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arXiv · 2206.13062

High-frequency solutions to the constraint equations

Arthur Touati
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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.

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Arthur Touati. 2022-06-27. High-frequency solutions to the constraint equations. https://doi.org/10.1007/s00220-023-04715-8

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