Excitation region of Kerr black hole quasinormal modes from Stokes geometry

Technical Summary: Excitation Region of Kerr Black Hole Quasinormal Modes from Stokes Geometry

Problem Statement
Black hole (BH) spectroscopy relies on modeling the ringdown waveform of perturbed BHs as a superposition of quasinormal modes (QNMs). However, a longstanding issue in this framework is the "starting-time problem": it remains unclear from what time or spatial location the QNM description becomes valid for merger waveforms. Specifically, the physical location where QNMs are excited—the "QNM excitation radius"—has not been definitively identified in Kerr spacetime. While previous studies in the Schwarzschild limit identified a characteristic excitation location near $r \approx 2.557M$ (distinct from the light ring at $r=3M$), the behavior in rotating (Kerr) spacetime, particularly regarding spin dependence and the near-extremal limit, requires further investigation.

Methodology
The author employs two complementary approaches to investigate the QNM excitation radius in Kerr spacetime (mass $M$, dimensionless spin $\chi$):

1. Analytical Approach (Stokes Geometry):

   • The study utilizes the Stokes phenomenon within the Teukolsky equation for gravitational perturbations ($s=-2$).
   • In the high-overtone limit ($\omega_I \to -\infty$), the radial equation is approximated, and the behavior of WKB solutions is analyzed.
   • The excitation radius ($r_{\text{Stokes}}$) is identified as the intersection of an anti-Stokes line with the real-$r$ axis. At this intersection, the dominance between incoming and outgoing WKB modes switches, suggesting this is the site where QNMs are effectively excited and begin to propagate toward the horizon and spatial infinity.
   • The analysis is extended to the near-extremal limit ($\chi \to 1$) specifically for zero-damping modes (ZDMs), where the high-damping approximation fails.

2. Numerical Approach (Convergence Test):

   • The author performs a numerical convergence test of the QNM-plus-tail expansion using the Mano-Suzuki-Takasugi (MST) method.
   • By reconstructing the ringdown waveform ($\Psi_{\text{RD}}$) from a localized source and comparing it to the full waveform ($\Psi$), the study determines the convergence radius ($r_{\text{con}}$). This is defined as the location where the expansion ceases to converge within a specified error threshold.
   • The comparison between $r_{\text{Stokes}}$ and $r_{\text{con}}$ serves as a validation of the Stokes geometry interpretation.

Key Contributions and Results

• Identification of Excitation Radius via Stokes Geometry: The paper proposes that the QNM excitation radius is determined by the anti-Stokes line intersection. In the Schwarzschild limit ($\chi=0$), this yields $r_{\text{Stokes}}/M = 2.5569291$, reproducing the known value derived from the root of the tortoise coordinate ($x_{\text{Sch}}=0$) and confirming it is distinct from the light ring ($r=3M$).
• Spin Dependence in High-Overtone Limit: For Kerr black holes, the excitation radius $r_{\text{Stokes}}$ decreases as the spin $\chi$ increases. For $\chi \lesssim 0.4$, the radius lies below the prograde light ring; for $\chi \gtrsim 0.4$, it lies between the retrograde and prograde light rings.
• Independence from Angular Mode ($\ell$): In the high-overtone limit, the analytical derivation shows that $r_{\text{Stokes}}$ is independent of the angular mode $\ell$. Numerical results for $\ell=2, 3, 4$ in the Schwarzschild case confirm this, showing a stable convergence radius of $r_{\text{con}} \approx 2.55M$.
• Near-Extremal Regime and Zero-Damping Modes:
  • In the near-extremal limit, the high-overtone approximation ($\omega_I \to -\infty$) fails to match the numerical convergence radius ($r_{\text{con}}$), as the ringdown is dominated by long-lived zero-damping modes (ZDMs) rather than highly damped modes.
  • By applying Stokes geometry to the ZDMs ($\omega \to m\Omega_H$) in the extremal limit, the author identifies an innermost excitation radius of $r_{\text{Stokes}}/M \approx 1.105$. This value aligns with the numerical $r_{\text{con}}$, which also approaches the horizon in this limit.
• Overtone Dependence: The discrepancy between $r_{\text{Stokes}}$ (derived from asymptotic limits) and $r_{\text{con}}$ in intermediate and rapid spin regimes suggests that the excitation radius is not universal but depends on the QNM overtone number. Different frequencies generate different Stokes geometries and anti-Stokes lines.

Significance and Claims
The paper argues that the Stokes geometry provides a powerful analytical tool for identifying the physical origin of QNM excitation, offering a resolution to the "starting-time problem" by linking the onset of ringdown to a specific spatial radius.

• Physical Interpretation: The intersection of the anti-Stokes line with the real axis is interpreted as the location where QNMs are excited and subsequently dissipate toward the horizon and infinity.
• Bridge to Dynamics: The identification of a specific excitation radius allows for a more direct connection between the ringdown onset and the dynamics of the merger process, particularly in the extreme-mass-ratio limit where a companion object might pass through this radius.
• Modesty on Universality: The author explicitly notes that the excitation radius is generally overtone-dependent. The study does not claim a single universal excitation radius for all modes in all regimes but rather demonstrates that different overtones are excited at different radii, with the dominant mode (e.g., ZDMs in the extremal limit) dictating the effective convergence region.
• Limitations: The analytical results rely on approximations (high-damping or eikonal limits). The author acknowledges that a more precise, non-approximated numerical treatment of the complex potential could reveal further frequency dependence and refine the determination of the excitation radius.