Alphabeta MathLinearized coupled equations and gauge identities
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1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Absolute Continuity and the Sharp Fundamental Theorem of Calculus
- Algebraic Closure, Embeddings, and Separability
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Bounded Variation and the Riemann–Stieltjes Integral
- Categories, Functors and Natural Transformations
- Causal And Initial Value Boundaries
- Chain Complexes and Homology
- Cm Math Continuum
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Lp Spaces and Test-Function Conventions
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Connectedness
- Connections And Free Fall
- Connections Levi Civita and Parallel Transport
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Curvature And Tides
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Differentiation of Monotone Functions and the Vitali Covering Theorem
- Distributions Test Functions and Differentiation
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Einstein And Matter
- Electrovacuum tensor Cauchy system
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Exterior Powers, Orientation and Hodge Duality
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Framework And Geometry
- Free Modules, Exact Sequences, Projective and Injective Modules
- Fubini and Change of Variables
- Function Space Topologies and the Exponential Law
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Geometric electrovac equations and flux normalization
- Gr14 Full High Regularity Symmetric Hyperbolic Proof
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hereditary and Productive Behaviour of the Separation Axioms
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Improper Integrals
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Integration of Forms and the General Stokes Theorem
- Lebesgue Measure on Euclidean Space
- Limits and Colimits
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Local potentials and smooth circle-bundle connection data
- Manifolds with Boundary Collars and Orientations
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modes of Convergence Egorov and Lusin
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Morse Critical Points Hessians and Indices
- Normal Subgroups and Quotient Groups
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- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Oriented Lorentzian fields, observers and coordinate conventions
- Orthonormal Bases, Parseval and Fourier Series
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Positive symmetric Maxwell field-strength reduction
- Power Series and Real-Analytic Functions
- Preadditive and Additive Categories and Biproducts
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Rank Theorems and Embedded Submanifolds
- Relations, Functions, and Quotients
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Metrics Length Distance and Volume
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Schwartz Space and the Plancherel Theorem
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- Simple Field Extensions and the Construction of the Complex Numbers
- Simplicial Complexes and Simplicial Homology
- Sine, Cosine, and the Definition of Pi
- Singular Chains and Singular Homology
- Singular Cochains Mayer Vietoris and Smooth Singular Comparison
- Singular Cohomology and Coefficient Theorems
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Sourced Maxwell sign and current conservation
- Spacetime Primitives
- Spherical Vacuum Model
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- Tensor Products of Modules
- Tensors And Observers
- The Analytic Hahn Banach Theorem
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The De Rham Complex Homotopy and Mayer Vietoris
- The de Rham Theorem and Degree
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Exterior Derivative and Cartan Calculus
- The Field of Fractions and Localisation
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Uniform Spaces: the Three Definitions
- Vector Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Volumes of Elementary Solids and Solids of Revolution
- Weak Derivatives and Sobolev Spaces
2 · Logical flowchart
3 · Items
Definition
Linearized geometric electrovacuum operator and gauge equivalence
Lemma
Variation of the Lorentzian Hodge map and the linearized Maxwell equation
Proposition
Pure-gauge and exact-family perturbations solve the linearized electrovacuum system
Lemma
Finite-slab energy control and subsidiary propagation for the linearized electrovacuum system
Sources
- Sean M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019v1 (1997), Chapter 6 and Chapter 7, equations (7.106)–(7.113)
- Elena Giorgi, The linear stability of Reissner–Nordström spacetime for small charge, arXiv:1904.04926v2 (2020)
- Sean M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019v1 (1997), Chapter 1, equations (1.87)–(1.93)
- Elena Giorgi, The linear stability of Reissner–Nordström spacetime for small charge, arXiv:1904.04926v2 (2020), Sections 6.1–6.2
- Sean M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019v1 (1997), Chapter 6, equations (6.10)–(6.14)
- Helmut Friedrich and Alan Rendall, The Cauchy Problem for the Einstein Equations, Living Reviews in Relativity 3 (2000), article 1, arXiv:gr-qc/0002074, Section 3.1
- Elena Giorgi, The linear stability of Reissner–Nordström spacetime for small charge, arXiv:1904.04926v2 (2020), Sections 1 and 6