Alphabeta MathArticle items · 06 — Null normal pairs and screen geometry
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Covariant derivative along a curve
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields
- The Axiom of Countable Choice (AC_ω)
- Smooth covariant 2-tensor fields on a smooth manifold
- Linear functionals and the algebraic dual V^*=L(V,F)
- The differential of a smooth map
- Coordinate derivations form a basis of the tangent space
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
- The timelike set has exactly two convex components, described in orthonormal coordinates
- The Lie derivative of a tensor field
- A smooth vector field is a smooth section of the tangent bundle
- Observer congruences, their observer curves and adapted charts
- Observers, rest spaces and the spatial projection at an event
- Cʳ and smooth maps between smooth manifolds
- The Lie bracket of smooth vector fields
- Coordinate formula for the Lie bracket
- Affine connection on a smooth manifold
- Torsion tensor of an affine connection
- Smoothness and transformation of the inverse-metric components
- Christoffel symbols of an affine connection
- Leibniz rules for the Lie bracket with function multiples
- Smoothness of a vector field is equivalent to smooth coordinate components
- Curvature of an affine connection
- Curvature is a type (1,3) tensor
- Coordinate vector fields commute
- Oriented smooth manifolds and oriented charts
- The wedge product of differential forms
- Interior product of a form by a vector field
- Local coordinate expression for a differential form
- Integral of a form over a smooth singular simplex
- eta-orthogonality and orthogonal complements in a Lorentz vector space
- Euclidean spheres and closed balls as subspaces of ℝⁿ
- The round metric on the sphere as an induced metric
- Products of smooth manifolds have a canonical product smooth structure
- Smooth manifolds and their smooth charts
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Rational powers aʳ of a positive base
- Derivations at a point and the tangent space
- Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space
- Timelike, null, spacelike and causal vectors of a Lorentz vector space
- Time orientation, future and past timelike/null/causal vectors, and the closed cones
- The velocity derivation of a smooth curve
- Continuity and derivatives of positive-base real powers
- Every continuous function on an interval has a primitive; two primitives differ by a constant; and ∫ₐᵇ f = G(b)-G(a) for any primitive G
- 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)
- Choice-free smooth inverse function theorem in Euclidean space
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)
- The lower and upper Darboux integrals of a bounded f on [a,b] as sup_P L(f,P) and inf_P U(f,P), Darboux integrability as their equality, and the notation ∫ₐᵇ f
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- 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
- A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions
- Oriented Lorentzian fields, observers and coordinate conventions
- The exterior derivative by the invariant vector-field formula
- A smooth differential k-form
- Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane
- The invariant exterior-derivative formula is C^∞-multilinear
- Differential forms form a graded commutative algebra
- Clairaut--Schwarz theorem for continuous second partial derivatives
- The local coordinate formula for the exterior derivative
- The exterior derivative is a graded derivation
- Causal curves, chronological and causal reachability, and inextendibility
- Absolute continuity on a compact interval
- Fundamental theorem of calculus for absolutely continuous functions
- Absolutely continuous functions have integrable derivatives
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Pointwise local causality: closed cones, openness of the local chronological future, no local closed causal curves, corner smoothing and closedness of the local causal relation
- Future sets, past sets, achronal boundaries and their elementary properties
- Embedded submanifolds and slice charts
- Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique
- The smooth inverse function theorem on manifolds
- If f : [a,b] → ℝᵐ is differentiable with integrable f' then ∫ₐᵇ f' = f(b)-f(a); and a bounded derivative makes f Lipschitz
- Coordinate balls form a basis of a topological manifold
- For same-sized finite square matrices over a commutative ring, det(AB)=det(A)det(B)
- Chart independence and the transformation law of a smooth covariant 2-tensor field
- Torsion freedom and metric compatibility in chart components
- The derivative of the determinant along an invertible C¹ matrix curve: (det A)'=det A tr(A⁻¹A')
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Spacelike initial data, the constraint equations, developments and maximal developments
- The observer rest-space decomposition and the spatial projection
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Chronology, causality, strong causality, global hyperbolicity and time functions
- Pairing of an observer with future causal vectors
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
- The flow-box theorem
- Tensoriality of the curvature, its coordinate formula and its point-normal metric formula
- Causal solutions, Killing horizons, escape and metric extensions
- Explicit rational and real power regimes
- Classification of future and past causal vectors by a timelike field
- The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors
- Kerr-Newman metric and potential on the exact Boyer-Lindquist domain
- Fully verified electric Kerr-Newman electrovacuum
- Geometric electrovac equations and flux normalization
- Cauchy sets are closed, split the manifold into past, set and future, and are locally Lipschitz graphs
