The Global Nonlinear Stability of Minkowski Space for Self-Gravitating Massive Fields

This book is devoted to the Einstein's field equations of general relativity for self-gravitating massive scalar fields. We formulate the initial value problem when the initial data set is a perturbation of an asymptotically flat, spacelike hypersurface in Minkowski spacetime. We then establish the existence of an Einstein development associated with this initial data set, which is proven to be an asymptotically flat and future geodesically complete spacetime.

Contents:

• Introduction

• Overview of the Hyperboloidal Foliation Method

• Functional Analysis on Hyperboloids of Minkowski Spacetime

• Quasi-Null Structure of the Einstein-Massive Field System on Hyperboloids

• Initialization of the Bootstrap Argument

• Direct Control of Nonlinearities in the Einstein Equations

• Direct Consequences of the Wave Gauge Condition

• Second-Order Derivatives of the Spacetime Metric

• Sup-Norm Estimate Based on Characteristics

• Low-Order Refined Energy Estimate for the Spacetime Metric

• Low-Order Refined Sup-Norm Estimate for the Metric and Scalar Field

• High-Order Refined L² Estimates

• High-Order Refined Sup-Norm Estimates

• Low-Order Refined Energy Estimate for the Scalar Field

• Appendices:

  • Revisiting the Wave-Klein-Gordon Model
  • Sup-Norm Estimate for the Wave Equations
  • Sup-Norm Estimate for the Klein-Gordon Equation
  • Commutator Estimates for the Hyperboloidal Frame

• Bibliography

Readership: Graduate students and researchers interested in mathematical general relativity.
Key Features:

• This is the first mathematical result on the global nonlinear stability of matter fields in general relativity
• Provide a framework to treat other matter fields such as the massive Yang-Mills fields which are of particular importance in physics