标题： Kerr时空引力扰动的新规范 I：线性化理论 显示英文标题

标题： A new gauge for gravitational perturbations of Kerr spacetimes I: The linearised theory

摘要： 我们提出了一种新的几何框架，以在完整的次极端范围内处理Kerr解对引力扰动的稳定性$|a|<M$。 我们的框架的核心是对Kerr非线性引力扰动的新表述，其两个新要素是选择一个几何规范和不可积分的零框架，这些都专门针对Kerr的出射主零测地线。 在我们的规范下，扰动的真空爱因斯坦方程被表述为相对于所选框架的连接系数和曲率分量的方程组。 当相对于Kerr进行归一化时，具有出射传输方程形式的零结构方程在右边不包含归一化连接系数的任何导数。 在本工作中，我们推导了新框架下围绕Kerr的线性真空爱因斯坦方程。 我们的新框架旨在有效捕捉红移传输方程的稳定特性，从而隔离问题的一个关键结构。 这一特性表明未来分析可能会有简化。 作为例证，我们的配套工作\cite{benomio_schwarzschild}采用线性化引力及其增强的红移传输方程，并专门针对$|a|=0$情况，以产生一个新的简化证明，证明Schwarzschild解的线性稳定性。 完整的次极端范围$|a|<M$的线性稳定性分析将留待未来的工作。 如\cite{benomio_schwarzschild}所示，我们的框架将允许将传输方程中的新结构与系统的椭圆部分结合，建立一个没有导数损失的线性轨道稳定性结果，表明该框架可能非常适合解决完整次极端范围$|a|<M$的非线性稳定性问题。 显示英文摘要

摘要： We propose a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range $|a|<M$. Central to our framework is a new formulation of nonlinear gravitational perturbations of Kerr, whose two novel ingredients are the choice of a geometric gauge and non-integrable null frames both tailored to the outgoing principal null geodesics of Kerr. The vacuum Einstein equations for the perturbations are formulated in our gauge as a system of equations for the connection coefficients and curvature components relative to the chosen frames. When renormalised with respect to Kerr, the null structure equations with the form of outgoing transport equations do not possess any derivatives of renormalised connection coefficients on the right hand side. In this work, we derive the linearised vacuum Einstein equations around Kerr in the new framework. Our new framework is designed to effectively capture the stabilising properties of the red-shifted transport equations, thereby isolating one of the crucial structures of the problem. Such a feature is suggestive of future simplifications in the analysis. As an illustration, our companion work \cite{benomio_schwarzschild} employs the system of linearised gravity and its enhanced red-shifted transport equations, specialised to the $|a|=0$ case, to produce a new simplified proof of linear stability of the Schwarzschild solution. The full linear stability analysis in the full sub-extremal range $|a|<M$ is deferred to future work. As already apparent from \cite{benomio_schwarzschild}, our framework will allow to combine the new structure in the transport equations with the elliptic part of the system to establish a linear orbital stability result without loss of derivatives, indicating that the framework may be well suited to address nonlinear stability in the full sub-extremal range $|a|<M$.