Obstruction-free gluing for the Einstein equations

TL;DR

This paper introduces a novel nonlinear approach to the gluing problem in General Relativity, enabling the matching of solutions along hypersurfaces and removing previous obstructions, thus broadening the scope of initial data configurations.

Contribution

The authors develop a new nonlinear method that eliminates obstructions in the null and spacelike gluing problems, allowing for flexible matching of initial data in Einstein's equations.

Findings

Any asymptotically flat spacelike initial data can be glued to large-mass Schwarzschild data.

The method allows choosing Kerr initial data for gluing, extending previous results.

A new technique combines low-frequency linear analysis with high-frequency nonlinear control.

Abstract

In this paper we develop a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to the previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the $10$-dimensional spaces of obstructions to the null and spacelike (asymptotically flat) gluing problems, previously known in the literature. In particular, we show that any asymptotically flat spacelike initial data can be glued to the Schwarzschild initial data of mass $M$ for any $M>0$ sufficiently large. More generally, compared to the celebrated result of Corvino-Schoen, our methods allow us to choose ourselves the Kerr spacelike initial data that is being glued onto. As in our earlier work, our primary focus is the analysis of the null problem, where we develop a new technique of combining low-frequency linear analysis with high-frequency nonlinear control. The corresponding spacelike results are derived a posteriori by solving a characteristic initial value problem.