r), and it is unbounded
- Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
- Open cover, subcover, compact subset of ℝ (every open cover has a finite subcover), and sequentially compact subset
- Cʳ and smooth maps between smooth manifolds
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- An open subset of a smooth manifold has a canonical restricted smooth structure
- Smooth maps are continuous
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- Causal exterior, chronological boundaries and Cauchy data
- Embedded smooth submanifolds with boundary
- Codimension and hypersurfaces
- Diffeomorphisms and local diffeomorphisms of manifolds
- The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold
- Local and global flows generated by a vector field
- Chruściel–Wald, Maximal Hypersurfaces in Asymptotically Stationary Spacetimes · Section 2 opening conventions and Definition 2.1, printed/PDF pp. 8–9, equations (2.1)–(2.5).
- Bifurcate Killing horizons, rotation and normalization
- Spacelike sections, maximality and lapse–shift data
- Differentiated asymptotic bounds and parameter uniformity
- The Axiom of Choice
- S. Alexakis, A. D. Ionescu and S. Klainerman, Rigidity of stationary black holes with small angular momentum on the horizon · §1.1.1–§1.1.2, (1.1)–(1.4), SBS, PDF pp. 3–4; §1.2 (1.6)–(1.10), PDF pp. 5–6
- E. Minguzzi, Lorentzian causality theory · Definition 2.130, PDF p. 76; Definition 3.35, PDF p. 95
- Complete regular static exterior data and its normalized end
- Gradient hessian and divergence connection formulas
- Christoffel formula for the levi civita connection
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- Scalar curvature
- Ricci curvature
- Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ
- The Axiom of Countable Choice (AC_ω)
- 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, Assumption 1, equations (4)–(5), the following derivative conditions and definition of the weighted space
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
Pathwise exhaustion by anchored Ernst sublevels
Definition
Exact asymptotically stationary end contract
Lemma
Black and white exclusion is an ambient reachability condition
Definition
Exact AIK global, horizon and quantitative hypotheses
Lemma
Vacuum curvature of a globally static product
Lemma