High-frequency solutions to the constraint equations

TL;DR

This paper develops a method to construct high-frequency initial data for Einstein's vacuum equations using a geometric optics-inspired expansion, enabling the analysis of gravitational waves in a high-frequency regime.

Contribution

It introduces a novel high-frequency expansion approach to solve the Einstein constraint equations, extending the conformal method to this context.

Findings

Constructed a family of high-frequency initial data close to null dust data.

Adapted the conformal method for high-frequency Einstein constraints.

Provided groundwork for analyzing high-frequency gravitational waves.

Abstract

We construct high-frequency initial data for the Einstein vacuum equations in dimension 3+1 by solving the constraint equations on $\mathbb{R}^3$. Our family of solutions $(\bar{g}_\lambda,K_\lambda)_{\lambda\in(0,1]}$ is defined through a high-frequency expansion similar to the geometric optics approach and is close in a particular sense to the data of a null dust. In order to solve the constraint equations, we use their conformal formulation and the main challenge of our proof is to adapt this method in the high-frequency context. The main application of this article is our companion article \cite{Touati2022a} where we construct high-frequency gravitational waves in generalised wave gauge.