Quasinormal modes of black holes. The improved semianalytic approach (1704.00361v1)

Abstract: We have extended the semianalytic technique of Iyer and Will for computing the complex quasinormal frequencies of black holes, $\omega,$ by constructing the Pad\'e approximants of the (formal) series for $\omega^{2}$. It is shown that for the (so far best documented) quasinormal frequencies of the Schwarzschild and Reissner-Nordstr\"om black holes the Pad\'e transforms $P_{6}^{6}$ and $P_{7}^{6}$ are, within the domain of applicability, always in excellent agreement with the numerical results. We argue that the method may serve as the black box with the "potential" $Q(x)$ as an input and the accurate quasinormal modes as the output. The generalizations and modifications of the method are briefly discussed as well as the preliminary results for other classes of the black holes.

• The paper refines quasinormal mode calculations by integrating Padé transformations with extended WKB approximations.
• It shows high accuracy for Schwarzschild and Reissner-Nordström black holes, particularly for modes where overtones meet multipole limits.
• The improved method is adaptable to diverse black hole models, offering practical insights for gravitational wave astrophysics.

Analysis of the Improved Semianalytic Approach to Quasinormal Modes of Black Holes

The paper by Jerzy Matyjasek and Michal Opala advances the semianalytic calculation of quasinormal modes (QNMs) in black holes, particularly focusing on the Schwarzschild and Reissner-Nordström black hole configurations. The authors notably refine the established Iyer and Will approach, augmenting it with Padé transformations to extend the calculation potential beyond the sixth-order WKB approximation. This method attains high precision in complex frequency calculations by solving perturbation equations formulated around the notion of a potential function $Q(x)$.

Methodology

The technique espoused in the paper incorporates iterative mathematical constructs to better estimate QNM frequencies. The authors employ an algorithmic strategy that specifically leverages the properties of the Padé approximants, enhancing the predictive accuracy of the quasinormal frequencies for spherical and charged black holes. Notably, they compute these frequencies up to the 13th order of the WKB approximation. The choice to rely on Padé approximants is substantiated by their capacity to effectively stabilize the series expansions involved in the computation, particularly for cases where overtone numbers approach or exceed multipole numbers.

Key Findings

1. Accuracy of the Padé Transformation: For Schwarzschild and Reissner-Nordström black holes, the Padé transformations, $P_6$ and $P_7$, provide frequencies with deviations in the real and imaginary components that are significantly minimized compared to the sixth-order WKB approximation traditionally used.
2. Numerical Comparisons: The devised approach yields results that exhibit excellent congruence with established numerical data, especially for low-lying modes where overtone numbers are close to or slightly surpass the multipole number. This validation underscores the enhanced reliability of the method.
3. Applicability Across Diverse Black Hole Models: The paper delineates the potential of the improved semianalytic method to be adapted to various black hole models without fundamental alterations to the core computational algorithm. This adaptability is crucial for exploring quasinormal modes in higher-dimensional or asymptotically (anti)-de Sitter black holes.

Implications

This research signifies a substantial technical contribution to the computational methods utilized in determining quasinormal modes associated with black holes. The enhanced accuracy and broad applicability suggest practical implications for gravitational wave astrophysics, offering potential insights into black hole detection mechanisms and characterization through emitted gravitational wave signatures.

Speculation on Future Research

Future research could further refine or extend this approach to encompass other gravitational scenarios or explore the potential integration with quantum field theoretical frameworks in curved spacetime. Additionally, expanding upon the current results to incorporate rotational black holes, such as Kerr black holes, may further elucidate the role of angular momentum in QNM behavior.

In conclusion, Matyjasek and Opala’s work exemplifies a methodical enhancement in the semianalytic computation of black hole quasinormal modes. This advancement affords other researchers a tested and potent tool for probing the complex dynamical properties of black holes as understood in the context of both theoretical and observational astrophysics.