Exceptional Points in Quasinormal Spectra of Hairy Black Holes

Abstract

Exceptional points (EPs) in quasinormal mode (QNM) spectra are non-Hermitian degeneracies at which both the eigenvalues and eigenfunctions coalesce. In this paper, we identify an EP in the scalar QNM spectrum of hairy black holes in the Einstein-Maxwell-scalar theory by scanning the parameter space. We then investigate its implications for ringdown signals by extracting QNMs from time-domain waveforms. Our results show that an EP ansatz, which includes a resonant contribution containing a term linear in time, provides a more robust description of ringdown at the EP than the standard ansatz based on a superposition of independent damped modes. In particular, it captures the resonant contribution associated with spectral coalescence more naturally, and even when the waveform is fitted with the standard ansatz, the resulting fit may still exhibit characteristic features of the EP ansatz.

Figures (8)

• Figure 1: QNM spectrum of the test scalar field at $\alpha=\alpha_{\mathrm{EP}}$ for $l=2$. Left: Trajectories of QNM frequencies in the complex $\omega$ plane as the BH charge $q$ is varied. The color bar indicates the value of $q$. The spectrum is organized into five branches, two of which meet at an EP at $q=q_{\mathrm{EP}}$. Near the EP, each of the two participating branches splits into an upper and a lower segment; the two upper (lower) segments recombine into the branch labeled by $n_{p}=0$ ($n_{o}=1$). Peak and off-peak modes are indexed by $n_{p}$ and $n_{o}$, respectively. Upper right: Squared real part of the QNM frequencies, $\omega_{R}^{2}$, as a function of $q$. The gray dotted line marks $q_{\mathrm{EP}}$, at which the real parts of the $n_{p}=0$ and $n_{o}=1$ branches cross. The panel also shows the effective potential $V_{\mathrm{eff}}$ for several representative values of $q$, with each curve aligned so that its highest peak corresponds to the selected $q$. As $q$ increases, $V_{\mathrm{eff}}$ transitions from a single-peaked to a double-peaked structure. In the single-peak regime, peak and off-peak modes are distinguished by the location of $\omega_{R}^{2}$: the former lie close to the peak value of $V_{\mathrm{eff}}$, whereas the latter take values appreciably below it. Lower right: Imaginary part of the QNM frequencies as a function of $q$. At $q=q_{\mathrm{EP}}$, the imaginary parts of the $n_{p}=0$ and $n_{o}=1$ modes coincide.
• Figure 2: Squared difference between the QNM frequencies $\omega_{0}^{p}$ and $\omega_{1}^{o}$ for the $n_{p}=0$ and $n_{o}=1$ modes (Left), and the $L^{2}$ norm of the difference between the corresponding eigenfunctions $\psi_{1}^{o}$ and $\psi_{0}^{p}$ (Right), as functions of the BH charge $q$. In both panels, the gray dotted line marks the EP location at $q=q_{\mathrm{EP}}$. As $q$ approaches $q_{\mathrm{EP}}$, the squared frequency difference decreases approximately linearly to an extremely small value, while the eigenfunction difference drops sharply, indicating the near coalescence of the two eigenfunctions. The simultaneous strong suppression of the eigenvalue and eigenfunction differences provides direct numerical evidence for the EP in the QNM spectrum.
• Figure 3: Time-domain ringdown waveform $\psi$ of the scalar field at $\alpha=\alpha_{\mathrm{EP}}$ and $q=q_{\mathrm{EP}}$, plotted as a function of $(t-t_{\mathrm{peak}})/M$. Here $t_{\mathrm{peak}}$ denotes the time at which the waveform amplitude reaches its maximum. The observer is located at $r=27.452M$.
• Figure 4: Extracted amplitudes $A_{n}$ (Left) and phases $\phi_{n}$ (Right) obtained with the standard ansatz, shown as functions of the start time $(t_{0}-t_{\mathrm{peak}})/M$. The fit includes the four least damped modes, namely the $n_{p}=0,1$ peak modes and the $n_{o}=0,1$ off-peak modes, with frequencies fixed by the frequency-domain analysis. The two EP modes, $n_{p}=0$ and $n_{o}=1$, have nearly equal amplitudes and dominate over the other fitted modes, while their phases are approximately opposite, indicating destructive interference between their contributions.
• Figure 5: Extracted amplitudes $A_{n}$ (Left) and phases $\phi_{n}$ (Right) obtained with the EP ansatz, shown as functions of the start time $(t_{0}-t_{\mathrm{peak}})/M$. The fit includes two nonresonant modes, $n_{o}=0$ and $n_{p}=1$, together with the EP contribution, with frequencies fixed by the frequency-domain analysis. The EP contribution is decomposed into a time-independent term (solid black) and a term linear in time (dashed black). Their amplitudes are substantially smaller than the separate amplitudes of the EP modes obtained with the standard ansatz in Fig. \ref{['fig:4']}, indicating that the EP ansatz provides a more economical description of the near-EP ringdown.