论文标题
KERR间距的重力扰动的新规格I：线性化理论
A new gauge for gravitational perturbations of Kerr spacetimes I: The linearised theory
论文作者
论文摘要
我们提出了一个新的几何框架，以解决KERR解决方案在完整的次级范围$ | a | <m $中对重力扰动的稳定性。我们框架的核心是Kerr非线性重力扰动的新表述，其两种新颖的成分是选择几何规格和不可融合的空框架，均针对Kerr即将出发的主要无原理大地测量学量身定制。扰动的真空爱因斯坦方程在我们的仪表中被配制为相对于所选框架的连接系数和曲率成分的方程系统。当对Kerr的重新规定时，具有传出传输方程形式的空结构方程在右侧不具有任何重量化连接系数的导数。在这项工作中，我们在新框架中得出了Kerr周围的线性真空爱因斯坦方程。 我们的新框架旨在有效地捕获红移传输方程的稳定特性，从而隔离了问题的关键结构之一。这样的功能暗示了分析中的未来简化。作为例证，我们的同伴工作\ cite {benomio_schwarzschild}采用了线性重力系统及其增强的红移传输方程，专门针对$ | a | = 0 $ case，以产生新的简化的Schwarzschild解决方案线性稳定性的简化证明。完整的次级范围$ | a | <m $中的完整线性稳定性分析将延迟到将来的工作。从\ cite {benomio_schwarzschild}中可以看出，我们的框架将允许将新结构与系统的椭圆形部分结合起来，以建立线性轨道稳定性而不会丢失衍生物，这表明该框架可以很好地解决非线性稳定性，可以在整个子范围内解决非线性稳定性。
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 $|a|<M$.
