Alphabeta MathSpherical electrovacuum mass and local rigidity
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- 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
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Curvature And Tides
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Einstein And Matter
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Exterior Powers, Orientation and Hodge Duality
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Framework And Geometry
- Fubini and Change of Variables
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Geometric electrovac equations and flux normalization
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hereditary and Productive Behaviour of the Separation Axioms
- 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
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
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- Manifolds with Boundary Collars and Orientations
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Oriented Lorentzian fields, observers and coordinate conventions
- Partitions of Unity and Paracompactness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rank Theorems and Embedded Submanifolds
- Relations, Functions, and Quotients
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- 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
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Simplicial Complexes and Simplicial Homology
- Singular Chains and Singular Homology
- Singular Cochains Mayer Vietoris and Smooth Singular Comparison
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- Spherical Vacuum Model
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- The de Rham Theorem and Degree
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- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Exterior Derivative and Cartan Calculus
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- 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
- Vector Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Volumes of Elementary Solids and Solids of Revolution
2 · Logical flowchart
3 · Items
Definition
Spherical orbit geometry and rotational invariance
Lemma
Source-free rotational Maxwell fields have constant fluxes
Lemma
Ricci tensor and volume form on a Lorentzian surface
Definition
Spherical electrovacuum quotient and charged mass
Lemma
Warped two-sphere curvature and charged mass conservation
Theorem
Local charged Birkhoff rigidity away from critical areal gradients
Sources
- K. A. Bronnikov, S.-W. Kim and M. V. Skvortsov, The Birkhoff theorem and string clouds, arXiv:1604.04905v1 (2016)
- David Tong, General Relativity, complete author-hosted lecture notes
- P. T. Chruściel, J. Lopes Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond, arXiv:1205.6112v1 (2012)