Quasi-Normal Modes and Nonlinear Electrodynamics in Black Hole Phase Transitions

TL;DR

The paper addresses whether thermodynamic phase transitions of charged black holes in $F(R)$-Euler-Heisenberg gravity are encoded in their dynamical QNMs. It computes QNMs for massless scalar perturbations using a pseudo-spectral method and analyzes thermodynamic quantities such as the heat capacity $C_Q$ across the nonlinear Euler-Heisenberg parameter $\lambda$, curvature $R_0$, and charge $Q$. A key finding is that the transition in $C_Q$ from a single to a double divergence occurs at a $\lambda$ value nearly coinciding with a structural transition in the QNM slope $K$, with a typical discrepancy of about a few percent, indicating a deep dynamical-thermodynamic link. The study also reveals that larger angular number $l$ obscures this correspondence while higher overtone number $n$ can restore it, suggesting that QNMs carry embedded thermodynamic information that could inform gravitational-wave observations and holographic studies of black-hole phase transitions.$

Abstract

We investigate the connection between thermodynamic phase transitions and quasi-normal modes (QNMs) in charged black holes with a positive curvature constant, within the framework of $F(R)$-Euler-Heisenberg gravity. Nonlinear electromagnetic fields lead to rich thermodynamic phase structures and significantly affect the QNMs of massless scalar fields. By analyzing the QNMs spectrum, we find that the transition point marking the disappearance of the divergence in the QNMs slope parameter $K$ aligns with the change of the thermodynamic phase structure described by the heat capacity, within the bounds of computational uncertainty. This precise matching holds under variations of the curvature parameter and charge. Furthermore, we show that larger angular quantum number $l$ diminishes this correspondence, while higher overtone number $n$ restores it beyond a threshold. These findings demonstrate that thermodynamic phase transitions of black holes carry embedded dynamical information, uncovering a fundamental link between black hole thermodynamic and dynamical properties.

Figures (13)

• Figure 1: The blackening factor $h(r)$ of $F(R)$-Euler-Heisenberg black holes with $R_0=1$, $q=0.5$, $\lambda=0.03$ and $f_{R_0}=0.1$ for different $r_{\text{h}}$. The red dashed line does not correspond to an actual black hole solution, while the black dashed line marks the event horizon radius $r_{\text{h}}$ of the black hole solution.
• Figure 2: The Hawking temperature $T$ as a function of the black hole event horizon $r_{\text{h}}$ with $R_0=1$, $q=0.5$, $\lambda=0.03$ and $f_{R_0}=0.1$. The two black dashed lines represent the Nariai limit and the extremal limit, respectively, where the Hawking temperature drops to zero $T = 0$.
• Figure 3: Three distinct profiles of the heat capacity $C_Q$ as a function of the rescaled horizon radius $\tilde{r}_{\text{h}}$ under variation of the Euler-Heisenberg parameter $\lambda$, in which (a) $\lambda=0.1$, (b) $\lambda=0.3$ and (c) $\lambda=0.5$, while the other parameters are fixed as $R_0=1$, $Q=0.5$, $f_{R_0}=0.1$.
• Figure 4: The real part $\omega_{\text{R}}$ and the imaginary part $\omega_{\text{I}}$ of the QNMs frequencies as a function of the rescaled horizon radius $\tilde{r}_{\text{h}}$ with $l=0$. The specific black hole parameter settings are (a) $\lambda=0.1$, (b) $\lambda=0.2$ and (c) $\lambda=0.3$, while the other parameters are fixed as $R_0=1,\ Q=0.5,\ f_{R_0}=0.1$.
• Figure 5: Three distinct profiles of the QNMs frequencies slope $K = \frac{\mathrm{d}\omega_{\text{I}}/\mathrm{d}\tilde{r}_{\text{h}}}{\mathrm{d}\omega_{\text{R}}/\mathrm{d}\tilde{r}_{\text{h}}}$ as a function of the rescaled horizon radius $\tilde{r_h}$ under variation of $\lambda$. Specifically, the panels show the full range $[0,1]$ of $\tilde{r}_{\text{h}}$ , while the small panel in (b) corresponds to the region $[0,0.1]$, where structural transitions occur. The detail parameter settings are $R_0=1,$$Q=0.5$, $f_{R_0}=0.1$, and (a) $\lambda=0.1$, (b) $\lambda=0.2$, (c) $\lambda=0.3$.