- Failure of strong causality yields causal curves escaping one fixed neighbourhood with endpoints accumulating at the point
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- Two-endpoint limit curve theorem: bounded lengths give joining limits, unbounded lengths give two inextendible tails with the closure relation
- Basic limit curve lemma: arclength-parametrized causal curves have causal limit curves on compact parameter intervals and future-inextendible limits on half-lines
- The auxiliary Riemannian metric h=g+2n-flat tensor n-flat is complete after a proper-exhaustion conformal factor, and its closed bounded sets are compact
- Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal
- Equivalence of the absolutely-continuous and continuous causal conventions, and one-sided inextendible extensions
- For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
- The riemannian distance topology is the manifold topology
- Riemannian distance is a metric
- Local comparison of a riemannian metric with the euclidean metric
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Convexity of the squared Euclidean radius along small contained geodesics: segments with endpoints in a small ball stay in the ball
- Speed bounds along geodesics in a chart, and small initial velocity of every geodesic segment contained in a small ball
- Normal neighborhoods, convex normal sets, the position vector and the squared-displacement function
- Point-normal and exponential coordinate charts of the metric-selected connection
- Coordinate geodesic equation
- Smooth dependence of ODE solutions on parameters
- The coordinate formula for the Lie derivative of a covariant tensor
- Tensor fields of type (r,s) on a Lorentzian four-manifold and their components
- Geodesic of an affine connection
- Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed
- Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current
- Finite chart localization gives choice-free integration and compact Stokes
- The Hilbert stress-energy tensor of a metric-dependent action
- A continuous symmetric coefficient orthogonal to every compactly supported variation vanishes
- Lorentzian metrics are stable under small compactly supported symmetric perturbations
- Compactly supported smooth vector fields are complete
- Complete vector fields
- Local and global flows generated by a vector field
- Picard-Lindelöf local existence and uniqueness for first-order systems
- The fundamental theorem for autonomous smooth ODEs
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover
- A vector field acts as a derivation of smooth functions
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- Dual connection
- Product connection on tensor and hom bundles
- Finite tensor products of smooth vector bundles
- A type (r,s) tensor on a finite-dimensional vector space
- The contraction of a mixed tensor
- The differential of the squared-displacement function and the position vector of the reversed geodesic
- Length monotonicity of the squared displacement: once a future causal curve in a normal neighborhood has a future timelike position vector, the displacement radius grows at least as fast as the accumulated Lorentzian length
- Future causal position vectors along a curve in a convex normal set
- A Lipschitz function after an absolutely continuous function is absolutely continuous
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- ℚ is countably infinite
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
DefinitionLibrary prerequisite
Null normal pairs and screen geometry
DefinitionLibrary prerequisite
The inverse metric, musical isomorphisms and index raising
DefinitionLibrary prerequisite
Killing fields of a Lorentzian metric
DefinitionLibrary prerequisite
Local frame fields, coordinate frames and tetrad fields
DefinitionLibrary prerequisite
Metric compatibility, torsion freedom and the metric-selected connection class
DefinitionLibrary prerequisite
The metric volume form, current flux through 3-faces and finite-chain charges
DefinitionLibrary prerequisite
The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation
DefinitionLibrary prerequisite
Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
DefinitionLibrary prerequisite
Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold
LemmaLibrary prerequisite
Lorentzian exterior calculus and Hodge star
LemmaLibrary prerequisite
Chart independence, concatenation, reparametrization and transitivity for causal curves
LemmaLibrary prerequisite
Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
LemmaLibrary prerequisite
Local null normal pairs and the positive screen quotient
LemmaLibrary prerequisite
The joint exponential map: smoothness, invertible differential at the diagonal, and the small-velocity two-point inverse
LemmaLibrary prerequisite
The derivative of the metric density and the chart formula for the divergence of a vector field
LemmaLibrary prerequisite
The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
LemmaLibrary prerequisite
Temporal functions are time functions, and time functions forbid closed causal curves
LemmaLibrary prerequisite
Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)
LemmaLibrary prerequisite
Natural tensors under Killing isometries
PropositionLibrary prerequisite
KN Killing generators, surface gravity, area and potential
PropositionLibrary prerequisite
Null Raychaudhuri and focal-point bound
TheoremLibrary prerequisite
Regular ingoing KN metric, field and sphere axes
TheoremLibrary prerequisite
The existence of a Cauchy set implies strong causality and compact causal diamonds
TheoremLibrary prerequisite
Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points
TheoremLibrary prerequisite
Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor
TheoremLibrary prerequisite
The geodesic initial-value problem and the exponential map of the metric-selected connection
TheoremLibrary prerequisite
Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
TheoremLibrary prerequisite