Complete quasinormal modes of Type-D black holes (2506.14635v3)

Abstract: Quasinormal mode (QNM) spectra of black holes exhibit two open problems [Conf. Proc. C 0405132, 145 (2004); CQG 26, 163001 (2009)]: (i) the discontinuity in highly damped QNMs between Schwarzschild and Kerr solutions as $a \to 0$, and (ii) the unexplained spectral proximity between QNMs and algebraically special (AS) frequencies, particularly the anomalous multiplet splitting for Kerr $\ell=2$, $m \geq 0$ modes. We develop a novel method to compute complete QNM spectra for Type-D black holes, solving both problems and establishing a mathematical framework for boundary value problems of dissipative systems. Using analytic continuation of radial eigenvalue equations, our method eliminates the dependence on auxiliary parameters in the connection formulas for confluent Heun solutions. This breakthrough overcomes the long-standing challenge of calculating QNMs that cross or lie on the negative imaginary axis (NIA). For Schwarzschild and Kerr spacetimes ($0 \leq \hat{n} \leq 41$, $2 \leq \ell \leq 16$), we present complete spectra validated through scattering amplitudes with errors $<10^{-10}$. The results provide definitive solutions to both open problems: (i) The inability of conventional methods to compute QNMs crossing or residing on the NIA leads to apparent discontinuities in Kerr spectra as $a \to 0$. (ii) When a QNM exactly coincides with the AS frequency, an additional QNM (unconventional mode) appears nearby. For the Kerr case with $\ell=2$, overtone sequences from both unconventional and AS modes exhibit precisely $2\ell+1$ branches, without multiplets or supersymmetry breaking at AS frequencies. Our method confirms Leung's conjecture of high-$\ell$ unconventional mode deviations from the NIA through the first $\ell=3$ calculation. Moreover, this paradigm surpasses the state-of-the-art Cook's continued fraction and isomonodromic methods in computational efficiency.

• The paper introduces a novel analytic continuation method with confluent Heun functions to compute a complete, precise spectrum of QNMs for both Schwarzschild and Kerr spacetimes.
• The paper achieves error margins below 10^-10 by eliminating auxiliary parameters, thereby overcoming numerical instabilities and oscillatory convergence failures near the negative imaginary axis.
• The paper resolves longstanding discontinuities and clarifies the spectral proximity between quasinormal modes and algebraically special frequencies, offering new insights for gravitational wave tests and quantum gravity research.

Overview of the Research on Quasinormal Modes of Type-D Black Holes

This paper presents a novel approach to addressing longstanding issues in the paper of quasinormal modes (QNMs) related to Type-D black holes. Specifically, it tackles two unresolved problems: the discontinuity in highly damped QNMs between Schwarzschild and Kerr solutions as the spin parameter approaches zero, and the unexplained spectral proximity between QNMs and algebraically special (AS) frequencies.

1. Analytical and Numerical Framework

The authors introduce a comprehensive method based on analytic continuation and the confluent Heun function for constructing eigenvalue equations suitable for studying QNMs across various Type-D black holes. This method addresses susceptibilities inherent in traditional numerical techniques, notably crossing or lying on the negative imaginary axis (NIA) in the complex frequency plane. By overcoming the limitations of traditional approaches, which exhibit oscillatory convergence failures near the NIA, this research achieves the first complete and precise spectrum of QNMs for both Schwarzschild and Kerr spacetimes, marking a notable advance in the computational methodologies for gravitational perturbations.

2. Handling Complex Frequency Challenges

Key to the paper is its novel analytic continuation technique that sidesteps the auxiliary parameters often needed in numerical algorithms, which typically slow convergence or make higher overtone modes computationally expensive. By effectively calculating the connection coefficients for the confluent Heun function without these parameters, the paper achieves comprehensive spectra with errors below 10^-10 over a range of overtone numbers and modes. This methodological improvement allows for a more accurate characterization of both low-lying (weakly damped) QNMs and high-overtone (highly damped) QNMs.

3. Addressing Open Problems

The paper provides definitive answers to the previously mentioned spectral problems:

• Problem (i): The work clarifies that previously observed discontinuities in Kerr spectra as the spin parameter tends to zero were due to the inability of former methods to compute modes crossing the NIA. This development, supported by robust numerical data exhibiting no such discontinuities, reinforces the mathematical continuity between Schwarzschild and Kerr solutions.
• Problem (ii): On the mysterious proximity between QNMs and AS modes, the paper shows that whenever a QNM frequency coincides with an AS frequency, an "unconventional mode" crystallizes nearby. In this manner, the overtone sequences from both unconventional and AS modes reveal precise branching in accordance with the theoretically expected $2\ell+1$ structure, without multiplet discrepancies or supersymmetry violations at AS frequencies. Thus, it confirms longstanding conjectures regarding asymmetric mode multiplets.

4. Computational Superiority

Moreover, this innovative approach brings efficiency and accuracy that surpass traditional techniques, such as Cook’s continued fraction and isomonodromic methods, particularly highlighted by its capacity to solve the complete quasinormal spectrum what the authors denote as a step towards a comprehensive exploration of highly damped QNMs in Type-D spacetimes.

5. Implications and Future Directions

The implications of this research extend both practically and theoretically. Practically, the precise computation of QNMs enhances the potential for observational tests of general relativity, especially in the context of gravitational wave astronomy. Theoretically, this methodological advancement paves the way for potential explorations in quantum gravity scenarios, where spacetime modifications could be imprinted in the QNM spectra.

The research suggests several avenues for future work, such as extending these computational techniques to more exotic or modified gravity backgrounds where black hole solutions deviate from Type-D spacetimes, thereby transforming our understanding and modeling of astrophysical phenomena.

In summary, the paper not only solves existing theoretical dilemmas but sets the stage for rigorous investigations into the physics of black holes, potentially informing both the interpretation of empirical data and the development of alternative theories of gravity.