On the Asymptotics for the Vacuum Einstein Constraint Equations

TL;DR

This paper develops a method to modify initial data for the vacuum Einstein equations, matching given data inside a domain and Kerr solutions outside, with the set of such data being dense in the solution space.

Contribution

It generalizes previous work by constructing solutions that interpolate between arbitrary data and Kerr solutions outside a domain, expanding the class of initial data for Einstein's equations.

Findings

Constructed solutions match given data inside a domain and Kerr outside.

The set of such solutions is dense in the space of asymptotically flat solutions.

Extended previous results from the time-symmetric case to the general vacuum case.

Abstract

Given asymptotically flat initial data on M^3 for the vacuum Einstein field equation, and given a bounded domain in M, we construct solutions of the vacuum constraint equations which agree with the original data inside the given domain, and are identical to that of a suitable Kerr slice (or identical to a member of some other admissible family of solutions) outside a large ball in a given end. The data for which this construction works is shown to be dense in an appropriate topology on the space of asymptotically flat solutions of the vacuum constraints. This construction generalizes work of the first author, where the time-symmetric case was studied.