Schwarzschild-德西特时空的极限几何与谱不稳定性
Limiting geometry and spectral instability in Schwarzschild--de Sitter spacetimes
AI总结：
研究Schwarzschild-德西特时空的极限几何与谱不稳定性，通过统一的基础设施分析quasi-normal模式的极限行为及稳定性特征。
AI中文摘要：
我们重新审视Schwarzschild-德西特时空中的 quasi-normal 模式（QNM）问题，提供了一个统一的基础设施，专门用于研究极限配置。从几何上讲，我们采用双曲面框架，明确实施Geroch的严格极限程序，用于研究时空家族。这使得能够受控地过渡到Schwarzschild、德西特和Nariai几何。从数值上讲，我们将分析网格细化技术引入quasi-normal模式计算中，成功地在适当的极限情况下恢复了已知的quasi-normal模式家族：复光环模式和纯虚德西特模式。我们将这些结果解释为谱不稳定性，其中稳定和不稳定模式的概念取决于所考虑的具体时空极限。在Schwarzschild极限下，德西特模式作为连续分支切口在ω=0处的不稳定性效应出现。相反，分支切口可以理解为在过渡区域中无限积累的离散模式在ω=0处的出现。我们提出了一种启发式的QNM密度度量来表征这种积累，并强调了对潜在分支切口不稳定性更严谨研究的必要性——这在晚期引力波信号的背景下尤其相关。所提出的基础设施为更复杂的时空研究提供了一个通用且可扩展的框架，如Reissner-Nordström-德西特或Kerr-Newman-德西特时空。
英文摘要：
We revisit the quasinormal mode (QNM) problem in Schwarzschild--de Sitter spacetimes providing a unified infrastructure tailored for studying limiting configurations. Geometrically, we employ the hyperboloidal framework to explicitly implement Geroch's rigorous limiting procedures for families of spacetimes. This enables a controlled transition between Schwarzschild, de Sitter, and Nariai geometries. Numerically, we introduce the analytical mesh refinement technique into quasinormal mode calculations, successfully recovering -- within the appropriate limiting scenarios -- both known families of quasinormal modes: complex light ring modes and purely imaginary de Sitter modes. We interpret these results in terms of spectral instability, where the notions of stable and unstable modes depends on the specific spacetime limit under consideration. In the Schwarzschild limit, de Sitter modes appear as a destabilizing effect on the continuous branch cut at $ω= 0$. Conversely, the branch cut can be understood as emerging from an infinite accumulation of discrete modes at $ω= 0$ in the transitional regime. We propose a heuristic measure of QNM density to characterize this accumulation and highlight the need for a more rigorous study of potential branch cut instabilities -- especially relevant in the context of late-time gravitational wave signals. The proposed infrastructure provides a general and extensible framework for investigations in more complex spacetimes, such as Reissner--Nordström--de Sitter or Kerr--Newman--de Sitter.
