[Paper Review] A short remark (with a long title) on the universality of the quasinormal spectrum of near-extremal Kerr-Newman black holes

Shahar Hod|arXiv (Cornell University)|Oct 8, 2014

Black Holes and Theoretical Physics3 citations

TL;DR

This paper demonstrates that the quasinormal mode spectrum of near-extremal Kerr-Newman black holes—governed by coupled gravitational and electromagnetic perturbations—is universally described by a single analytical formula involving the horizon angular velocity $\Omega_{\text{H}}$ and Bekenstein-Hawking temperature $T_{\text{BH}}$. The formula accurately predicts numerical time evolutions from nonlinear simulations, including cases where prior universality based on $a/a_{\text{max}}$ failed, confirming its robustness across diverse black hole parameters.

ABSTRACT

In a recent paper (arXiv:1410.0694) Zilhão, Cardoso, Herdeiro, Lehner, and Sperhake have studied the nonlinear stability of Kerr-Newman black holes. We show that their numerical results for the time evolutions of the spacetime deformations of near-extremal Kerr-Newman black holes are described extremely well by a {\it universal} formula for the quasinormal resonances of the black holes. This formula is expressed in terms of the black-hole physical parameters: the horizon angular velocity $Ω_{ ext{H}}$ and the Bekenstein-Hawking temperature $T_{ ext{BH}}$.

Motivation & Objective

• To test the universality of quasinormal mode behavior in near-extremal Kerr-Newman black holes beyond scalar perturbations.
• To resolve discrepancies in prior claims of universality based on $a/a_{\text{max}}$ by evaluating all available numerical time evolutions.
• To verify whether the analytical formula for quasinormal resonances applies to coupled gravitational-electromagnetic perturbations in Kerr-Newman black holes.
• To demonstrate that the formula $\omega_n = m\Omega_{\text{H}} - i2\pi T_{\text{BH}}(n + \frac{1}{2} - i\delta)$ universally describes nonlinear time evolutions in near-extremal black holes.

Proposed method

• Analytical derivation of the quasinormal mode formula $\omega_n = m\Omega_{\text{H}} - i2\pi T_{\text{BH}}(n + \frac{1}{2} - i\delta)$ for scalar, electromagnetic, and gravitational perturbations of near-extremal Kerr black holes.
• Application of the same formula to Kerr-Newman black holes, treating $\Omega_{\text{H}}$ and $T_{\text{BH}}$ as physical parameters derived from $a$ and $Q$.
• Comparison of numerically computed time periods $T_{\text{num}}$ from ZCHLS simulations with analytically predicted periods $T_{\text{ana}} = 2\pi / (m\Omega_{\text{H}})$.
• Evaluation of agreement between $T_{\text{num}}$ and $T_{\text{ana}}$ across four distinct Kerr-Newman black hole configurations with varying $a/M$ and $Q/M$.
• Use of the formula to explain both overlapping and non-overlapping time evolutions in ZCHLS’s Fig. 4, showing consistency across all cases.

Experimental results

Research questions

• RQ1 Does the quasinormal mode formula $\omega_n = m\Omega_{\text{H}} - i2\pi T_{\text{BH}}(n + \frac{1}{2} - i\delta)$ accurately describe coupled gravitational-electromagnetic perturbations in near-extremal Kerr-Newman black holes?
• RQ2 Why do some numerical time evolutions in ZCHLS’s simulations show non-overlapping behavior despite prior claims of $a/a_{\text{max}}$-based universality?
• RQ3 Can the formula explain the time evolution of the fourth black hole in ZCHLS’s Fig. 4, which deviates from the other three?
• RQ4 Is the universality of quasinormal modes in near-extremal Kerr-Newman black holes better explained by $\Omega_{\text{H}}$ and $T_{\text{BH}}$ than by the ratio $a/a_{\text{max}}$?

Key findings

• The time evolution of the three Kerr-Newman black holes with $a/M \approx 0.907, 0.944, 0.99$ and $Q/M = 0.4, 0.3, 0$ shows a $2\%$ deviation between numerically computed and analytically predicted periods, confirming strong agreement with the formula.
• The time evolution of the fourth black hole with $(a/M, Q/M) = (0.594, 0.8)$ shows a $0.3\%$ deviation between $T_{\text{num}} \simeq 8.11M$ and $T_{\text{ana}} \simeq 8.09M$, indicating excellent agreement with the analytical formula.
• The formula $\omega_n = m\Omega_{\text{H}} - i2\pi T_{\text{BH}}(n + \frac{1}{2} - i\delta)$ accurately describes quasinormal resonances across all four simulated black hole configurations, including the one that previously appeared to break universality.
• The horizon angular velocity $\Omega_{\text{H}} \simeq 0.432M^{-1}$ for the first three black holes and $\Omega_{\text{H}} \simeq 0.388M^{-1}$ for the fourth are key to explaining the observed differences in oscillation periods, not the $a/a_{\text{max}}$ ratio.
• The results suggest that the formula is universally valid for coupled gravitational-electromagnetic perturbations in near-extremal Kerr-Newman black holes, despite the lack of a formal analytical proof for such coupled modes.