Perturbative Solutions of the Extended Constraint Equations in General Relativity

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.