Master variables and Darboux symmetry for axial perturbations of the exterior and interior of black hole spacetimes

TL;DR

This paper unifies the Hamiltonian analysis of axial perturbations in black hole spacetimes, clarifies the relation between gauge invariants and master functions, and explores Darboux transformations as canonical symmetries within this framework.

Contribution

It extends the Hamiltonian formalism for axial perturbations from black hole interiors to exteriors and interprets Darboux transformations as canonical transformations.

Findings

Unified Hamiltonian framework for interior and exterior black hole perturbations.

Clarified the relation between gauge invariants and master functions.

Identified Darboux transformations as canonical transformations in the Hamiltonian setting.

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.