黑弦和黑环的稳定俘获现象及其对线性波衰减的阻碍
The Stable Trapping Phenomenon for Black Strings and Black Rings and its Obstructions on the Decay of Linear Waves
摘要：
高维爱因斯坦真空方程的几何解提出了四维空间中不存在的一些方面，其中最引人注目的是在渐近平坦的黑洞外部存在稳定捕获的零测地线。本文研究了两类高维黑洞，即黑弦和黑环的稳定捕获现象，以及这种捕获结构是如何导致其外部线性波缓慢衰减的.更准确地说，我们研究的标量，线性波动方程$\Box_{g_{\textup{ring}}} Ψ=0$，其中$g_{\textup{ring}}$是一个固定的黑环解决方案的五维爱因斯坦真空方程的度量的解决方案的能量衰减特性。对于一类$\mathfrak{g}$黑环度量，利用$g_{\textup{ring}}\in\mathfrak{g}$证明了$(\mathcal{D},g_{\textup{ring}})$黑环外均匀能量衰减率的一个对数下界.证明推广的扰动参数和准模式建设Holzegel-Smulevici \cite{SharpLogHolz}的情况下，一个不可分的波动方程，并依赖于稳定的陷获零测地线$\mathcal{D}$的存在。作为一个副产品，同样的对数下限可以建立任何五维黑弦。我们的结果是第一个数学上严格的声明支持的期望，黑环是动态不稳定的一般扰动。特别是，我们猜想一个新的\textit{nonlinear}不稳定性的五维黑弦和薄黑环，这是已经存在的标量扰动的水平，并明显不同于由著名的格雷戈里-Laflamme不稳定性驱动的机制。
英文摘要：
The geometry of solutions to the higher dimensional Einstein vacuum equations presents aspects that are absent in four dimensions, one of the most remarkable being the existence of stably trapped null geodesics in the exterior of asymptotically flat black holes. This paper investigates the stable trapping phenomenon for two families of higher dimensional black holes, namely black strings and black rings, and how this trapping structure is responsible for the slow decay of linear waves on their exterior. More precisely, we study decay properties for the energy of solutions to the scalar, linear wave equation $\Box_{g_{\textup{ring}}} Ψ=0$, where $g_{\textup{ring}}$ is the metric of a fixed black ring solution to the five-dimensional Einstein vacuum equations. For a class $\mathfrak{g}$ of black ring metrics, we prove a logarithmic lower bound for the uniform energy decay rate on the black ring exterior $(\mathcal{D},g_{\textup{ring}})$, with $g_{\textup{ring}}\in\mathfrak{g}$. The proof generalizes the perturbation argument and quasimode construction of Holzegel--Smulevici \cite{SharpLogHolz} to the case of a non-separable wave equation and crucially relies on the presence of stably trapped null geodesics on $\mathcal{D}$. As a by-product, the same logarithmic lower bound can be established for any five-dimensional black string. Our result is the first mathematically rigorous statement supporting the expectation that black rings are dynamically unstable to generic perturbations. In particular, we conjecture a new \textit{nonlinear} instability for five-dimensional black strings and thin black rings which is already present at the level of scalar perturbations and clearly differs from the mechanism driven by the well-known Gregory--Laflamme instability.
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