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Perturbative Solutions of the Extended Constraint Equations in General Relativity
Authors:Adrian Butscher
Institution:(1) Department of Mathematics, University of Toronto, Toronto, ON, M1C 1A4, Canada
Abstract:The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface $${\mathcal{Z}}$$ in an asymptotically simple space-time satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on $${\mathcal{Z}}$$ , and are equivalent to the usual constraint equations that $${\mathcal{Z}}$$ satisfies as a space-like hypersurface in a space-time satisfying Einstein’s vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the ‘classical’ method of Lichnerowicz and York that is used to solve the usual constraint equations.
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