Global Existence for the Einstein Vacuum Equations in Wave Coordinates

By Hans Lindblad, Igor Rodnianski. 2003

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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.

Subfields: Astronomy and Astrophysics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/44f94ece22848661dca3ef74d46354af8ad5f6da

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Citations in the corpus

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations (2013, background): “Phase mixing also shares certain similarities with scattering in the theory of dispersive wave equations (see for instance [50, 34, 58]) as already pointed out in [27, 18].”
Landau Damping: Paraproducts and Gevrey Regularity (2013, unknown): “pagating controls on both “high” and “low” norms of regularity – the “high” ones being mildly growing in time, a general scheme which is common in the study of wave and dispersive equations (see e.g. [41, 49, 26] and the references therein). 2.2 Gevrey functional setting As discussed in the previous”
Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov) (2010, methodology): “On perturbations of Minkowski space itself, the full non-linear analysis was done in the path-breaking work of Christodoulou and Klainerman [9], and Lindblad and Rodnianski simplified the analysis [34, 35], Bieri [2, 3] succeeded in relaxing the decay conditions, while Wang [59] obtained additional,”
Future stability of the Einstein-non-linear scalar field system (2008, methodology): “We refer the reader to [22] for a careful discussion of these issues, cf.”
The global non-linear stability of the Kerr-de Sitter family of black holes (2016, methodology): “Non-linear problems on exterior Λ = 0 black hole spacetimes were studied by Dafermos, Holzegel and Rodnianski [DHR13] who constructed backward solutions of the Einstein vacuum equations settling down to Kerr exponentially fast (regarding this point, see also [DSR15]); for forward problems, Dafermos ”
Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009, methodology): “More recently, a simpler proof of the nonlinear stability of Minkowski space has been developed [35]; however, it also makes use of the vector-field method.”
The Global Nonlinear Stability of Minkowski Space for Self-gravitating Massive Fields (2015, methodology): “Our method covers self-gravitating massive scalar fields (as will be presented in full details in [23]), while earlier works were restricted to vacuum spacetimes or to spacetimes with massless scalar fields; cf. Christodoulou and Klainerman [7], and Lindblad and Rodnianski [26,27], as well as Bieri ”
Extensions of the Stability Theorem of the Minkowski Space in General Relativity (2009, background): “Rodnianski [30], [31], the latter for EV (scalar field) equations.”
The global stability of the Minkowski space-time in harmonic gauge (2004, background): “ocal system of coordinates {xµ}µ=0,...,3 satisfying the wave coordinate condition gxµ = 0 at x0 = t= 0. This procedure is standard and its slightly simpler version can be found e.g. in [Wa], see also [L-R2]. The smallness condition EN(0)+M≤ ǫfor the initial data (g0ij,k0ij,ψ0,ψ1) implies the smallne”
A new proof of long-range scattering for critical nonlinear Schrödinger equations (2010, background): “For some results concerning nonlinear asy mptotic behavior of solution of wave equations see [18, 1, 17] and references therein.”