Exceptional Points and Resonance in Black Hole Ringdown

Abstract

We propose an exceptional-point (EP) framework for black-hole ringdown beyond the standard quasinormal-mode (QNM) paradigm. It provides a first-principles characterization of the resonance associated with avoided crossings near EPs, an effect that conventional QNM analysis cannot fully capture. Employing a phenomenological environmental black-hole model with the hyperboloidal framework, we identify near-coalescence of both QNM eigenvalues and eigenfunctions, and directly demonstrate that the resonance produces enhanced mode contributions in the time domain, resulting in characteristic departures from exponentially damped oscillations. Our formulation further reveals that the EP frequency, given by the averaged value of the resonant modes, emerges as the physically relevant observable in the near-EP regime, and offers a robust foundation for modeling and extracting resonant ringdown signals.

Figures (3)

• Figure 1: Panels (a) and (b): Migration of the QNMs for parameter range $a/r_h\in[0,15]$ with $\epsilon=\epsilon_*(\simeq 0.00204 \,r_h^{-2})$. The fundamental mode $n=0$ (purple) and the overtones $n=1$--$3$ (green, blue, yellow) are displayed, with black markers denoting the initial Schwarzschild values. One observes overtaken transitions between overtones and the avoided crossing between the resonant $n=0$ and $n=1$ (originally $n=3$) modes --- see zoom in Panel (b) around $a=a_*(\simeq 11.5722\, r_h)$. Panel (c): Trajectories of the amplitudes of the individual tones of the time-domain signals, reconstructed from constant initial data and measured at future null infinity. The fundamental mode and the first overtone trace lemniscate (figure-eight-like) trajectories, attaining maxima at $a=a_*$. Panel (d):$L^2$-norm $\left\Vert \bar{\psi}_0(\sigma) - \bar{\psi}_n(\sigma) \right\Vert_2$ between fundamental and overtone QNM eigenfunctions. The fundamental and first overtone eigenfunctions nearly coincides at $a=a_*$, confirming an EP signature.
• Figure 2: Wave signal at future null infinity $\mathscr{I}^+$. Panel (a): Decomposition of the reconstructed resonant waveform that evolves from the constant initial data $(\Psi,\partial_\tau\Psi)|_{\tau=0}=(1,0)$. The fundamental mode and first overtone are significantly excited compared to the others. The inset highlights the fundamental mode and first overtone, showing they are nearly out of phase. Panel (b): Comparison among the full signal (solid black), the reconstructed fundamental mode (dashed blue) and first overtone (dashed red), and their superposition (solid orange). Panel (c): Comparison among the full signal (solid black), the reconstructed constant amplitude term (dashed navy) and linearly growing term (dashed dark magenta) in the EP model \ref{['eq:fittingfunc']} and their superposition (solid dark red).
• Figure 3: Extracted frequencies obtained using the 2QNM (purple and green dots), EP linear (blue dots), and EP total models (orange dots) for bump positions $a/r_h\in[11.54,11.6]$ with $\epsilon = \epsilon_*$. The curves denote the corresponding QNM trajectories for the fundamental mode (light purple), first overtone (light green), and their average (magenta), with thicker segments indicating the parameter range where the fitting is performed.