论文标题
爱因斯坦真空方程的几何光学近似
Geometric optics approximation for the Einstein vacuum equations
论文作者
论文摘要
我们通过构建一个$（g_λ）_ {λ\ in（0,1]} $，在3+1个维度的3+1个维度的情况下，$λ$λ$λ$λ$λ$λ$λ\ t $λ$，我们在一般相对论中显示了几何光学近似值的稳定性。尘埃系统，说明了倒退现象。传输方程及其耦合会引起衍生物的损失，以解决它，我们利用了与$ G_0 $相关的无效叶面，以及适用于我们的ANSATZ的傅立叶截止。
We show the stability of the geometric optics approximation in general relativity by constructing a family $(g_λ)_{λ\in(0,1]}$ of high-frequency metrics solutions to the Einstein vacuum equations in 3+1 dimensions without any symmetry assumptions. In the limit $λ\to 0$ 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 $g_λ$ 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}.
