Master Variables and Darboux Symmetry for Axial Perturbations of the Exterior and Interior of Black Hole Spacetimes (2512.10664v2)

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.

• The paper formalizes the extraction of master variables using a Hamiltonian approach, revealing Darboux symmetry as a canonical transformation.
• It employs ADM decomposition with triad-connection variables to unify the treatment of interior and exterior Schwarzschild regions.
• The work provides an explicit scheme for metric reconstruction and highlights the implications of isospectrality and gauge invariance in perturbation theory.

Master Variables and Darboux Symmetry for Axial Black Hole Spacetime Perturbations

Introduction and Motivation

This work provides a comprehensive Hamiltonian formulation for axial perturbations in spherically symmetric spacetimes, focusing on both the exterior and interior regions of Schwarzschild black holes. The analysis leverages the Kantowski-Sachs spacetime as a unifying framework, so that both the 'interior' (traditionally requiring analytic continuation or Wick rotation) and the 'exterior' regions can be treated with identical formal machinery via a complex canonical transformation on phase space variables. This generalizes and refines previous Lagrangian and canonical approaches, clarifying the symmetry structures and gauge-invariant formulation of axial perturbations for static (or analytically continued) spherically symmetric backgrounds.

The paper's dual objectives are to formalize the extraction of master variables in the Hamiltonian setting and to elucidate how the Darboux covariance symmetry—a previously identified hidden symmetry that yields infinitely many isospectral master equations—emerges naturally as a canonical transformation symmetry in this context. Furthermore, the results tie the canonical forms to well-known constructions such as the Regge-Wheeler (RW), Cunningham-Price-Moncrief (CPM), and Gerlach-Sengupta (GS) master functions.

Hamiltonian Framework for Perturbations

Starting from the 3+1 Arnowitt-Deser-Misner (ADM) decomposition, the analysis employs triad-connection variables instead of the conventional metric formulation. The phase space is described by the densitized triad and SU(2) connection variables, facilitating compatibility with gauge formalism and possible coupling to matter. The symmetry reduction to Kantowski-Sachs retains spatial homogeneity with anisotropy (i.e., a preferred direction), making it suitable to describe both the Schwarzschild interior and exterior via complexification of the canonical variables.

The constrained structure of general relativity is manifest: the Hamiltonian and momentum constraints remain at leading order. Perturbations are introduced at the action level, expanded and decomposed into axial (odd-parity) and polar (even-parity) components using scalar, vector, and tensor spherical harmonics. Focusing on the axial sector, the perturbative Hamiltonian is expressed in terms of mode coefficients for both configuration and conjugate momentum variables and their associated constraints.

Construction of Gauge-Invariant Variables

Gauge-invariant master variables are produced via systematic canonical transformations. For each multipole, conditions are imposed to ensure that the new configuration variable commutes with the (linearized) gauge constraints, thus rendering it physical. The symplectic structure is preserved under this transformation. The canonical variables are then further manipulated so that the Hamiltonian governing their dynamics is diagonal and the coefficient of the conjugate momentum is constant. These requirements are motivated not by aesthetics but by the aim to ensure clear dynamical interpretation, avoid nonlocality, and render quantization straightforward.

Perturbation dynamics thus reduce to a 1+1-dimensional wave type equation with an effective potential, exactly matching the form of the well-known Regge-Wheeler equation (and its Darboux-related variants). The formalism also subsumes the construction of the CPM and GS master functions, with explicit relations between their variables established via canonical and scale transformations.

Darboux Covariance as Canonical Symmetry

A central technical achievement is the realization that the Darboux covariance symmetry, which allows infinite "master variable + potential" pairs leading to spectrally equivalent wave equations, acquires a canonical interpretation: Darboux transformations act as canonical transformations relating the master function variables and their Hamiltonians. The subclass of physically meaningful transformations in general backgrounds is isolated by the requirement that the effective potential be frequency-independent, precluding nonlocal structure and yielding a tower of local, isospectral master equations.

The explicit construction of the Darboux canonical transformation demonstrates that for each axial sector, the infinite family of gauge-invariant Hamiltonian descriptions forms a symmetry class under Darboux-generated transformations. The structure is analogous to supersymmetric quantum mechanics, with associated Riccati equations relating partner potentials. The mapping between CPM, RW, and GS master functions is shown to be a sequence of canonical and conformal (scale) transformations, the latter inducing non-trivial contributions to the potential via the Schwarzian derivative, further linking the formalism to integrability structures in mathematical physics.

Metric Reconstruction

Importantly, the work provides an explicit inverse to the gauge-invariant transformation pipeline, yielding systematic 'metric reconstruction' formulas: given a solution to the master equation for the appropriate gauge-invariant variable, the physical metric perturbations (and momenta) can be reconstructed algorithmically via the chain of canonical inverses. This procedure is crucial for both the interpretation of physical observables and the eventual quantization of the theory.

Theoretical and Practical Implications

By unifying the treatment of the black hole exterior and interior and providing off-shell (i.e., background-independent) prescriptions for the dynamics of axial perturbations, the research has broad implications:

• General background applicability: The framework is not tied specifically to vacuum (Schwarzschild) solutions; it applies to generic spherically symmetric effective backgrounds, admitting extensions with quantum corrections, alternative gravity terms, or environmental effects as long as they preserve spherical symmetry.
• Isospectrality and symmetry structure: The canonical/Darboux symmetry elucidates why distinct master equations (with different master functions and potentials) yield identical spectra in Schwarzschild, but the formalism also clarifies that isospectrality may break down in more general backgrounds.
• Foundations for quantization: The diagonal and canonical structure of the Hamiltonian is favorable for canonical quantization schemes, including hybrid or polymer quantization strategies for quantum gravity scenarios.
• Algorithmic metric reconstruction: The machinery allows for explicit reconstruction of all metric perturbation fields from master function solutions, streamlining the calculation of observable quantities or initial data for numerical relativity and perturbation theory in modified gravity.

Conclusion

This work rigorously systematizes the Hamiltonian description of axial perturbations for the full Schwarzschild black hole spacetime and effective spherically symmetric backgrounds. It reveals the Darboux covariance symmetry as a canonical transformation symmetry, connects the master function formalism to the structure of the underlying constrained Hamiltonian system, and provides explicit transformations relating various classical master variables. The formalism is fully off-shell, gauge-invariant, and algorithmically implements metric reconstruction. The results pave the way for analogous treatment of the polar sector and for canonical quantization of black hole perturbations, with broad applicability to models beyond classical general relativity.

Reference: "Master Variables and Darboux Symmetry for Axial Perturbations of the Exterior and Interior of Black Hole Spacetimes" (2512.10664).