Long-lived quasinormal modes in the Euler-Heisenberg electrodynamics

TL;DR

The paper investigates massive scalar perturbations in a charged black-hole background of Einstein–Euler–Heisenberg nonlinear electrodynamics, focusing on quasinormal modes (QNMs) and late-time behavior. It combines the Frobenius/Leaver method with time-domain integration, using Prony fits and WKB-Padé guesses to obtain QNM spectra for neutral and charged fields in this EEH background. A key finding is that as the field mass $\mu$ grows, the damping rate $\mathrm{Im}(\omega)$ can approach zero, producing quasi-resonances whose presence is often masked by oscillatory late-time tails that decay as a power law; increased field charge $q$ raises the oscillation frequency and reduces damping, improving the quality factor. The study also reveals that the late-time tails remain oscillatory with a power-law envelope (roughly $t^{-5/6}\sin(\mu t)$ in many cases), with possible subleading corrections and phase modulations when $q\neq0$, thereby enriching the understanding of EEH-induced perturbation dynamics and offering benchmarks for gravitational-wave and holographic contexts.

Abstract

Using the precise Leaver method and time-domain integration, we analyze the quasinormal modes and late-time behavior of massive neutral and charged scalar fields in the background of a charged, asymptotically flat black hole in the presence of Euler-Heisenberg nonlinear electrodynamics. We show that as the field mass increases, the damping rate decreases significantly, approaching arbitrarily long-lived states known as quasi-resonances. However, these modes cannot be identified in time-domain profiles due to the dominance of asymptotic tails in this regime, which decay slowly and exhibit oscillations with a power-law envelope. We observe that a larger field charge leads to a significantly higher quality factor, as it increases the oscillation frequency while reducing the damping rate.

Figures (7)

• Figure 1: Effective potential for scalar perturbations. Left panel: $\ell=0$, $r_{h}=1$, $Q=0.5$, $a=1$, $q=0$, $\mu=0$ (bottom), $\mu=0.3$ and $\mu=0.5$ (top). Right panel: $\ell=1$, $r_{h}=1$, $Q=0.5$, $a=0.5$, $q=0$, $\mu=0$ (bottom), $\mu=0.6$, $\mu=1$ (top).
• Figure 2: Imaginary part of $\omega$ (right panel), $n=0$ for $\ell=1$ scalar perturbations: $M=1$, $Q=0.5$, $a=0.5$, $q=-0.1$, $n=1$ (top), $q=0$, $n=0$ (middle) $q=0.1$, $n=0$ (bottom). Real part of $\omega$ for the same values of the parameters (left panel); $q=-0.1$ (bottom), $q=0$ (middle), $q=0.1$ (top). Notice that in the spectrum there are both positive and negative values of $\mathrm{Re}(\omega)$, which can be obtained by replacing $q \mapsto-q$, $\omega \mapsto - \omega^{*}$.
• Figure 3: Real (left panel) and imaginary (right panel) parts of $\omega$ for the first overtone $n=1$ with $\ell=1$ scalar perturbations: $M=1$, $Q=0.5$, $a=0.5$, and $q=0$.
• Figure 4: Real (left panel) and imaginary (right panel) parts of $\omega$ for the second overtone $n=2$ with $\ell=1$ scalar perturbations: $M=1$, $Q=0.5$, $a=0.5$, and $q=0$.
• Figure 5: The logarithmic time-domain profile for $\ell=1$ scalar perturbations: $r_{h}=1$, $Q=0.5$, $a=0.5$, $q=0$, $\mu=0$, (left), $\mu=1$, $r_{h}=1$ (middle) and $M=1$, $\mu=5$ (right). For the massless case on the left plot, the asymptotic late-time tail is $\sim t^{-5}$. The Prony fit gives $\omega = 0.553432 - 0.16884 i$, while the precise Leaver method gives $\omega = 0.553431 - 0.168843 i$. For the massive case $\mu=1$ and $\mu=5$, the long-lived QNM $\omega =0.766690 - 0.006224 i$ cannot be extracted because of the early dominance of the intermediate late-time tails. Here, we used the units $r_{h}=1$ for the first two plots to allow for the straightforward comparison with QNMs obtained by the Leaver method. The last plot is obtained in units $M=1$, which is convenient for comparison with the known asymptotic tails.