论文标题
Kerr公制上线性化的爱因斯坦方程的模式分析：大$ \ Mathfrak {a} $ case
Mode analysis for the linearized Einstein equations on the Kerr metric : the large $\mathfrak{a}$ case
论文作者
论文摘要
我们对线性化爱因斯坦方程的模式解决方案进行了完整的分析，以及在大型$ \ mathfrak {a} $ case中的Kerr Metric上的$ 1- $ form Wave Operator。通过模式解决方案，我们的意思是$ e^{ - it_*σ} \ tilde {h}（r，θ，φ）$的解决方案，其中$ t _*$是合适的时间变量。相应的傅立叶变换$ 1- $形式波动算子和线性化的爱因斯坦运算符在合适的功能空间之间是弗雷德·霍尔姆（Fredholm），$ \ tilde {h} $必须躺在这些操作员的范围内。这些空间是按照Vasy的一般框架构建的。 $ {\ im} \，σ\ ge 0，\，σ\ neq 0 $都不存在模式解决方案。对于$σ= 0 $模式解决方案是$ 1- $形式波动算子的库仑解决方案和线性化的kerr解决方案以及在线性化的爱因斯坦方程的情况下，纯仪表术语。如果我们修复了de turck/Wave Map仪表，则线性化的爱因斯坦方程的零模式解决方案位于固定的$ 7- $维空间中。证明取决于第三作者所示的Teukolsky方程的模式以及由于Aksteiner等人而导致的Kerr Spacetime上线性重力的量规分类。
We give a complete analysis of mode solutions for the linearized Einstein equations and the $1-$form wave operator on the Kerr metric in the large $\mathfrak{a}$ case. By mode solutions we mean solutions of the form $e^{-it_*σ}\tilde{h}(r,θ,φ)$ where $t_*$ is a suitable time variable. The corresponding Fourier transformed $1-$form wave operator and linearized Einstein operator are shown to be Fredholm between suitable function spaces and $\tilde{h}$ has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy. No mode solutions exist for ${\Im}\, σ\ge 0,\, σ\neq 0$. For $σ=0$ mode solutions are Coulomb solutions for the $1-$form wave operator and linearized Kerr solutions plus pure gauge terms in the case of the linearized Einstein equations. If we fix a De Turck/wave map gauge, then the zero mode solutions for the linearized Einstein equations lie in a fixed $7-$dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by the third author and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime due to Aksteiner et al.
