Communications in Mathematical Physics

, Volume 256, Issue 1, pp 43–110 | Cite as

Global Existence for the Einstein Vacuum Equations in Wave Coordinates

  • Hans Lindblad
  • Igor Rodnianski


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 the Schwarzschild 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 weak null condition and showed that the Einstein equations in harmonic 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.


Einstein Equation Global Existence Minkowski Space Global Stability Null Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hans Lindblad
    • 1
  • Igor Rodnianski
    • 2
  1. 1.Mathematics DepartmentUniversity of California at San DiegoLa JollaUSA
  2. 2.Department MathematicsFine Hall, Princeton UniversityPrincetonUSA

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