Hamiltonian formalism, master functions and Darboux transformations for perturbed (interiors and exteriors of) nonrotating black holes
Hamiltonian formalism, master functions and Darboux transformations for perturbed (interiors and exteriors of) nonrotating black holes
AI总结：
本文研究了非旋转黑洞内外扰动的Hamiltonian形式、主函数和Darboux变换，探讨了主函数与canonical变换之间的双射对应关系，并扩展了该对应关系到极向扰动的情况。
摘要：
受其在非旋转黑洞内部的相关性启发，经典和量子Kantowski-Sachs宇宙学近期引起了越来越多的关注。这种兴趣导致了轴向和极向扰动的Hamiltonian形式的发展，可以扩展到外部区域的应用。该形式还提供了背景物理自由度的描述。此外，它允许构造所有物理扰动的规范不变量，这些可以排列成与主函数相关的canonical对。在本文中，我们回顾了该Hamiltonian形式的基础，强调其基础和基本步骤，而不是涉及的计算细节。我们的讨论集中在经典和有效方面，尽管我们还简要评论了其在扰动黑洞量子化中的自然作用。采用该形式，我们提出了主函数对之间的Darboux变换的几何解释，将它们描述为保持扰动哈密顿结构的广义canonical变换，这些扰动被看作受某些势能约束的谐振子。这种canonical变换与Darboux变换之间的双射对应关系，最近已被证明适用于轴向扰动，现在扩展到极向扰动的情况。此外，我们还展示了存在某些canonical变换，它们类似于Darboux变换，混合轴向和极向主函数。
英文摘要：
Motivated by their relevance to the interior of nonrotating black holes, classical and quantum Kantowski-Sachs cosmologies have recently attracted increasing attention. This interest has led to the development of a Hamiltonian formalism for axial and polar perturbations, which can be extended to applications in the exterior region. The formalism provides also a description of the background physical degrees of freedom. Moreover, it allows for the construction of all physical perturbative gauge invariants, which can be arranged into canonical pairs associated with master functions. In this work, we review the basis of this Hamiltonian formalism, putting the emphasis on its foundations and fundamental steps rather than on details of the involved calculations. Our discussion focuses on classical and effective aspects, although we also briefly comment on its natural role in the quantization of perturbed black holes. Adopting this formalism we present a geometric interpretation of Darboux transformations between pairs of master functions, characterizing them as generalized canonical transformations that preserve the Hamiltonian structure of the perturbations as harmonic oscillators subject to certain potentials. This bijective correspondence between such canonical transformations and Darboux transformations, which was recently proved to hold for axial perturbations, is here extended to the case of polar perturbations. In addition, we demonstrate the existence of canonical transformations that, similarly to Darboux transformations, mix axial and polar master functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
