arXiv 1010.32162010-10-15gr-qcmath.AP

终极Schwarzschildean时空与黑洞稳定性问题

Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem

Gustav Holzegel

备注：: 118 pages

摘要：

本文引入了一类满足真空爱因斯坦方程的时空$\left(\mathcal{M},g\right)$，它动态地逼近质量$M$的史瓦西解，我们称之为\emph{ultimately Schwarzschildean spacetimes} .该方法被捕获的有界性和衰减假设适当的时空规范的Ricci系数和时空曲率。鉴于这样的假设在水平上的$k$衍生物的Ricci系数（因此$k-1$衍生物的曲率），我们证明有界性和衰减估计的$k$衍生物的\emph{curvature}。这个证明采用了向量场乘子和贝尔-罗宾逊张量的复乘子的框架，这是由Christodoulou-Klainerman在闵可夫斯基空间稳定性的背景下开创的。我们提供乘数模拟捕捉必不可少的衰变机制（已确定以前的标量波方程黑洞背景）的Bianchi方程。特别地，得到了视界附近红移效应的公式。此外，我们确定了一定的比安奇方程的层次，这导致强$r$加权时空曲率范数无穷大附近的控制。这允许避免使用经典的共形Morawetz乘子$K$，因此推广Dafermos和Rodnianski最近的工作的波动方程的上下文中。最后，证明需要详细了解内部误差项的结构。这是特别复杂的，因为这两个现象的陷阱轨道和事实，不像在闵可夫斯基空间的稳定性，不是所有的曲率分量衰减到零。

英文摘要：

In this paper, we introduce a class of spacetimes $\left(\mathcal{M},g\right)$ which satisfy the vacuum Einstein equations and dynamically approach a Schwarzschild solution of mass $M$, a class we shall call \emph{ultimately Schwarzschildean spacetimes}. The approach is captured in terms of boundedness and decay assumptions on appropriate spacetime-norms of the Ricci-coefficients and spacetime curvature. Given such assumptions at the level of $k$ derivatives of the Ricci-coefficients (and hence $k-1$ derivatives of curvature), we prove boundedness and decay estimates for $k$ derivatives of \emph{curvature}. The proof employs the framework of vectorfield multipliers and commutators for the Bel-Robinson tensor, pioneered by Christodoulou-Klainerman in the context of the stability of the Minkowski space. We provide multiplier analogues capturing the essential decay mechanisms (which have been identified previously for the scalar wave equation on black hole backgrounds) for the Bianchi equations. In particular, a formulation of the redshift-effect near the horizon is obtained. Morever, we identify a certain hierarchy in the Bianchi equations, which leads to the control of strongly $r$-weighted spacetime curvature-norms near infinity. This allows to avoid the use the classical conformal Morawetz multiplier $K$, therby generalizing recent work of Dafermos and Rodnianski in the context of the wave equation. Finally, the proof requires a detailed understanding of the structure of the error-terms in the interior. This is particularly intricate in view of both the phenomenon of trapped orbits and the fact that, unlike in the stability of Minkowski space, not all curvature components decay to zero.

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