具有正宇宙学常数的Linet-Tian度规的线性微扰
Linear perturbations of the Linet-Tian metrics with a positive cosmological constant
摘要：
Linet-Tian度规是爱因斯坦方程的解，它的宇宙学常数为$Λ$，可以是正的，也可以是负的。在$Λ<0$的情况下，这些度量的线性不稳定性已经建立。在$Λ>0$的情况下，在最近的分析中发现，扰动方程允许不稳定的模式。分析是基于建设的规范不变函数的度量扰动系数，这里称为$W(y)$。该函数满足线性二阶方程，该方程可用于建立边界值问题，以确定扰动的允许频率、实频率或纯虚频率。然而，这些解与全谱扰动的关系，以及与任意扰动的演化的关系，仍然是开放的。本文再次考虑Linet-Tian度规对$Λ>0$的扰动，并利用达布变换的一种形式，证明了可以把扰动方程与一个自伴问题联系起来，它给出了扰动的完备性和谱的解。这也被用来构造$W(y)$的规范不变方程的解与任意初始数据的演化之间的显式关系，从而解决了在以前的研究中仍然开放的问题。然后用数值方法证实了不稳定模的存在作为扰动的完整谱的一部分，从而建立了线性引力不稳定性与$Λ>0$的Linet-Tian度规。
英文摘要：
The Linet-Tian metrics are solutions of the Einstein equations with a cosmological constant, $Λ$, that can be positive or negative. The linear instability of these metrics in the case $Λ<0$, has already been established. In the case $Λ>0$, it was found in a recent analysis that the perturbation equations admit unstable modes. The analysis was based on the construction of a gauge invariant function of the metric perturbation coefficients, called here $W(y)$. This function satisfied a linear second order equation that could be used to set up a boundary value problem determining the allowed, real or purely imaginary frequencies for the perturbations. Nevertheless, the relation of these solutions to the full spectrum of perturbations, and, therefore, to the evolution of arbitrary perturbations, remained open. In this paper we consider again the perturbations of the Linet-Tian metric with $Λ>0$, and show, using a form of the Darboux transformation, that one can associate with the perturbation equations a self adjoint problem that provides a solution to the completeness and spectrum of the perturbations. This is also used to construct the explicit relation between the solutions of the gauge invariant equation for $W(y)$, and the evolution of arbitrary initial data, thus solving the problem that remained open in the previous study. Numerical methods are then used to confirm the existence of unstable modes as a part of the complete spectrum of the perturbations, thus establishing the linear gravitational instability of the Linet-Tian metrics with $Λ>0$.
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