Quasinormal modes of a static black hole in nonlinear electrodynamics

TL;DR

This work computes the axial electromagnetic quasinormal modes of a static, AdS black hole in Plebański-type nonlinear electrodynamics by reformulating the perturbation problem as a linear generalized eigenvalue problem via an ingoing Eddington–Finkelstein (IEF) approach and a Chebyshev–Lobatto pseudospectral discretization. The authors show that nonlinear electrodynamics breaks electric–magnetic isospectrality and that increasing the nonlinearity parameter $\beta$ or the effective charge $Q$ raises both the oscillation frequency $\mathrm{Re}(\omega)$ and the damping rate $-\mathrm{Im}(\omega)$, leading to faster but more strongly damped ringdowns. In the magnetic sector, they find systematically smaller $\mathrm{Re}(\omega)$ and damping rates than in the electric sector, with the possibility of purely imaginary fundamental modes for large $\beta$ and small $Q_m$, indicating overdamped perturbations and absence of a near-horizon barrier. The results provide qualitative and quantitative signatures of NLED in black hole ringdown and establish a robust numerical framework for exploring QNMs in nonlinear electrodynamics backgrounds, with potential implications for strongly magnetized astrophysical environments.

Abstract

We investigate the axial electromagnetic quasinormal modes of a static, asymptotically Anti--de Sitter (AdS) black hole sourced by a nonlinear electrodynamics model of Plebański type. Starting from the master equation governing axial perturbations, we impose ingoing boundary conditions at the event horizon and normalizable (Dirichlet) behavior at the AdS boundary. Following the approach of Jansen, we recast the radial equation into a linear generalized eigenvalue problem by using an ingoing Eddington--Finkelstein formulation, compactifying the radial domain, and regularizing the asymptotic coefficients. The resulting problem is solved using a Chebyshev--Lobatto pseudospectral discretization. We compute the fundamental quasinormal mode frequencies for both the purely electric ($Q_m=0$) and purely magnetic ($Q_e=0$) sectors, emphasizing the role of the nonlinearity parameter $β$ and the effective charge magnitude $Q$. Our results show that increasing either $β$ or $Q$ raises both the oscillation frequency $ω_R$ and the damping rate $-ω_I$, leading to faster but more rapidly decaying ringdown profiles. Nonlinear electrodynamics breaks the isospectrality between electric and magnetic configurations: magnetic modes are systematically less oscillatory and more weakly damped than their electric counterparts. For sufficiently large $β$ and small $Q_m$, the fundamental mode becomes purely imaginary ($ω_R \approx 0$), in agreement with the absence of a trapping potential barrier in this regime. These findings reveal qualitative signatures of nonlinear electromagnetic effects on black hole perturbations and may have implications for high-field or high-charge astrophysical environments.

Figures (6)

• Figure 1: Radial profile of the lapse function for two representative values of the nonlinearity parameter $\beta$ and several values of $F_0$. The units on both axes are normalized by the mass parameter $M$.
• Figure 2: Radial behavior of the lapse function for (a) $\beta = 0.2$, where the extremal horizon formation occurs at $r_+ = 1.771$, and (b) $\beta = 0.01$, where increasing $F_0$ leads to the disappearance of horizons and the emergence of naked singularities. Units are normalized by $M$.
• Figure 3: Contours of the equation $f(r)=0$ in the $(F_0,\beta)$-plane. Panel (a) shows a broad parameter range illustrating the overall structure, while panel (b) highlights the restricted region in which extremal black holes may form.
• Figure 4: Panels (a-d): radial profiles of the electromagnetic potential for (a) fixed electric/magnetic charges with varying nonlinearity; (b) purely electric black holes with varying nonlinearity; (c) varying magnetic charge; and (d) fixed nonlinearity with different multipole index $\ell$. Panels (e-f): comparison between the electromagnetic potential $V_{\mathrm{em}}(r)$ and the scalar potential $V(r)$ for (e) fixed charges with varying nonlinearity and (f) fixed electric charge and nonlinearity with varying magnetic charge. Solid curves correspond to scalar perturbations and dashed curves to electromagnetic perturbations, with $\ell=2$ in all cases. The mass $M$ is used as the unit of length.
• Figure 5: Radial profiles of the effective potential for purely electric and purely magnetic configurations. Each row corresponds to a fixed nonlinearity parameter, with $\beta=0.30$ (a,b), $0.60$ (c,d), and $0.90$ (e,f). The left column shows purely electric cases ($Q_m=0$), while the right column presents purely magnetic ones ($Q_e=0$). Within each panel, the curves denote $Q\in\{0.3,0.6,0.9\}$ for $M=1$. The vertical marker indicates the horizon radius $r_+$, and the horizontal line marks $V_{\mathrm{em}}=0$. The spherical-harmonic index is fixed to $\ell=2$, and all length scales are normalized by the black hole mass $M$. In the electric sector (a,c,e), the potential develops a horizon-centered barrier followed by the characteristic NLED-induced growth at large $r$. In the magnetic sector (b,d,f), the NLED angular contribution suppresses the near-horizon barrier and may produce a shallow well; for $\beta=0.90$ and $Q_m=0.30$, the potential becomes monotonic, with neither barrier nor well.