Kerr Black Hole Ringdown in Effective Field Theory

Abstract

We develop a systematic effective field theory calculation of the quasinormal modes of Kerr black holes valid for arbitrary spin, providing model-independent corrections to their ringdown spectrum directly relevant for gravitational-wave observations. Close to extremality, the effective field theory corrections in the grand-canonical ensemble exhibit an oscillatory dependence on $\log τ_{H}$, with $τ_H \equiv T_H/Ω_H$ a dimensionless measure of the black hole temperature. This behaviour signals an underlying discrete-scale-invariant structure.

Figures (6)

• Figure 1: $M_{\rm K}\,{\rm Re}\,\delta\omega^{(1)\;+}_{220}$ (left) and $M_{\rm K}\,{\rm Im}\,\delta\omega^{(1)\;+}_{220}$ (right) as functions of $j/j_{\rm ext}$. Black squares show our numerical results; grey discs reproduce the data of Cano:2024ezp, publicly available in the https://github.com/pacmn91/BeyondKerrQNM/tree/main?tab=readme-ov-file. The vertical dashed line at $j/j_{\rm ext}=\sqrt{2\sqrt{3}-3}$ indicates the critical spin where the small‑spin expansion breaks down.
• Figure 2: Real (black disks) and imaginary (grey squares) parts of the QNM EFT shifts in the grand‑canonical ensemble versus $\tau_H \equiv T_H/\Omega_H$ (log–log scale). The curves develop logarithmic oscillations toward $\tau_H\to0$, with the real‑part "echoing period" roughly twice that of the imaginary part.
• Figure 3: Maximum constraint violation $\chi_{\infty}$ as a function of the number of collocation points $N$ for the EFT deformation proportional to $d_1$, evaluated at the near‑extremal spin $j/j_{\rm ext}=0.998686$. The clear exponential decay demonstrates the expected spectral convergence of the Chebyshev–Gauss–Lobatto scheme.
• Figure 4: Convergence diagnostic $\widehat{\Delta}_N$ for the QNM frequency shift $\delta\omega_{220}$ in the $d_1$ deformation at near‑extremal spin $j/j_{\rm ext}=0.998686$. The exponential decay with $N$ reflects the spectral accuracy of the Chebyshev–Gauss–Lobatto scheme.
• Figure 5: $M_{\rm K}\,\mathrm{Re}\,\delta\omega^{(k)\pm}_{220}$ (black disks) and $M_{\rm K}\,\mathrm{Im}\,\delta\omega^{(k)\pm}_{220}$ (grey squares) as functions of $j/j_{\rm ext}$. The different sectors are indicated on the right of each plot. The data are shown in the microcanonical ensemble, where the EFT black holes and the Kerr black holes are compared at fixed energy and angular momentum.