Hamiltonian formalism, master functions and Darboux transformations for perturbed (interiors and exteriors of) nonrotating black holes

TL;DR

This paper reviews a Hamiltonian formalism for black hole perturbations, highlighting its geometric interpretation, and extends the understanding of Darboux transformations and canonical transformations between master functions.

Contribution

It extends the Hamiltonian formalism to include polar perturbations and their Darboux transformations, providing a geometric interpretation and exploring axial-polar mixing.

Findings

Extended the bijective correspondence between Darboux and canonical transformations to polar perturbations.

Demonstrated the existence of canonical transformations mixing axial and polar master functions.

Provided a geometric interpretation of Darboux transformations as generalized canonical transformations.

Abstract

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.