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
- Geometric electrovacuum equations and charge convention
- Null expansions, screen quotients and focal data
- Smooth formal matter, EM tensors and observer quantities
- Pointwise Maxwell and matter energy-condition vocabulary
- EM, dust and perfect-fluid energy inequalities
- The Axiom of Choice
- Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique
- Stationary Lorentzian data and tensor conventions
- Lorentzian exterior calculus and Hodge star
- Lecture Notes on General Relativity · §5, equations (5.41)–(5.42), PDF p. 147
- Complex self-dual forms and metric-dependent contraction
- Point-normal and exponential coordinate charts of the metric-selected connection
- The metric volume form, current flux through 3-faces and finite-chain charges
- Relative homotopy primitive on a normal bundle
- Periodic isometries and invariant normal averaging
- Cartan's magic formula
- The pullback of a differential form
- Pullback of forms is smooth functorial and preserves wedges
- The exterior derivative commutes with pullback
- Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral
- The local coordinate formula for the exterior derivative
- The Lie derivative of a tensor field
- A smooth function with zero differential is constant on each connected component
- Local and global flows generated by a vector field
- Interior product of a form by a vector field
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- The chain rule for differentials of smooth maps
- A smooth tensor field
- The pullback of a covariant tensor field
- Differentiated asymptotic bounds and parameter uniformity
- Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts
- Improper integrals over unbounded intervals
- Differentiation under an improper multiple integral under an integrable derivative bound
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- Comparison tests for improper integrals
- Sum, product, quotient, and chain rules from difference quotients
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Smooth partition of unity on a manifold with boundary
- A smooth differential k-form
- The Lie derivative of a differential form
- Closed and exact differential forms
- The exterior derivative of a function is its differential
- The exterior derivative squares to zero
- The exterior derivative is local
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Embedded submanifolds and slice charts
- The n-sphere with its standard smooth atlas
- A vector field acts as a derivation of smooth functions
- An open subset of a smooth manifold has a canonical restricted smooth structure
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
Maxwell algebra supplies timelike and null convergence
Lemma
The Killing derivative is an exact closed two-form
Lemma
Vanishing divergence implies closed Hodge dual
Lemma
Equivariant homotopy and forced contraction constant
Lemma
Radial primitive and five differentiated decay bounds
Lemma