Global Existence for the Einstein Vacuum Equations in Wave Coordinates

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 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. We shall use below the standard convention of summing over repeated indices and the notation ∂α = ∂/∂x α 2 For the definitions of global hyperbolicity and maximal Cauchy development see [H-E], [Wa] 3 The stability result of [C-K] was proved for strongly asymptotically flat data g0 ij = (1 + 2M/r)δij + o(r −3/2 ),where c α J γ are constants such thatThe last identity is a consequence of the relation between c I γ J α and the commutator constants c αβ = [∂ α , Z] β for which we have established that c LL = 0. It therefore follows that 41Decomposing relative to the null frame (L, L, S 1 , S 2 ) we obtainWe now contract the above identity with one of the tangential vector fields T ν , T ∈ {L, S 1 , S 2 } to obtain|Z Im H| · · · |Z I 2 H||∂Z I 1 H|, and (12.11) |∂Z I H| LL |J|≤|I| |∂Z J H| + |J|≤|I|−1 |∂Z J H| LT + |K|≤|I|−2 |∂Z J H| + |I 1 |+...+|Im|≤|I|, m≥2|Z Im H|· · · |Z I 2 H||∂Z I 1 H|.The same estimates also hold for H replaced by h.Proof. This follows directly by the previous lemma with the help of the identities m LT = 0 and c I L J L = 0.

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

“…The Einstein equations do not satisfy the Klainerman null condition completely because of the nonzero coupling between gravitational waves and a mean gravitational field [6]. Nevertheless, it is the vanishing of the nonlinear Λ-coefficients in the equations for the amplitude-waveform functions at the Minkowski metric, for example, that permits the existence of global smooth, small-amplitude perturbations [8,18,22]. A similar result would not be true for (1 + 3)-dimensional variational wave equations in which Λ = 0.…”

“…where H µν is a quadratic form in the metric derivatives ∂g with coefficients depending on g. (See [22], for example, for an explicit expression.) The weakly nonlinear expansion described in Section 2.3 for variational systems of wave equations corresponds to the expansion of Choquet-Bruhat [4] and Isaacson [15] for the Einstein equations.…”

Geometrically motivated hyperbolic coordinate conditions for numerical relativity: Analysis, issues and implementations

“…which can be reinterpreted as a four-dimensional generalization of the positive definite 3D lagrangian (4). Following the analogy, the HAKE equation (14) can be seen as the 4D generalization of the minimal densitized strain (6).…”

“…From the early well posedness result by Choquet-Bruhat [1] to recent global existence proofs [2,3,4] a judicious choice of coordinates was key to yielding a tractable problem.…”

Perspectives on geometric analysis

“…The Cauchy problem was considered by many people: A. Lichnerowicz, Y. Choquet-Bruhat, J. York, V. Moncrief, H. Friedrich, D. Christodoulou, S. Klainerman, H. Lindblad, M. Dafermos (see, e.g., [452], [157], [156], [233], [158], [382], [459], [180]). But the global behavior is still far from being understood.…”