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Journal Article

Perturbative Solutions of the Extended Constraint Equations in General Relativity


Butscher,  Adrian
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Butscher, A. (2007). Perturbative Solutions of the Extended Constraint Equations in General Relativity. Communications in Mathematical Physics, 272(1), 1-23. doi:10.1007/s00220-007-0204-8.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-4851-4
The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface $Z$ in an asymptotically simple spacetime 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 $Z$, and are equivalent to the usual constraint equations that $Z$ satisfies as a spacelike hypersurface in a spacetime 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.