Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities
Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities
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DOI：
10.1007/s40818-020-00083-x
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发表时间：
2018-09
期刊：
影响因子：
2.8
通讯作者：
Pei-Ken Hung;Jordan Keller;Mu-Tao Wang
中科院分区：
文献类型：
--
作者：
Pei-Ken Hung;Jordan Keller;Mu-Tao Wang
本文研究了以高维史瓦西度规为中心的线性化真空爱因斯坦方程的解。我们采用霍奇分解的解决方案分为标量，共矢量，和两张量片;前两个部分分别对应于封闭和共封闭，或极轴，解决方案的情况下，四维时空，而两张量部分是一个新的功能，在更高的维度设置。改写早期的工作Kodama-Ishibashi-濑户在我们的霍奇分解的语言，我们产生解耦规范不变的主量满足Regge-Wheeler型波动方程的三个部分。标量和共矢量分别推广了在四维时空中发现的Moncrief-Zerilli和Regge-Wheeler量;在这些量之外，我们在共矢量部分发现了Cunningham-Moncrief-Price量的高维模拟。此外，我们的工作提供了第一个验证，标量主量满足其假定的Regge-Wheeler方程。在主量的分析中，我们加强了模式稳定性的结果Kodama-Ishibashi在所有维度上的统一有界性估计;此外，我们证明了在六个或更少的时空维度的情况下的衰减估计。在超过六个时空维度的情况下，我们发现了一个障碍Morawetz型估计所产生的负势项在时空维度二次增长。最后，我们提供了一个严格的论点，线性化的解决方案的低角频率是可分解的纯规范解决方案和线性化的Myers-Perry解决方案，后者的解决方案推广线性化克尔解决方案在四维时空。
In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild metrics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively correspond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting. Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master quantities satisfying Regge-Wheeler type wave equations in each of the three portions. The scalar and co-vector quantities respectively generalize the Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four spacetime dimensions; beyond these quantities, we discover a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector portion. In addition, our work provides the first verification that the scalar master quantity satisfies its putative Regge-Wheeler equation. In the analysis of the master quantities, we strengthen the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate in all dimensions; further, we prove decay estimates in the case of six or fewer spacetime dimensions. In the case of more than six spacetime dimensions, we discover an obstruction to Morawetz type estimates arising from negative potential terms growing quadratically in spacetime dimension. Finally, we provide a rigorous argument that linearized solutions of low angular frequency are decomposable as a sum of pure gauge solution and linearized Myers-Perry solution, the latter solutions generalizing the linearized Kerr solutions in four spacetime dimensions.