Noncommutative quasinormal modes of Schwarzschild black hole

TL;DR

This paper investigates how noncommutative geometry affects Schwarzschild black hole perturbations, calculating quasinormal modes and revealing that spacetime quantization breaks classical isospectrality, with implications for quantum gravity models.

Contribution

It introduces noncommutative modifications to gravitational perturbations, derives new polar solutions, and computes quasinormal modes using advanced WKB methods with detailed error analysis.

Findings

Noncommutative geometry alters quasinormal mode spectra.

Classical isospectrality is broken by spacetime quantization.

Isospectrality is restored in the eikonal limit.

Abstract

We study gravitational perturbations of the Schwarzschild metric in the context of noncommutative gravity. $r-\varphi$ and $r-t$ noncommutativity are introduced through a Moyal twist of the Hopf algebra of diffeomorphisms. Differential geometric structures such as curvature tensors are also twisted. Noncommutative equations of motion are derived from the recently proposed NC vacuum Einstein equation. Here, in addition to previously calculated axial NC potential, we present the polar solution which generalizes the work done by Zerilli. Quasinormal mode frequencies of the two potentials are calculated using three methods: WKB, P\"oschl-Teller and Rosen-Morse. Notably, we apply the WKB method up to the 13th order and determine the optimal order for each noncommutative parameter value individually. Additionally, we provide comprehensive error estimations for the higher-order WKB calculations, offering insights into the accuracy of our results. By comparing the spectra, we conclude that the classical isospectrality of axial and polar modes is broken upon spacetime quantization. Isospectrality is restored in the eikonal limit.