Quasinormal modes of Kerr-Newman black holes: revisiting the Dudley-Finley approximation

TL;DR

The Kerr–Newman quasinormal mode problem is studied within the Dudley–Finley decoupled perturbation framework, and its accuracy is assessed against the full gravitoelectromagnetic KN spectrum. The paper derives analytic and WKB criteria for zero-damped and damped mode coexistence near extremality, and establishes connections between the DF ZDMs/DMs and the NH–PS modes of the full KN solution. It also analyzes near-extremal spectra and highly damped large-n modes, providing a comprehensive picture of spectral structure in the DF approximation. The results clarify when the DF approach provides reliable guidance for KN perturbations and reveal the spectral organization near extremality and in the highly damped regime, with implications for gravitational-wave phenomenology and tests of gravity.

Abstract

We present a comprehensive study of the Kerr-Newman quasinormal mode spectrum in the Dudley-Finley approximation, where the linear gravitoelectromagnetic perturbations are decoupled by "freezing" either one of the fields to its background value. First, we reassess the accuracy of this approximation by comparing it to calculations that solve the coupled system of gravitoelectromagnetic perturbation equations across the subextremal spin-charge parameter space. We find that for the $(\ell,m,n) = (2,2,0)$, $(2,2,1)$, and $(3,3,0)$ modes, the agreement is typically within $10\%$ and $1\%$ for the real and imaginary parts of the frequencies, respectively. Next, we investigate the spectrum in the near-extremal limit, and study the family of long-lived ("zero-damped") gravitational modes. We find that the near-extremal parameter space consists of subregions containing either only zero-damped modes, or zero-damped modes alongside modes that retain nonzero damping. We derive analytic expressions for the boundaries between these regions. Moreover, we discuss the connection between the zero-damped and damped modes in the Dudley-Finley approximation and the "near-horizon/photon-sphere" modes of the full Kerr-Newman spectrum. Finally, we analyze the behavior of the quadrupolar gravitational modes with large overtone numbers $n$, and study their trajectories in the complex plane.

Figures (18)

• Figure 1: Logarithmic absolute error between the $(\ell, m, n) = (2, 2, 0)$ quasinormal frequencies calculated in the Kerr--Newman problem and with the Dudley--Finley equation for $s=-2$, for points on the line $a = q$ in the spin-charge parameter space. We show the errors in the real ("Re") and imaginary ("Im") parts of the frequencies. The dashed vertical line at $a = q = (2 \sqrt{2})^{-1}$ indicates the extremal limit.
• Figure 2: Logarithmic absolute error between the quasinormal frequencies calculated using the Bayesian fitting formula and the Dudley–Finley approximation for $s=-2$, for points on the line $a = q$ in spin-charge parameter space. We show the results for two sets of $(\ell, m, n)$ values: $(3,3,0)$ in the top panel and $(2,2,1)$ in the bottom panel. As in Fig. \ref{['fig:l2m2n0exactvsdf']}, we show the errors in the real ("Re") and imaginary ("Im") parts of the frequencies. The dashed vertical line at $a = q = (2 \sqrt{2})^{-1}$ indicates the extremal limit.
• Figure 3: Logarithmic absolute error between the quasinormal frequencies computed using the Bayesian fitting formula and the Dudley--Finley approximation for $s = -2$, for points within the quarter-circular region defined by $a^2 + q^2 \leq 1/4$. We show the results for $(\ell,m,n) = (2,2,0), (2,2,1)$, and $(3,3,0)$ as heat maps. The dashed line corresponds to the $a = q$ line that we studied in Figs. \ref{['fig:l2m2n0exactvsdf']} and \ref{['fig:knfit_vs_df_aeq']}. In the near-extremal region, the errors increase as we move from the high-spin to the high-charge limit. The Dudley--Finley approximation yields about $1\%$ errors for the imaginary parts of all the mode frequencies that we studied for subextremal Kerr--Newman black holes. Moreover, for small values of $q$, the errors in the real parts of these frequencies are also restricted to about $1\%$.
• Figure 4: Comparison between $\delta^2$ and $\mathcal{F}_s^2$ for the fundamental $(\ell,m) = (2,2)$ and $(2,1)$ gravitational modes (top and bottom panel, respectively). We show the behavior of $\delta^2$ and $\mathcal{F}_{s}^2$ as we move along the extremal curve $a_{\rm ext}^2 + q_{\rm ext}^2 = 1/4$. In general, the two curves follow each other closely. In the top panel, $\delta^2$ and $\mathcal{F}_s^2 > 0$ cross over from negative to positive values almost simultaneously, indicating a transition from a regime where ZDMs and DMs coexist, to one with only ZDMs. In the bottom panel, both curves remain negative valued, indicating that both ZDMs and DMs are always present in the spectrum. Hence, $\delta^2$ and $\mathcal{F}_s^2$ effectively convey the same information about the boundary between the regime with only ZDMs and the regime where ZDMs and DMs coexist.
• Figure 5: Comparison between the $\delta^2$ and $\mu$-criteria for the fundamental $\ell = m = 2$ mode. In the left column we show a high-spin case, $\theta = 10^{\circ}$, while in the right column we show a high-charge case, $\theta = 80^{\circ}$. Close to extremality ($\epsilon \lesssim 10^{-3}$), when $\theta = 10^\circ$, $\delta^2 > 0$ and $\mu > \mu_c$, indicating a ZDM-only regime; when $\theta = 80^\circ$, $\delta^2 < 0$ and $\mu < \mu_c$, indicating a regime where ZDMs and DMs coexist. Thus, for practical purposes, both criteria are equivalent.