Near-horizon aspects of black holes in quantum spacetime

TL;DR

This paper explores the effects of noncommutative quantum spacetime on Schwarzschild black hole perturbations, analyzing near-horizon behavior and mode differences to understand quantum-induced deviations from classical black hole spectra.

Contribution

It introduces a formalism for gravitational perturbations in noncommutative spacetime and examines the near-horizon behavior of polar and axial modes, highlighting quantum effects on black hole spectra.

Findings

Differences between polar and axial mode potentials near the horizon.

Asymptotic solutions for Schrödinger-type equations at the horizon.

Insights into polar-axial isospectrality violation due to quantum spacetime.

Abstract

We give a short introduction to the formalism of noncommutative (twisted) differential geometry that is used to derive the equations of motion for the gravitational perturbation of the Schwarzschild black hole in quantized spacetime. Special attention is given to quantum spacetime arising from $r - \varphi$ noncommutativity. Tortoise coordinate and near-horizon regions of the effective potentials are analyzed for both polar and axial modes. By carefully examining the associated Schr\"odinger-type equations, we provide the asymptotic solutions at the horizon and illustrate some differences between the polar and axial modes. These findings give further insight into the polar-axial isospectrality violation in the presence of the quantum structure of spacetime.