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
- Null expansions, screen quotients and focal data
- Geodesic of an affine connection
- Causal curves, chronological and causal reachability, and inextendibility
- S. Alexakis, A. D. Ionescu and S. Klainerman, Hawking’s local rigidity theorem without analyticity · §1 setup, pp. 1–2; §2, pp. 4–7, equations (2.1)–(2.4), Definition 2.1 and equation (2.11) with its following sentence.
- Real and imaginary parts, complex conjugation, and modulus
- Embedded submanifolds and slice charts
- Cʳ and smooth maps between smooth manifolds
- Continuity of a map of topological spaces at a point and globally
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 3.1, equations (3.2), (3.5), (3.7); Section 3.2, equation (3.17) and the positivity paragraph after (3.19).
- Geometric electrovacuum equations and charge convention
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- The standard flat function
- The standard flat function is smooth and flat at zero
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Proposition 3.1; Appendix A.2, opening paragraph and the paragraph immediately after (A.18).
- Complex self-dual forms and metric-dependent contraction
- Interior product of a form by a vector field
- The metric volume form, current flux through 3-faces and finite-chain charges
- Smooth embeddings
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §4.2, (4.11), (4.15), (4.17) and the normalization paragraph after (4.18), printed pp. 22–23; §4.3, (4.19)–(4.21), (4.27), Definition 4.3 (4.29), printed pp. 23–24; §7, (7.3), printed p. 35; Appendix A.3, opening curvature-component formula, printed p. 54 (PDF p. 54).
- Differentiated asymptotic bounds and parameter uniformity
- The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a)
- Bifurcate Killing horizons, rotation and normalization
- Local and global flows generated by a vector field
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Integral curves of a vector field
- A vector field tangent to an embedded submanifold restricts to a vector field on it
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- Through each point there is a unique maximal integral curve
- Spacelike sections, maximality and lapse–shift data
- Diffeomorphism pushforward preserves Lie brackets
- Diffeomorphisms preserve interior and boundary
- The fundamental theorem on flows
- A vector field is complete if and only if its flow is global
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- The timelike set has exactly two convex components, described in orthonormal coordinates
- The pullback of a covariant tensor field
- Immersions and embeddings for manifolds with boundary
- Embedded smooth submanifolds with boundary
- Codimension and hypersurfaces
- The logarithm grows more slowly than every positive real power
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
Regular bifurcate null horizon data before rigidity
Lemma
Positive Harrison denominator and its parameter identity
Lemma
Scope of smooth field equations at an exterior horizon boundary
Definition
Ernst sublevels and the Mars–Simon tensor interface
Lemma
Differentiated stationary end bounds imply the weighted Hölder interface
Lemma
A complete stationary flow preserves its compact tangent boundary
Lemma
Isometries preserve maximality and transported boundary
Lemma