A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions
A Master Equation for Gravitational Perturbations of Maximally Symmetric Black Holes in Higher Dimensions
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DOI：
10.1143/ptp.110.701
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发表时间：
2003-05
影响因子：
--
通讯作者：
H. Kodama;A. Ishibashi
中科院分区：
文献类型：
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作者：
H. Kodama;A. Ishibashi
我们表明，在四维或四维以上的时空中，爱因斯坦方程的引力扰动的最大对称真空黑洞可以减少到一个单一的二阶波动方程在一个二维静态时空，不管模式的扰动。我们的出发点是规范不变的形式主义的扰动在任意数量的维度由本作者开发的，和变量的最终二阶主方程是由一个简单的组合规范不变的变量在这种形式主义。我们的公式适用于宇宙学常数Λ不为零和为零的情况。等势面的每个空间截面的截面曲率K的符号也保持一般性。在Λ = 0，K = 1的四维Schwarzschild背景中，标量扰动的主方程与极模的Zerilli方程相同，矢量扰动的主方程与轴模的Regge-Wheeler方程相同。此外，在Λ< 0，K =0，1的四维Schwarzschild-anti-de Sitter背景下，我们的方程与Cardoso和Lemos最近导出的方程一致.作为一个简单的应用，我们证明了四维非极值球对称黑洞对任意Λ的微扰稳定性和唯一性。我们还指出，在高维空间中，标量型扰动和矢量型扰动之间不存在简单的关系，这与四维空间不同。虽然在本文中，我们只处理的情况下，视界几何是最大对称的，最终的主方程是有效的，即使当视界几何是由一个通用的爱因斯坦流形描述，如果我们采用适当的曲率K和调和张量的特征值的重新解释。
We show that in four or more spacetime dimensions, the Einstein equations for gravitational perturbations of maximally symmetric vacuum black holes can be reduced to a single second-order wave equation in a two-dimensional static spacetime, irrespective of the mode of perturbations. Our starting point is the gauge-invariant formalism for perturbations in an arbitrary number of dimensions developed by the present authors, and the variable for the final second-order master equation is given by a simple combination of gauge-invariant variables in this formalism. Our formulation applies to the case of non-vanishing as well as vanishing cosmological constant Λ. The sign of the sectional curvature K of each spatial section of equipotential surfaces is also kept general. In the four-dimensional Schwarzschild background with Λ = 0 and K = 1, the master equation for a scalar perturbation is identical to the Zerilli equation for the polar mode and the master equation for a vector perturbation is identical to the Regge-Wheeler equation for the axial mode. Furthermore, in the four-dimensional Schwarzschild-anti-de Sitter background with Λ< 0 and K =0 , 1, our equation coincides with those recently derived by Cardoso and Lemos. As a simple application, we prove the perturbative stability and uniqueness of four-dimensional non-extremal spherically symmetric black holes for any Λ. We also point out that there exists no simple relation between scalar-type and vector-type perturbations in higher dimensions, unlike in four dimension. Although in the present paper we treat only the case in which the horizon geometry is maximally symmetric, the final master equations are valid even when the horizon geometry is described by a generic Einstein manifold, if we employ an appropriate reinterpretation of the curvature K and the eigenvalues for harmonic tensors.