Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5.
Phys Rev Lett. 2012 Sep 14;109(11):111101. doi: 10.1103/PhysRevLett.109.111101.
Linear field perturbations of a black hole are described by the Green function of the wave equation that they obey. After Fourier decomposing the Green function, its two natural contributions are given by poles (quasinormal modes) and a largely unexplored branch cut in the complex frequency plane. We present new analytic methods for calculating the branch cut on a Schwarzschild black hole for arbitrary values of the frequency. The branch cut yields a power-law tail decay for late times in the response of a black hole to an initial perturbation. We determine explicitly the first three orders in the power-law and show that the branch cut also yields a new logarithmic behavior T(-2ℓ-5)lnT for late times. Before the tail sets in, the quasinormal modes dominate the black hole response. For electromagnetic perturbations, the quasinormal mode frequencies approach the branch cut at large overtone index n. We determine these frequencies up to n(-5/2) and, formally, to arbitrary order. Highly damped quasinormal modes are of particular interest in that they have been linked to quantum properties of black holes.
黑洞的线性场扰动可以用它们所遵循的波动方程的格林函数来描述。对格林函数进行傅里叶分解后，它的两个自然贡献分别由极点（准正则模式）和复频率平面上一个尚未被广泛探索的分支切割给出。我们提出了新的分析方法，可以针对任意频率的 Schwarzschild 黑洞计算分支切割。分支切割会导致黑洞对初始扰动的响应在后期呈现幂律衰减。我们明确地确定了幂律的前三个阶数，并表明分支切割也会导致后期的新对数行为 T(-2ℓ-5)lnT。在尾部出现之前，准正则模式主导着黑洞的响应。对于电磁扰动，准正则模式的频率在大过频率指数 n 时趋近于分支切割。我们确定了这些频率直到 n(-5/2)，并且在形式上可以任意阶展开。高度阻尼的准正则模式特别有趣，因为它们与黑洞的量子性质有关。
