The causal structure of microlocalized rough Einstein metrics

Abstract

This is the second in a series of three papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates.By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities.In this paper we develop the geometric analysis of the Eikonal equation for microlocalized rough Einstein metrics.This is a crucial step in the derivation of the decay estimates needed in the first paper.1 To go beyond our result will require the development of bilinear techniques for the Einstein equations; see the discussion in the introduction to [Kl-Ro1].2 We denote by R αβ the Ricci curvature of g. 3 In wave coordinates the Einstein equations take the reduced form g αβ ∂α∂ β gμν = Nμν (g, ∂g) with N quadratic in the first derivatives ∂g of the metric. 4We assume however that T stays sufficiently small, e.g.T ≤ 1.This a purely technical assumption which one should be able to remove.5 More precisely, for a given function of the spatial variables x = x 1 , x 2 , x 3 , the Littlewood Paley projectionThe definition of the projector P <λ in [Kl-Ro1] was slightly different from the one we are using in this paper.There P <λ removed all the frequencies above 2 -M 0 λ for some sufficiently large constant M0.It is clear that a simple rescaling can remedy this discrepancy.

Locations

• arXiv (Cornell University) - PDF
• Annals of Mathematics - View - PDF