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 (Gleiser 2017 Class. Quantum Grav. 34 065010). In the case Λ > 0, it was found in a recent analysis that the perturbation equations admit unstable modes (Gleiser 2018 arXiv:1810.07296v2[gr-qc]). 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 adjointproblem 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.

Linet-Tian 度量是爱因斯坦方程的解，其宇宙常数 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确定（Gleiser 2017 Class. Quantum Grav . 34 065010）。在 Λ > 0 的情况下，最近的一项分析发现扰动方程允许不稳定模式（Gleiser 2018 arXiv:1810.07296v2[gr-qc]）。该分析基于度量扰动系数的规范不变函数的构建，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，确定扰动的允许、实数或纯虚数频率。然而，这些解决方案与全谱扰动的关系，因此，与任意扰动的演变，仍然是开放的。在本文中，我们再次考虑 Λ > 0 的 Linet-Tian 度量的扰动，并使用 Darboux 变换的一种形式表明，可以将扰动方程与自伴随问题联系起来，该问题提供了完备性和谱的解的扰动。这也用于构造W ( y)，以及任意初始数据的演化，从而解决了之前研究中仍然存在的问题。然后使用数值方法来确认不稳定模式的存在，作为完整扰动谱的一部分，从而建立 Λ > 0 的 Linet-Tian 度量的线性重力不稳定性。