标题： 真空爱因斯坦方程的几何光学近似 显示英文标题

标题： Geometric optics approximation for the Einstein vacuum equations

摘要： 我们通过构造一组高频率度规解$(g_\lambda)_{\lambda\in(0,1]}$来展示广义相对论中几何光学近似的稳定性，这些解是3+1维真空爱因斯坦方程的解，且没有任何对称性假设。 在极限$\lambda\to 0$下，这一族解趋近于一个固定的背景$g_0$解，这是爱因斯坦-零压尘埃系统的解，说明了反作用现象。 我们引入了一个精确的二阶高频率假设，并在每一阶上确定了广义波规范以及极化条件。 我们的假设的有效性由里奇张量中二次非线性项满足的弱极化零条件来保证。 真空爱因斯坦方程对于$g_\lambda$被重新表述为传输方程和波动方程的层次结构，它们的耦合导致了导数的损失。 为了求解它，我们利用了与$g_0$相关的零叶层以及适应我们假设的傅里叶截断。 构造满足约束方程的高频率初始数据的内容是我们另一篇论文\cite{Touati2023a}的内容。 显示英文摘要

摘要： We show the stability of the geometric optics approximation in general relativity by constructing a family $(g_\lambda)_{\lambda\in(0,1]}$ of high-frequency metrics solutions to the Einstein vacuum equations in 3+1 dimensions without any symmetry assumptions. In the limit $\lambda\to 0$ this family approaches a fixed background $g_0$ solution of the Einstein-null dust system, illustrating the backreaction phenomenon. We introduce a precise second order high-frequency ansatz and identify a generalised wave gauge as well as polarization conditions at each order. The validity of our ansatz is ensured by a weak polarized null condition satisfied by the quadratic non-linearity in the Ricci tensor. The Einstein vacuum equations for $g_\lambda$ are recast as a hierarchy of transport and wave equations, and their coupling induces a loss of derivatives. In order to solve it, we take advantage of a null foliation associated to $g_0$ as well as a Fourier cut-off adapted to our ansatz. The construction of high-frequency initial data solving the constraint equations is the content of our companion paper \cite{Touati2023a}.