Perturbative Solutions of the Extended Constraint Equations in General Relativity

TL;DR

This paper develops a novel perturbative method to find asymptotically flat solutions of the extended constraint equations in general relativity, expanding the toolkit beyond classical approaches.

Contribution

It introduces a new perturbative approach for solving the extended constraint equations near flat space, differing from classical methods.

Findings

Successfully constructs perturbative solutions near flat space.

Provides a new method applicable to the extended constraint equations.

Enhances understanding of initial data in asymptotically simple spacetimes.

Abstract

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.