The initial value problem for linearized gravitational perturbations of the Schwarzschild naked singularity
The initial value problem for linearized gravitational perturbations of the Schwarzschild naked singularity
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DOI：
10.1088/0264-9381/26/21/215002
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发表时间：
2009-11-07
影响因子：
3.5
通讯作者：
Gleiser, Reinaldo J.
中科院分区：
文献类型：
--
作者：
Dotti, Gustavo;Gleiser, Reinaldo J.
围绕史瓦西时空的线性化爱因斯坦方程的标量模式的耦合方程被 Zerilli 简化为 (1+1) 波动方程偏导数(2)Psi(z)/偏导数 t(2) + H Psi(z) = 0，其中 H = 偏导数(2)/偏导数 x(2) + V (x) 是 Zerilli 的“哈密尔顿”，x 是乌龟径向坐标。根据其定义，对于平滑扰动，场 Psi(z) 在 r(s) = -6M/(l-1)(l+2) 处是奇异的，其中 l 是模谐波数。方程 Psi(z) 遵循的也是奇异的，因为 V 在 rs 处有一个二阶极点。这与黑洞外部稳定性问题无关，其中 r > 2M > 0，且 r(s) < 0，但它在裸奇点情况下引入了一个非平凡问题，其中 M < 0，则 rs > 0，并且奇点出现在 r 的相关范围内（0 < r < 无穷大）。我们通过开发一种新的偶模态演化方法来解决这个问题，该方法基于新的规范不变函数 (Psi) over cap，它是任何 M 值的度量扰动的正则函数。(Psi) over cap 与 Psi(z) 的关系由交织算子提供。 (Psi) over cap 和 Psi(z) 服从的 (1+1) 波动方程的空间部分作为一对超对称量子哈密顿量 H 和 (H) over cap 相关。对于 M < 0，在由 r = 0 处的物理激励边界条件定义的域 D 中具有正则势和唯一的自伴扩张。这使我们能够解决这种非全局双曲背景中的引力扰动的演化问题。该公式用于完成史瓦西裸奇点线性不稳定性的证明，通过表明先前发现的不稳定模式属于 D 中 (H) 上限的完整基础，因此可由通用初始数据激发。通过针对适当选择的初始数据对线性化方程进行数值求解来进一步说明这一点。
The coupled equations for the scalar modes of the linearized Einstein equations around Schwarzschild's spacetime were reduced by Zerilli to a (1+1) wave equation partial derivative(2)Psi(z)/partial derivative t(2) + H Psi(z) = 0, where H = partial derivative(2)/partial derivative x(2) + V (x) is the Zerilli 'Hamiltonian' and x is the tortoise radial coordinate. From its definition, for smoothmetric perturbations the field Psi(z) is singular at r(s) = -6M/(l-1)(l+2), with l being the mode harmonic number. The equation Psi(z) obeys is also singular, since V has a second- order pole at rs. This is irrelevant to the black hole exterior stability problem, where r > 2M > 0, and r(s) < 0, but it introduces a non- trivial problem in the naked singular case where M < 0, then rs > 0, and the singularity appears in the relevant range of r (0 < r < infinity). We solve this problem by developing a new approach to the evolution of the even mode, based on a new gauge invariant function, (Psi) over cap, that is a regular function of the metric perturbation for any value of M. The relation of (Psi) over cap to Psi(z) is provided by an intertwiner operator. The spatial pieces of the (1+1) wave equations that (Psi) over cap and Psi(z) obey are related as a supersymmetric pair of quantum Hamiltonians H and (H) over cap. For M < 0, has a regular potential and a unique self-adjoint extension in a domain D defined by a physically motivated boundary condition at r = 0. This allows us to address the issue of evolution of gravitational perturbations in this non- globally hyperbolic background. This formulation is used to complete the proof of the linear instability of the Schwarzschild naked singularity, by showing that a previously found unstable mode belongs to a complete basis of (H) over cap in D, and thus is excitable by generic initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data.