"This monograph presents a significant new result in general relativity. In particular, it provides a proof related to the Einstein-Klein-Gordon equation, a fundamental equation in mathematical physics that couples the Einstein equation of general relativity with a matter field described by the Klein-Gordon equation. The book begins with an introduction and history of the subject, proceeds to prove several auxiliary lemmas, and culminates in the central proof. This book represents a significant advance in mathematical physics, and provides the most cutting-edge treatment of the Einstein-Klein-Gordon equation to date"--.

• Cover
• Contents
• 1. Introduction
  • 1.1 The Einstein-Klein-Gordon Coupled System
    • 1.1.1 Wave Coordinates and PDE Formulation of the Problem
  • 1.2 The Global Regularity Theorem
    • 1.2.1 Global Stability Results in General Relativity
    • 1.2.2 Simplified Wave-Klein-Gordon Models
    • 1.2.3 Small Data Global Regularity Results
    • 1.2.4 Assumptions on the Initial Data
    • 1.2.5 The Mini-bosons
  • 1.3 Main Ideas and Further Asymptotic Results
    • 1.3.1 Global Nonlinear Stability
    • 1.3.2 Nonlinear Scattering
    • 1.3.3 Asymptotic Bounds and Causal Geodesics
    • 1.3.4 Weak Peeling Estimates
    • 1.3.5 The ADM Energy and the Linear Momentum
    • 1.3.6 The Bondi Energy
    • 1.3.7 Organization
    • 1.3.8 Acknowledgements
• 2. The Main Construction and Outline of the Proof
  • 2.1 Setup and the Main Bootstrap Proposition
    • 2.1.1 The Nonlinearities Nhαβ and NΨ
    • 2.1.2 The Fourier Transform and Frequency Projections
    • 2.1.3 Vector-fields
    • 2.1.4 Decomposition of the Metric Tensor
    • 2.1.5 Linear Profiles and the Z-norms
    • 2.1.6 The Main Bootstrap Proposition
  • 2.2 Outline of the Proof
    • 2.2.1 Chapter 3: Preliminary Estimates
    • 2.2.2 Chapter 4: the Nonlinearities Nhαβ and NΨ
    • 2.2.3 Chapter 5: Improved Energy Estimates
    • 2.2.4 Chapter 6: Improved Profile Bounds
• 3. Preliminary Estimates
  • 3.1 Some Lemmas
    • 3.1.1 General Lemmas
    • 3.1.2 The Phases Φσµv
    • 3.1.3 Elements of Paradifferential Calculus
  • 3.2 Linear and Bilinear Estimates
    • 3.2.1 Linear Estimates
    • 3.2.2 Multipliers and Bilinear Operators
    • 3.2.3 Bilinear Estimates
    • 3.2.4 Interpolation Inequalities
  • 3.3 Analysis of the Linear Profiles
• 4. The Nonlinearities Nhαβ and NΨ
  • 4.1 Localized Bilinear Estimates
    • 4.1.1 Frequency Localized L2 Estimates
    • 4.1.2 The Classes of Functions Ga
  • 4.2 Bounds on the Nonlinearities Nhαβ and NΨ
    • 4.2.1 The Quadratic Nonlinearities
    • 4.2.2 The Cubic and Higher Order Nonlinearities
    • 4.2.3 Additional Low Frequency Bounds
    • 4.2.4 Additional Bounds on Some Quadratic Nonlinearities
  • 4.3 Decompositions of the Main Nonlinearities
    • 4.3.1 The Variables FL, FL, pL,w jL, ΩjL,v jkL
    • 4.3.2 Energy Disposable Nonlinearities
    • 4.3.3 Null Structures
    • 4.3.4 The Main Decomposition
• 5. Improved Energy Estimates
  • 5.1 Setup and Preliminary Reductions
    • 5.1.1 Energy Increments
    • 5.1.2 The Main Spacetime Bounds
    • 5.1.3 Poincaré Normal Forms
    • 5.1.4 Paralinearization of the Reduced Wave Operator
  • 5.2 Pure Wave Interactions
    • 5.2.1 Null Interactions
    • 5.2.2 Non-null Semilinear Terms
    • 5.2.3 Second Symmetrization and Paradifferential Calculus
  • 5.3 Mixed Wave-Klein-Gordon Interactions
• 6. Improved Profile Bounds
  • 6.1 Weighted Bounds
  • 6.2 Z-norm Control of the Klein-Gordon Field
    • 6.2.1 Renormalization
    • 6.2.2 Improved Control
  • 6.3 Z-norm Control of the Metric Components
    • 6.3.1 The First Reduction
    • 6.3.2 The Nonlinear Terms Rahαβ, a Є {1,2,4,6}
    • 6.3.3 Localized Bilinear Wave Interactions
    • 6.3.4 The Nonlinear Terms R3hαβ
    • 6.3.5 The Nonlinear Terms R5hαβ
• 7. The Main Theorems
  • 7.1 Global Regularity and Asymptotics
    • 7.1.1 Global Regularity
    • 7.1.2 Decay of the Metric and the Klein-Gordon Field
    • 7.1.3 Null and Timelike Geodesics
  • 7.2 Weak Peeling Estimates and the ADM Energy
    • 7.2.1 Peeling Estimates
    • 7.2.2 The ADM Energy
    • 7.2.3 The Linear Momentum
    • 7.2.4 Gauge Conditions and Parameterizations
  • 7.3 Asymptotically Optical Functions and the Bondi Energy
    • 7.3.1 Almost Optical Functions and the Friedlander Fields
    • 7.3.2 The Bondi Energy
    • 7.3.3 The interior energy
• Bibliography
• Index