Quasinormal Modes of Massive Scalar Perturbations in Slow-Rotation Bumblebee Black Holes with Traceless Conformal Electrodynamics

TL;DR

This work analyzes electrically charged, slowly rotating black holes in Einstein–Bumblebee gravity coupled to traceless ModMax electrodynamics, introducing a quadratic bumblebee potential that fixes Lorentz violation via the parameter $\ell$ and a nonlinear ModMax deformation parameter $\gamma$. The authors derive static and first-order rotating solutions, revealing how $\ell$ rescales the radial metric component while $\gamma$ modulates the effective charge, yielding a Kerr–Newman–like geometry with controlled Lorentz-violating and nonlinear electrodynamics effects. They then study a massive scalar field perturbation, obtaining a Schrödinger-like radial equation with an effective potential $\mathcal{V}_l$ that incorporates $\ell$, $\gamma$, rotation, and charge, and compute QNMs using both a matrix method and Leaver’s continued fraction method. The resulting QNM spectra display coherent, monotonic responses to the four governing parameters, suggesting potential observational implications for ringdown signals in strong-field gravity tests of Lorentz symmetry breaking and nonlinear electrodynamics. Overall, the paper provides a transparent four-parameter framework to explore how Lorentz violation and nonlinear electrodynamics shape black hole spacetimes and their dynamical perturbations.

Abstract

We study electrically charged, slowly rotating black hole solutions in Einstein-Bumblebee gravity coupled to the traceless (conformal) ModMax nonlinear electrodynamics. By adopting a quadratic bumblebee potential that fixes the vacuum expectation value of the Lorentz-violating vector, we derive both the static configuration and its first-order rotating extension and demonstrate how the bumblebee parameter $\ell$ and the ModMax deformation $γ$ modify the horizon structure and the effective electric charge. We further investigate the dynamical properties of this spacetime by considering a massive scalar field perturbation. Using two independent numerical techniques, we compute the quasinormal mode (QNM) spectra and perform a comprehensive analysis of the influence of all relevant parameters, including the black hole spin, the Lorentz-violating coupling, the ModMax deformation, and the scalar field mass. Our results reveal coherent trends in the QNM frequencies, highlighting the interplay between Lorentz-symmetry breaking and nonlinear electrodynamics effects in black hole dynamics.

Figures (5)

• Figure 1: Variation of the real and imaginary parts of the fundamental QNM frequency of the massive scalar field with $l=m=2$ as the parameter $\tilde{Q}$ increases toward the extremal limit.
• Figure 2: The percentage error between CFM (of orders 10 and 20) and MM (of order 15) results is analyzed. Data points are sampled at intervals of 0.01 for the dimensionless parameters $\tilde{Q}$.
• Figure 3: We display the complex scalar frequencies of the $l=m=2, n=0$ modes as functions of the spin, Lorentz-violation parameter, and field mass, for two cases of the effective charge parameter: $\tilde{Q}=0$ (left) and $\tilde{Q}=0.3$ (right).
• Figure 4: We display the complex scalar frequencies of the $l=m=2, n=0$ modes as functions of the spin, Lorentz-violation parameter, and field mass, for two cases of the effective charge parameter: $\tilde{Q}=0.3$ (left) and $\tilde{Q}=0.6$ (right).
• Figure 5: We display the complex scalar frequencies of the $l=m=2, n=0$ modes as functions of the spin, Lorentz-violation parameter, and field mass, for two cases of the effective charge parameter: $\tilde{Q}=0.6$ (left) and $\tilde{Q}=0.9$ (right).