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Ricci defects of microlocalized Einstein metrics. (English) Zbl 1063.53051

The rough solutions of the Einstein vacuum equations \({\mathbf R}_{\alpha\beta}({\mathbf g})= 0\) expressed in wave coordinates \(x^\alpha\): \[ \square_{{\mathbf g}} x^\alpha= (1/|{\mathbf g}|)\partial_\mu({\mathbf g}^{\mu\nu}|{\mathbf g}|\partial_\nu) x^\alpha \] are studied in detail. A statement concerning the Ricci defects of “microlocalized solutions” is proven which was previously used by the authors in the proof of the “asymptotic theorem” (in press) introduced by them. It is pointed out that this result is of independent interest in situations when one has to smooth out the given space-time metric in order to achieve good causality properties.
The paper is organized, as follows: introduction, preliminaries (background estimates, set-up and error terms), wave coordinate condition; first reduction, the algebraic structure of \({\mathbf R}_{\mu\nu}(H)\), the structure of the error and the principal terms as well as the estimates for them.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35Q75 PDEs in connection with relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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