Effective Regge-Wheeler equations of a hybrid loop quantum black hole

TL;DR

The paper develops a hybrid loop quantum gravity description of the Schwarzschild black hole interior, introducing gauge-invariant gravitational perturbations on a Kantowski-Sachs background quantized via loop quantum cosmology. By imposing the quantum zero-mode Hamiltonian constraint on background–perturbation factorized (BP) states and employing a relational time associated with a continuous mass variable, it derives an effective Hamiltonian for the perturbations that yields Regge-Wheeler-like equations with quantum corrections. These corrections appear through background expectation values such as $\langle |\hat{p}_c| \rangle_\Gamma$ and $\langle \hat{\mathcal{V}}^{\rm ax}_l \rangle_\Gamma$, providing a concrete framework to study LQG effects on gravitational-wave propagation in black-hole interiors. The results set the stage for extensions to the exterior region, discrete-mass scenarios, and the quasinormal-mode spectrum, with potential observational implications for gravitational-wave signals.

Abstract

A set of effective equations for the gauge-invariant gravitational perturbations in the interior of a spherically symmetric, non-rotating black hole is derived within the framework of hybrid loop quantum cosmology. The quantum zero-mode of the Hamiltonian constraint, obtained from a perturbative gauge-invariant canonical analysis, is explicitly imposed on a class of quantum states whose wavefunctions factorize into a background and a perturbative part, related through a geometric relational variable. These states naturally describe regimes with small perturbative backreaction and lead to an effective Hamiltonian and associated dynamics for the perturbations. The resulting equations take the form of Regge-Wheeler equations modified by expectation values of the quantum black hole geometry, providing a clear characterization of quantum corrections to the classical description of the black hole interior. This framework opens the way to investigating hybrid loop quantum gravity effects in the propagation of gravitational waves.