Master functions and hybrid quantization of perturbed nonrotating black hole interiors

TL;DR

The paper develops a Hamiltonian treatment of linear perturbations of nonrotating black holes starting from Kantowski-Sachs interior geometry and extends it to the exterior region, establishing a link between perturbative invariants and standard BH master functions. Through a sequence of background-dependent canonical transformations, axial and polar perturbations are recast into master-function variables, reproducing the Regge-Wheeler and Zerilli structures in Schwarzschild and yielding generalized wave equations for the master variables. A complex exterior formulation, interpreted as a radial-evolution problem, unifies interior and exterior analyses and enables a robust hybrid quantization: the background is quantized via loop quantum cosmology while the perturbations are treated with a Fock quantization, leading to a consistent total Hamiltonian constraint with quantum corrections. The framework recovers the CPM and ZM master variables in Schwarzschild and offers a practical pathway to quantum gravity phenomenology for BH ringdown and QNMs while remaining adaptable to effective background modifications and potential extensions to rotating BHs.

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.