Universality of Quasinormal-Mode Shifts from Small Nonlocal Effective Couplings

TL;DR

This paper derives a universal scaling law for shifts in black hole quasinormal modes caused by small nonlocal modifications, providing a potential observational signature of quantum gravity effects in gravitational wave spectra.

Contribution

It introduces an analytic framework for calculating fractional QNM shifts due to nonlocal operators, revealing a universal behavior largely independent of black hole specifics.

Findings

Derived an analytic expression for QNM frequency shifts.

Discovered a universal scaling law for fractional shifts.

Applied the formalism to various black hole models showing universality.

Abstract

We investigate perturbative quasinormal-mode (QNM) shifts of black holes arising from fractional, nonlocal modifications to the wave operator. Starting from a scalar master equation corrected by a small fractional Laplacian term $(-\Delta)^{s}$ with $0<s<1$, we derive an analytic expression for the complex frequency shift at first order in the nonlocal coupling $\varepsilon$. Evaluation of the fractional operator in both coordinate and momentum representations reveals a universal scaling law $\delta\omega/\omega \propto \varepsilon/M^{2s}$, largely independent of the field spin, with an additional $\ell^{2s}$ enhancement in the eikonal regime $\ell \gg 1$. Applying the formalism to Schwarzschild, slowly rotating Kerr, Hayward regular, and LQG-corrected black holes, we demonstrate that the leading-order fractional QNM shift is universal, with geometric details entering only through overlap integrals of the mode functions. This universality provides a model-independent signature of nonlocality in strong-gravity ringdown spectra and offers a potential observational window into quantum-gravity-inspired modifications.