We investigate quadratic algebraically special perturbations (ASPs) of the Schwarzschild black hole. Their dynamics are derived from the expansion up to second order in perturbation of the most general algebraically special twisting vacuum solution of general relativity. Following this strategy, we present analytical expressions for the axial-axial, polar-polar and polar-axial source terms entering in the dynamical equations. We show that these complicated inhomogeneous equations can be solved analytically and we present explicit expressions for the profiles of the quadratic ASPs. As expected, they exhibit exponential growth both at the past and future horizons even in the non-linear regime. We further use this result to analyze the quadratic zero modes and their interpretation in terms of quadratic corrections to mass and spin of the Schwarzschild black hole. The present work provides a direct extension beyond the linear regime of the original work by Couch and Newman.

中文翻译：

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史瓦西黑洞的二次扰动：代数特殊扇区


我们研究史瓦西黑洞的二次代数特殊扰动（ASP）。它们的动力学源自广义相对论最一般的代数特殊扭曲真空解的扰动扩展到二阶。按照这一策略，我们提出了进入动力学方程的轴-轴、极-极和极-轴源项的解析表达式。我们证明这些复杂的非齐次方程可以通过解析求解，并且我们提出了二次 ASP 轮廓的显式表达式。正如预期的那样，即使在非线性状态下，它们在过去和未来的视野中也表现出指数增长。我们进一步利用这一结果来分析二次零模式及其对史瓦西黑洞质量和自旋的二次修正的解释。目前的工作提供了超出库奇和纽曼原始工作的线性体系的直接延伸。