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Nonlinear stability of self-gravitating massive fields

Philippe G. LeFloch, Yue Ma

Abstract

We consider the global evolution problem for Einstein's field equations in the near-Minkowski regime and study the long-time dynamics of a massive scalar field evolving under its own gravitational field. We establish the existence of a globally hyperbolic Cauchy development associated with any initial data set that is sufficiently close to a data set in Minkowski spacetime. In addition to applying to massive fields, our theory allows us to cover metrics with slow decay in space. The strategy of proof, proposed here and referred to as the Euclidean-Hyperboloidal Foliation Method, applies, more generally, to nonlinear systems of coupled wave and Klein-Gordon equations. It is based on a spacetime foliation defined by merging together asymptotically Euclidean hypersurfaces (covering spacelike infinity) and asymptotically hyperboloidal hypersurfaces (covering timelike infinity). A transition domain (reaching null infinity) limited by two asymptotic light cones is introduced in order to realize this merging. On the one hand, we exhibit a boost-rotation hierarchy property (as we call it) which is associated with Minkowski's Killing fields and is enjoyed by commutators of curved wave operators and, on the other hand, we exhibit a metric hierarchy property (as we call it) enjoyed by components of Einstein's field equations in frames associated with our Euclidean-hyperboloidal foliation. The core of the argument is, on the one hand, the derivation of novel integral and pointwise estimates which lead us to almost sharp decay properties (at timelike, null, and spacelike infinity) and, on the other hand, the control of the (quasi-linear and semi-linear) coupling between the geometric and matter parts of the Einstein equations.

Nonlinear stability of self-gravitating massive fields

Abstract

We consider the global evolution problem for Einstein's field equations in the near-Minkowski regime and study the long-time dynamics of a massive scalar field evolving under its own gravitational field. We establish the existence of a globally hyperbolic Cauchy development associated with any initial data set that is sufficiently close to a data set in Minkowski spacetime. In addition to applying to massive fields, our theory allows us to cover metrics with slow decay in space. The strategy of proof, proposed here and referred to as the Euclidean-Hyperboloidal Foliation Method, applies, more generally, to nonlinear systems of coupled wave and Klein-Gordon equations. It is based on a spacetime foliation defined by merging together asymptotically Euclidean hypersurfaces (covering spacelike infinity) and asymptotically hyperboloidal hypersurfaces (covering timelike infinity). A transition domain (reaching null infinity) limited by two asymptotic light cones is introduced in order to realize this merging. On the one hand, we exhibit a boost-rotation hierarchy property (as we call it) which is associated with Minkowski's Killing fields and is enjoyed by commutators of curved wave operators and, on the other hand, we exhibit a metric hierarchy property (as we call it) enjoyed by components of Einstein's field equations in frames associated with our Euclidean-hyperboloidal foliation. The core of the argument is, on the one hand, the derivation of novel integral and pointwise estimates which lead us to almost sharp decay properties (at timelike, null, and spacelike infinity) and, on the other hand, the control of the (quasi-linear and semi-linear) coupling between the geometric and matter parts of the Einstein equations.

Paper Structure

This paper contains 263 sections, 114 theorems, 952 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The global evolution problem for self-gravitating matter
  3. Stability of vacuum spacetimes near Minkowski space.
  4. New methodology for massive matter fields.
  5. Numerical study of asymptotically flat spacetimes.
  6. Asymptotically Anti-de Sitter spacetimes.
  7. Background on Einstein's field equations
  8. Einstein equations.
  9. Massive scalar field.
  10. Nonlinear Klein-Gordon equation.
  11. Initial value problem.
  12. Vacuum spacetimes.
  13. Outline of the contribution in this paper
  14. The Euclidean-Hyperboloidal Foliation Method.
  15. Reference data and decay at spacelike infinity.
  16. ...and 248 more sections

Key Result

Theorem 2.7

Fix some sufficiently large integer $N$ ($N=20$ being sufficient) and consider an admissible light-bending initial data $(g_0,k_0,\phi_0,\phi_1)$ with parameters $(\lambda,\kappa,\epsilon_\star,N,\mu,\epsilon)$ satisfying Then there exists a small constant $c_0>0$ (determined by the Einstein system) such that for all the maximal globally hyperbolic Cauchy development of $(g_0,k_0,\phi_0,\phi_1)$

Figures (1)

  • Figure :

Theorems & Definitions (196)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: Nonlinear stability of self-gravitating massive fields
  • Theorem 2.8: Nonlinear stability of self-gravitating massive fields. A generalization
  • Lemma 3.1: The Euclidean--hyperboloidal time function
  • Lemma 3.2
  • ...and 186 more