Stability of Higher-Dimensional Schwarzschild Black Holes
Stability of Higher-Dimensional Schwarzschild Black Holes
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DOI：
10.1143/ptp.110.901
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发表时间：
2003-05
影响因子：
--
通讯作者：
Akihiro Ishibashi;Hideo Kodama
中科院分区：
文献类型：
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作者：
Akihiro Ishibashi;Hideo Kodama
在最大对称黑洞引力扰动的规范不变形式的框架下，我们研究了高维Schwarzschild黑洞相对于线性扰动的经典稳定性。这种形式主义是最近开发的本作者。根据扰动在背景度规球面截面上的张量行为，将扰动分为张量、矢量和标量三种模式。矢量和标量型模式分别对应于四维情况下的轴向和极向模式。我们表明，对于每一种模式的扰动，主方程的空间导数部分是一个积极的，自伴运营商在L 2 -Hilbert空间，因此，主方程的每一种类型的扰动没有正规化的负模式，将对应于不稳定的解决方案。在相同的史瓦西背景下，我们还分析了标量模式的静态扰动，并表明不存在静态扰动，这是经常性的事件视界外的任何地方，并表现良好的空间无穷远。这证实了在扰动框架内高维球对称静态真空黑洞的唯一性。我们的处理稳定性问题的策略也适用于其他高维最大对称黑洞与非零宇宙学常数。我们证明了所有可能类型的最大对称黑洞（包括高维Schwarzschild-de Sitter黑洞和Schwarzschild-anti-de Sitter黑洞）对于张量和矢量扰动都是稳定的.
We investigate the classical stability of higher-dimensional Schwarzschild black holes with respect to linear perturbations in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes. This formalism was recently developed by the present authors. The perturbations are classified into three types, those of tensor, vector and scalar modes, according to their tensorial behaviour on the spherical section of the background metric. The vector- and scalar-type modes correspond, respectively, to the axial and polar modes in the four-dimensional case. We show that for each mode of the perturbations, the spatial derivative part of the master equation is a positive, self-adjoint operator in the L 2 -Hilbert space, and hence that the master equation for each type of perturbation has no normalisable negative modes that would correspond to unstable solutions. In the same Schwarzschild background, we also analyse static perturbations of the scalar mode and show that there exists no static perturbation that is regular everywhere outside the event horizon and is well behaved at the spatial infinity. This confirms the uniqueness of the higher-dimensional spherically symmetric, static vacuum black hole, within the perturbation framework. Our strategy for treating the stability problem is also applicable to other higherdimensional maximally symmetric black holes with a non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (including the higherdimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable with respect to tensor and vector perturbations.