Master Variables and Darboux Symmetry for Axial Perturbations of the Exterior and Interior of Black Hole Spacetimes

TL;DR

This work develops a unified Hamiltonian framework for axial perturbations on Kantowski-Sachs backgrounds, linking the interior and exterior Schwarzschild geometries via complex canonical transformations. By isolating gauge-invariant axial degrees of freedom and diagonalizing the perturbative Hamiltonian, it reveals a Darboux covariance structure: an infinite family of master-variable Hamiltonians related by canonical Darboux transformations, with a physically meaningful subclass that yields the Regge-Wheeler and related master equations. The analysis unifies RW, CPM, and GS master functions within a single canonical scheme and provides explicit metric reconstruction formulas from master variables, highlighting how hidden symmetries emerge as canonical transformations. The results pave the way for applying this framework to beyond-GR backgrounds and for potential canonical quantization of BH perturbations, while clarifying the role of evolution parameter (timelike inside, radial outside) in a single geometric setup.

Abstract

Recent efforts have shown that Kantowski-Sachs spacetime provides a useful framework for analyzing perturbations inside a Schwarzschild black hole (BH). In these studies, the adoption of a Hamiltonian formulation offers an insightful perspective. The aim of this work is twofold. First, we revisit and elaborate the results obtained so far in Kantowski-Sachs, with the focus placed on axial perturbations. In particular, by exploiting the relation between this spacetime and the interior of a nonrotating BH, we consider the extension of those results to the exterior geometry of the BH. In this way, we clarify the relation between the axial perturbative gauge invariants emerging from the canonical analysis and the already well-established axial BH invariants, often referred to as master functions. We do so by providing a unified picture of the Hamiltonian formalism, which does not distinguish, formally, between exterior and interior geometries. The second objective is to explore the role of Darboux transformations, which were found as hidden symmetries in the context of BH perturbations, and their appearance in the Hamiltonian setting. Within this framework, the Hamiltonian formulation provides a clear geometric interpretation and characterization of Darboux transformations within the axial sector, viewing them as the set of canonical transformations between Hamiltonians for axial master functions.