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
- Differentiated asymptotic bounds and parameter uniformity
- Spacelike sections, maximality and lapse–shift data
- Laplace–Beltrami operator as the trace of the Hessian
- Smooth charts, atlases, and structures with boundary
- Riemannian distance on a connected manifold
- Causal curves, chronological and causal reachability, and inextendibility
- Chronology, causality, strong causality, global hyperbolicity and time functions
- Temporal functions are time functions, and time functions forbid closed causal curves
- AC implies DC implies countable choice
- The Axiom of Choice
- Bifurcate Killing horizons, rotation and normalization
- Complex self-dual forms and metric-dependent contraction
- Periodic isometries and invariant normal averaging
- Stokes theorem for smooth singular chains
- Relative homotopy primitive on a normal bundle
- Equivariant homotopy and forced contraction constant
- Radial primitive and five differentiated decay bounds
- Stationary lifting of a spatial primitive
- Stationary exact corrections preserve two prescribed gauges
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 GR/SBS and maximality; Lemma 2.1; §3.1 (3.2) and the following global-existence and extension paragraph; §3.2 (3.9), (3.13)–(3.15); Appendix A.1 (A.2)–(A.11), PDF pp. 29–30; Appendix A.3 Lemma A.1 at q=1
- The Killing derivative is an exact closed two-form
- The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors
- H. S. Reall, General Relativity, Part III Mathematical Tripos, October 12, 2022 · Conventions, printed p. 3 (PDF p. 4); §4.2, equations (4.7)–(4.8), printed p. 42 (PDF p. 43)
- Lorentzian exterior calculus and Hodge star
- Local and global flows generated by a vector field
- The Lie derivative of a tensor field
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- The exterior derivative commutes with pullback
- Pullback of forms is smooth functorial and preserves wedges
- Regular bifurcate null horizon data before rigidity
- Null expansions, screen quotients and focal data
- Affine connection on a smooth manifold
- Transverse embedded submanifolds intersect in the expected codimension
- A vector field tangent to an embedded submanifold restricts to a vector field on it
- Embedded submanifolds and slice charts
- The coordinate formula for the Lie derivative of a covariant tensor
- Positive Harrison denominator and its parameter identity
- The exterior derivative by the invariant vector-field formula
- The Lie derivative of a differential form
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Section 1.1, differentiated notation after (1.1); Section 3.2, equations (3.12)–(3.18).
- Causal exterior, chronological boundaries and Cauchy data
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Through each point there is a unique maximal integral curve
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum (arXiv:0711.0040v2) · Section 1.2, AF, equations (1.4)–(1.5), printed/PDF pp. 6–7, including footnote 4.
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
Complete regular static exterior data and its normalized end
Theorem
Smooth invariant Harrison primitive near the horizon and at infinity
Lemma
Ricci contraction of the Killing derivative
Lemma
Killing flow preserves its complex two-form
Lemma
Rescaling a null generator preserves zero expansion
Lemma
Null second fundamental form descends to the screen quotient
Lemma
A branch-tangent Killing field restricts to the bifurcation sphere
Lemma
Lifting a normalized stationary primitive to the Harrison parameter
Definition