Polar perturbations of dilaton-Euler-Heisenberg black holes

TL;DR

The paper investigates polar quasinormal modes of dilaton-Euler-Heisenberg black holes with dilaton hair in a two-coupling Einstein-Maxwell-dilaton framework. It derives the coupled metric-dilaton perturbation equations in the Regge-Wheeler gauge and computes the fundamental QNMs using direct integration and a matrix-valued continued fraction method, ensuring ingoing-horizon and outgoing-infinity boundary conditions. The results show all modes have negative imaginary parts, indicating stability, with distinct damping and oscillation patterns for $\varepsilon=1$ vs $\varepsilon=-1$, especially near extremality, and demonstrate strong agreement between the two numerical schemes. The findings offer potential observational signatures through gravitational waves and motivate future work on axial perturbations and electromagnetic perturbations in this nonlinear electrodynamics context.

Abstract

We investigate the quasinormal modes of polar metric-dilaton perturbations around the dilaton-Euler-Heisenberg (dEH) black holes with dilaton hair obtained from the Einstein-Maxwell-dilaton theory with two dilaton coupling constants ($α,β$) to the nonlinear Euler-Heisenberg term. We compute the quasinormal mode spectra by making use of two numerical techniques: direct integration and matrix values continued fraction methods. An excellent agreement is found between two approaches, confirming the robustness of our computation. We present the fundamental quasinormal frequencies for both gravitational and dilaton modes and analyze their dependence on the magnetic charge ($Q_m$), angular momentum quantum number $(l)$, and coupling parameter ($ε=α-β$). All negative imaginary quasinormal frequencies imply that the dEH black hole with dilaton hair is stable against polar metric-dilaton perturbations. Also, our results reveal distinct qualitative behaviors between $ε=1$ and $ε=-1 $, particularly in the damping rates near the extremality.

Figures (6)

• Figure 1: Graphs for the potential of the $l=0$ mode of the dilaton as function of $r_*$.
• Figure 2: Variation of fundamental ($n=0$) QNM frequencies (real and imaginary parts) with $Q_m$, for the $l=0$ mode dilaton around $M=1$ dEH black holes. Lines: solid ($\epsilon=1$), dashed ($\epsilon=-1$).
• Figure 3: Graphs for the potential of the $l=1$ mode of the dilaton as function of $r_*$.
• Figure 4: Variation of fundamental QNM frequencies (real and imaginary parts) with $Q_m$, for the $l=1$ mode dilaton around $M=1$ dEH black holes. Lines: solid ($\epsilon=1$), dashed ($\epsilon=-1$).
• Figure 5: Variation of real and imaginary parts for fundamental QNM frequencies with $Q_m$ for gravitational ($\Psi_g$: solid lines) and dilaton ($\Psi_d$: dashed lines) around $M=1$ bEH black holes at coupling $\epsilon=1$. Lines: black/red for gravitational mode with $l=2,3$ while blue/purple for dilaton mode with $l=2,3$.