The deep reason why the equations describing axial and polar perturbations of Schwarzschild black holes have the same spectrum is far from trivial. In this article, we revisit the original proof and try to make it clearer. Still focusing on uncharged and non-rotating black holes, we extend the results to spacetimes including a cosmological constant, which have so far mostly been investigated numerically from this perspective.

中文翻译：

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Schwarzschild-de Sitter 和 Schwarzschild-Anti de Sitter 时空准正态模式等谱性的分析证明

描述 Schwarzschild 黑洞轴向和极地扰动的方程具有相同光谱的深层原因远非微不足道。在本文中，我们重新审视原始证明并尝试使其更清晰。仍然专注于不带电和不旋转的黑洞，我们将结果扩展到时空，包括宇宙常数，到目前为止，大多数人都从这个角度进行了数值研究。