MATHEMATICAL ASPECTS OF GENERAL RELATIVITY

  • Introduction
    • Section 1: Geometric Quantities
    • Deterrmnant Fomrula
    • Section 2: Scalar Products
    • Section 3: Interior Multiplication
    • Section 4: Tensor Analysis
  • Geometric Quantities
  • Scalar Products
  • Interior Multiplication
  • Tensor Analysis
  • Lie Derivatives
  • Flows
  • Covariant Differentiation
  • Parallel Transport
  • Curvature
  • Semiriemannian Manifolds
  • The Einstein Equation
  • Decomposition Theory
  • Bundle Valued Forms
  • The Structural Equations
  • Transition Generalities
  • Metric Considerations
  • Submanifolds
  • Extrinsic Curvature
  • Hodge Conventions
  • Star Formulae
  • Metric Concomitants
  • Lagrangians
  • The Euler-Lagrange Equations
  • The Helmholtz Condition
  • Applications of Homogeneity
  • Questions of Uniqueness
  • Globalization
  • Functional Derivatives
  • Variational Principles
  • Splittings
  • Metrics on Metrics
  • The Symplectic Structure
  • Motion in a Potential
  • Constant Lapse, Zero Shift
  • Variable Lapse, Zero Shift
  • Incorporation of the Shift
  • Dynamics
  • Causality
  • The Standard Setup
  • Isolating the Lagrangian
  • The Momentum Form
  • Elimination of the Metric
  • Constraints in the Coframe Picture
  • Evolution in the Coframe Picture
  • Computation of the Poisson Brackets
  • Field Equations
  • Lovelock Gravity
  • The Palatini Formalism
  • Torsion
  • Extending the Theory
  • Evolution in the Palatini Picture
  • Expansion of the Phase Space
  • Extension of the Scalars
  • Selfdual Algebra
  • The Selfdual Lagrangian
  • Two Canonical Transformations
  • Ashtekar’s Hamiltonian
  • Evolution in the Ashtekar Picture
  • The Constraint Analysis
  • Densitized Variables
  • Rescaling the Theory
  • Asymptotic Flatness
  • The Integrals of Motion-Energy and Center of Mass
  • The Integrals of Motion-Linear and Angular Momentum
  • Modifying the Hamiltonian
  • The ~oincarg Structure
  • Function Spaces
  • Asymptotically Euclidean Riemannian Structures
  • Laplacians
  • Positive Energy
    • Sobolev Inequality