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The global stability of the Minkowski space-time in harmonic gauge

Hans Lindblad, Igor Rodnianski

TL;DR

This work proves global stability of Minkowski space for the Einstein vacuum and Einstein–scalar field equations under general asymptotically flat small data, using the harmonic gauge. The authors reduce to a quasilinear wave system in wave coordinates, decompose the metric into a Schwarzschild-like part h^0 and a dynamical remainder h^1, and develop a coordinated energy–decay framework enhanced by weighted energies and a wave-coordinate–driven commutator analysis. Central to the argument are robust decay estimates (via weighted Klainerman–Sobolev inequalities), an independent decay mechanism, and precise control of commutators, enabling closure of a bootstrap and yielding geodesic completeness with solutions converging to Minkowski space. The techniques leverage a detailed null-frame analysis, the weak null condition for the Einstein equations in wave coordinates, and a careful treatment of the long-range mass term, providing a relatively streamlined wave-equation approach to small-data global existence in general relativity.

Abstract

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach shows that the Einstein-vacuum and the Einstein-scalar field equations with general asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time. The proof utilizes the classical harmonic (also known as de Donder or wave coordinate) gauge.

The global stability of the Minkowski space-time in harmonic gauge

TL;DR

This work proves global stability of Minkowski space for the Einstein vacuum and Einstein–scalar field equations under general asymptotically flat small data, using the harmonic gauge. The authors reduce to a quasilinear wave system in wave coordinates, decompose the metric into a Schwarzschild-like part h^0 and a dynamical remainder h^1, and develop a coordinated energy–decay framework enhanced by weighted energies and a wave-coordinate–driven commutator analysis. Central to the argument are robust decay estimates (via weighted Klainerman–Sobolev inequalities), an independent decay mechanism, and precise control of commutators, enabling closure of a bootstrap and yielding geodesic completeness with solutions converging to Minkowski space. The techniques leverage a detailed null-frame analysis, the weak null condition for the Einstein equations in wave coordinates, and a careful treatment of the long-range mass term, providing a relatively streamlined wave-equation approach to small-data global existence in general relativity.

Abstract

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach shows that the Einstein-vacuum and the Einstein-scalar field equations with general asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time. The proof utilizes the classical harmonic (also known as de Donder or wave coordinate) gauge.

Paper Structure

This paper contains 25 sections, 45 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof and outline of the paper
  3. Notations and conventions
  4. Strategy of the proof
  5. The Einstein equations in wave coordinates
  6. The null-frame
  7. Vector fields
  8. Energy estimates in curved space-time
  9. Decay estimates in curved space-time
  10. The wave coordinate condition
  11. Statement of the Main Theorem and beginning of the proof
  12. Estimates for the inhomogeneous terms
  13. Decay estimates for the Einstein equations
  14. Proof of \ref{['eq:sharpdecayone']}-\ref{['eq:sharpdecaytwo']}.
  15. Proof of \ref{['eq:sharpdecaythree']}-\ref{['eq:sharpdecayfour']}.
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(\Sigma, g_0, k_0, \psi_0, \psi_1)$ be initial data for the Einstein-scalar field equations. Assume that the initial time slice $\Sigma$ is diffeomorphic to ${\Bbb R}^3$ and admits a global coordinate chart relative to which the data is close to the initial data for the Minkowski space-time. Mo where $\chi(s)\in C^\infty$ is $1$ when $s\geq 3/4$ and $0$ when $s\leq 1/2$. Set There is a const

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 2.1
  • Proposition 3.1
  • Remark 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • Corollary 4.4
  • proof
  • Lemma 5.1
  • ...and 74 more