Waveform stability of black hole ringdown with stochastic horizon structure

TL;DR

This work addresses whether black hole ringdown signals remain reliable when horizon-scale structure is stochastic rather than perfectly smooth. Using an effective-field framework, it models horizon fluctuations with a stochastic source within an Einstein-Langevin description, characterized by intensity $\epsilon$ and correlation length $L_c$, and analyzes both time-domain waveforms and frequency-domain phase shifts. The main finding is that phase averaging—via spatial integration of the wave equation—suppresses ultraviolet horizon details, yielding robust macroscopic waveforms even when the QNM spectrum is highly sensitive to small-scale perturbations; a resonance occurs near $L_c \sim M$ and the observed mismatch scales as $\mathcal{M} \propto \epsilon^2$, with a detectability threshold around $\epsilon_{\rm det} \gtrsim 2 \times 10^{-4}$. Consequently, incoherent, high-entropy quantum foam is observationally indistinguishable from classical backgrounds, and any significant ringdown deviation would indicate macroscopically coherent horizon structure (e.g., fuzzball geometries or exotic compact objects), with implications for the search of GW echoes and future extensions to rotation and microstate mapping.

Abstract

We examine the robustness of black hole ringdown to stochastic horizon-scale structure within an effective field framework. Consistent with the understanding that the spectral instability of quasinormal modes does not necessarily imply observational breakdown, our results demonstrate that the macroscopic gravitational waveform remains robust. We identify the phase averaging mechanism as the physical origin of this stability, demonstrating that the spatial integration of the wave equation efficiently attenuates ultraviolet geometric details below the resolution limit of the probing wavelength. Building on the scaling law $\mathcal{M} \propto ε^2$ and the characteristic mismatch profile with respect to $L_c$, we propose a geometric selection rule for observability: a detectable signal imposes a strict dual constraint requiring both macroscopic spatial coherence ($L_c \sim M$) and classical-level intensity ($ε\gtrsim 10^{-4}$). This criterion quantitatively rules out the observability of incoherent, high-entropy quantum foam, suggesting that any significant ringdown deviation would serve as definitive evidence for macroscopically coherent horizon structures.

Figures (6)

• Figure 1: Stochastic potential shattering on a Schwarzschild BH background with $\epsilon = 10^{-3}$ and $L_c = M$. The inset highlights the dense local extrema near the potential peak, where violet fluctuations (red line) break the smooth classical Regge-Wheeler barrier (dashed black line).
• Figure 2: Time-domain evolution and residual analysis under fluctuations.
• Figure 3: Frequency-domain scattering response for a Schwarzschild BH with $\epsilon = 10^{-3}$.
• Figure 4: Scaling of the mismatch $\mathcal{M}$ as a function of correlation length $L_c$. The "inverted U-shaped" profile characterizes the resonance window near $L_c \sim M$ and the linear suppression of observational deviations in the microscopic limit ($L_c \ll M$), quantifying the resolution limit of the horizon.
• Figure 5: Validation of the perturbative scaling law. The waveform mismatch $\mathcal{M}_t$ is plotted against the fluctuation intensity $\epsilon$ on a logarithmic scale. The red crosses represent numerical results obtained from our time-domain simulations at the resonance scale $L_c \sim M$. The dashed gray line indicates the theoretical prediction $\mathcal{M} \propto \epsilon^2$. The excellent agreement (slope $\simeq 2.0$) across two orders of magnitude confirms that the scattering process remains in the linear response regime even up to $\epsilon \sim 10^{-3}$, justifying our extrapolation strategy.