Master functions and hybrid quantization of perturbed nonrotating black hole interiors (2512.10692v2)

Abstract: Master functions of black holes are fundamental tools in gravitational-wave physics and are typically derived within a Lagrangian framework. Starting from the Kantowski-Sachs geometry, one can instead construct a perturbative Hamiltonian description for the interior region of an uncharged and nonrotating black hole. This approach provides a complementary perspective and enables a quantum treatment of the background geometry and its perturbations. In this work, we extend the application of this formulation to the exterior region and establish a correspondence between the perturbative invariants of the canonical approach and the master functions commonly used in black hole analyses. Once a consistent Hamiltonian description for their canonical counterparts is obtained, a hybrid quantization of the master functions follows naturally.

• The paper establishes a unified Hamiltonian formulation linking Kantowski-Sachs variables with classical CPM/ZM master functions for both interior and exterior black hole dynamics.
• The study implements a novel hybrid quantization approach that combines Loop Quantum Cosmology for the background with Fock quantization for perturbations, ensuring unitary evolution and singularity resolution.
• The work derives generalized master equations for the Regge-Wheeler and Zerilli potentials, paving the way for extracting quantum-corrected quasi-normal mode spectra and advancing black hole phenomenology.

Hybrid Quantization and Canonical Master Variables for Perturbed Nonrotating Black Hole Interiors

Introduction and Motivation

The study presents a Hamiltonian formulation for linear perturbations of uncharged, nonrotating black holes (BHs), rigorously connecting canonical perturbative gauge invariants with master functions used in classical black hole perturbation theory. The main technical contributions are twofold: (1) establishing a detailed correspondence between the canonical framework—employing Kantowski-Sachs (KS) variables—and the Cunningham-Price-Moncrief (CPM) and Zerilli-Moncrief (ZM) master functions; and (2) implementing a hybrid quantization scheme, leveraging results from Loop Quantum Cosmology (LQC), to combine background quantum gravity effects with standard Fock quantization for the perturbations.

Hamiltonian Framework and Extension to the Exterior

The analysis initiates from the KS geometry, capturing the spherically symmetric interior region of the black hole, formulated via Ashtekar-Barbero triad and connection variables. The KS background is encoded using two canonical pairs, $(b,p_b)$ and $(c,p_c)$, equipped with corresponding Hamiltonian constraints. The background metric, after implementing a complex canonical transformation ($b\to-ib$, $p_b\to ip_b$), extends naturally to describe the Schwarzschild exterior, with the canonical time evolution in the interior mapping to a radial evolution in the exterior. This approach unifies interior and exterior dynamics and is robust under modifications of the background, including semiclassical or effective quantum corrections.

Canonical Perturbation Theory and Mode Decomposition

Perturbations are parametrized through a decomposition using real Fourier and tensor harmonics on $S^2$, distinguishing between axial (odd) and polar (even) parity sectors. For each mode, the formulation yields six canonical perturbative pairs and four linear (gauge) constraints, with physical observables arising as gauge-invariant combinations. The framework handles background and perturbative variables without gauge fixing, facilitating symmetry analysis and mode mixing, and is well suited for subsequent quantization.

Construction and Identification of Master Functions

A sequence of canonical transformations abelianizes the perturbative gauge constraints and isolates two physical, gauge-invariant pairs: one axial and one polar. By further specializing these pairs (including background-dependent redefinitions permitted in the canonical structure), the configuration variables are shown to be direct generalizations of the master functions. For the Schwarzschild background, these reduce—up to normalization factors—to the CPM (axial) and ZM (polar) invariants which govern the standard Regge-Wheeler and Zerilli equations.

The Hamiltonian for the gauge-invariant sector assumes a diagonal (oscillator-like) form:

$$H = \sum_{\text{modes}} \frac{1}{2} \left[ P^2 + (\omega^2 - V) Q^2 \right]$$

where the potentials $V$ generalize the classical Regge-Wheeler and Zerilli potentials and are explicit functions of the dynamical background phase-space variables. The analysis reveals that physical master variables and their Darboux symmetry arise as canonical transformations within the perturbative Hamiltonian system.

Hybrid Quantization

The hybrid quantization method combines loop quantization (following LQC) for the KS background with Fock quantization for gauge-invariant perturbative modes. The LQC treatment uses holonomy and flux operators, resulting in superselection of triad eigenstates and the resolution of the central singularity via bounded, self-adjoint operators. The main features include:

• The background Hilbert space is constructed from tensor products of holonomy-flux algebras (for $b$ and $c$ sectors) and Schrödinger spaces for regularization parameters.
• The Fock space for perturbations is unique (up to unitary equivalence) due to the background isometry group and the requirement of unitary quantum dynamics.
• The total Hamiltonian constraint couples background and perturbations, with the Regge-Wheeler and Zerilli potentials promoted to quantum operators (including proper algebraic ordering and symmetrization).

This formalism is fully covariant with respect to the homogeneous background, supports dynamical backreaction up to second perturbative order, and provides explicit operator prescriptions for all geometric quantities, including inverse triads.

Strong Claims and Technical Features

• The construction yields, for the first time, explicit canonical master variables for the polar (even) sector that maintain full compatibility with the Hamiltonian phase-space structures and hybrid quantization. This closes the gap relative to the axial results and allows direct comparison of CPM/ZM approaches with canonical quantization.
• The interior-exterior unification via complex canonical transformation is robust and algorithmic, with no reliance on ad hoc coordinate choices.
• The generalized master equations hold for arbitrary (including quantum-corrected or effective) background geometries, not just classical Schwarzschild solutions.
• The approach ensures that the gauge-invariant sector's Fock quantization is unique modulo unitary maps, a nontrivial result given potential ambiguities in curved backgrounds and for polar perturbations.

Implications and Future Directions

Practically, this framework offers a path to jointly study quantum gravity effects (as realized by LQG/LQC) and gravitational wave phenomenology in extreme strong-field processes, such as black hole mergers. The explicit master variable framework lays a foundation for extracting quantum-corrected quasi-normal mode (QNM) spectra, potentially enabling the interpretation of future high-precision ringdown observations as quantum gravity probes.

On the theoretical side:

• The general Darboux symmetry is naturally recovered as a canonical invariance of the Hamiltonian structure, suggesting further applications in the analysis of master variables and parity-mixing canonical pairs.
• Extensions to rotating black holes and the canonical derivation of the Teukolsky equations (for both gravitational and matter perturbations) are natural sequels.

Additionally, the methodology enables rigorous analysis of quantum backreaction, including solutions for quantum-corrected background variables and their effect on observable perturbations.

Conclusion

The work provides a mathematically systematic treatment of linearized black hole perturbations in a fully canonical formalism, with explicit connections to conventional master functions and a robust hybrid quantization approach. The construction, applicable to both classical and quantum-corrected backgrounds, enables unified treatment of the interior and exterior, uniquely identifies gauge-invariant degrees of freedom, and sets a new standard for the interplay between quantum gravity phenomenology and gravitational wave physics. Future developments are expected to yield quantum-corrected master equations for QNMs and provide significant insight into quantum effects in strong gravity.