Gravitational perturbations of nonlocal black holes

TL;DR

The paper investigates how nonlocal gravity in the revised Deser--Woodard framework modifies gravitational perturbations around a static, spherically symmetric black hole, focusing on a DD BH that represents a first-order deviation from Schwarzschild via a small parameter $0<\alpha\ll1$. It derives axial and polar perturbation formalisms, obtaining a modified Regge--Wheeler equation for axial modes and a coupled, nonlocal Zerilli-type problem for polar modes, with an auxiliary field $\tilde{\mathbb{U}}$ that satisfies its own wave equation. At zeroth order in $\alpha$, GR is recovered; at first order, nonlocal corrections appear in the potentials and introduce couplings that preclude a universal Schrödinger form for axial perturbations, while polar perturbations remain describable by a Schrödinger-like equation with source terms. These results provide the first perturbative characterization of nonlocal black holes and establish a framework to compute quasi-normal mode spectra and potential observational signatures in gravitational waves and black hole imaging, guiding future tests of nonlocal gravity in the strong-field regime.

Abstract

We derive the master equations governing axial and polar gravitational perturbations of a generic static and spherically symmetric black hole spacetime within the framework of the revised Deser--Woodard nonlocal gravity theory. We then apply our general formalism to a one-parameter family of black hole solutions recently obtained by the present authors, representing small first-order deviations from the Schwarzschild geometry. We provide well-motivated arguments that allow us to render the analysis analytically tractable. Our results provide the first complete perturbative characterization of nonlocal black holes and lay the groundwork for future investigations.