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Global existence for the Einstein vacuum equations in wave coordinates

Hans Lindblad, Igor Rodnianski

TL;DR

The paper proves global existence for the Einstein vacuum equations in harmonic coordinates for a restricted class of asymptotically Schwarzschild data, establishing stability of Minkowski space in this gauge. Central to the argument are the weak null condition for the wave-coordinate reduced system, a vector-field method with Klainerman–Sobolev inequalities, and delicate decay/energy estimates that leverage the wave-coordinate constraint. The authors show that small, Schwarzschild-like data outside a unit ball produce a global, geodesically complete spacetime whose metric converges to Minkowski at late times, challenging prior beliefs about wave coordinates being unstable in the large. The work lays the groundwork for extending to general small data and coupling to scalar fields, via a framework that isolates tangential decay along null cones and controls transversal derivatives through the wave-coordinate condition.

Abstract

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with Schwartzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are "unstable in the large" and provides an alternative approach to the stability problem originally solved (for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman. Using the wave coordinate gauge we recast the Einstein equations as a system of quasilinear wave equations and, in absence of the classical null condition, establish a small data global existence result. In our previous work we introduced the notion of a eak nul condition and showed that the Einstein equations in wave coordinates satisfy this condition. The result of this paper relies on this observation and combines it with the vector field method based on the symmetries of the standard Minkowski space. In a forthcoming paper we will address the question of stability of Minkowski space for the Einstein vacuum equations in wave coordinates for all "small" asymptotically flat data and the case of the Einstein equations coupled to a scalar field.

Global existence for the Einstein vacuum equations in wave coordinates

TL;DR

The paper proves global existence for the Einstein vacuum equations in harmonic coordinates for a restricted class of asymptotically Schwarzschild data, establishing stability of Minkowski space in this gauge. Central to the argument are the weak null condition for the wave-coordinate reduced system, a vector-field method with Klainerman–Sobolev inequalities, and delicate decay/energy estimates that leverage the wave-coordinate constraint. The authors show that small, Schwarzschild-like data outside a unit ball produce a global, geodesically complete spacetime whose metric converges to Minkowski at late times, challenging prior beliefs about wave coordinates being unstable in the large. The work lays the groundwork for extending to general small data and coupling to scalar fields, via a framework that isolates tangential decay along null cones and controls transversal derivatives through the wave-coordinate condition.

Abstract

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with Schwartzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are "unstable in the large" and provides an alternative approach to the stability problem originally solved (for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman. Using the wave coordinate gauge we recast the Einstein equations as a system of quasilinear wave equations and, in absence of the classical null condition, establish a small data global existence result. In our previous work we introduced the notion of a eak nul condition and showed that the Einstein equations in wave coordinates satisfy this condition. The result of this paper relies on this observation and combines it with the vector field method based on the symmetries of the standard Minkowski space. In a forthcoming paper we will address the question of stability of Minkowski space for the Einstein vacuum equations in wave coordinates for all "small" asymptotically flat data and the case of the Einstein equations coupled to a scalar field.

Paper Structure

This paper contains 24 sections, 48 theorems, 421 equations.

Table of Contents

  1. Introduction
  2. The main results and the strategy of the proof
  3. The Einstein equations in wave coordinates
  4. The initial data
  5. The null-frame and null-forms
  6. The weak null condition and asymptotic expansion of Einstein's equation's in wave coordinates
  7. Vector fields and commutators
  8. Basic energy identities
  9. Poincaré and Klainerman-Sobolev inequalities
  10. Decay estimates for the wave equation on a curved space time
  11. Energy estimates for the wave equation on a curved space time
  12. Estimates from the wave coordinate condition
  13. Estimates for the inhomogeneous terms
  14. The decay estimates for Einstein's equations
  15. Proof of \ref{['eq:weakdecay2']} and \ref{['eq:tandecay']}
  16. ...and 9 more sections

Key Result

Theorem 2.1

Consider the reduced Einstein vacuum equations In what follows we shall introduce the reduced wave operator $\widetilde{\square}_g=g^{\alpha\beta} \partial^2_{\alpha\beta}$ and note that in wave coordinates $\widetilde{\square}_g = \square_g$, where $\square_g \phi=|g|^{-1/2} \partial_\alpha (g^{\al where $g_{\mu\nu}=m_{\mu\nu} + h_{\mu\nu}$ and the nonlinear term $F$ is as in Lemma Einstwavecquad

Theorems & Definitions (91)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Remark 5.1
  • ...and 81 more