Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations (1904.10333v2)

Abstract: The WKB approach for finding quasinormal modes of black holes, suggested in [1] by Schutz and Will at the first order and later developed to higher orders [2-4], became popular during the past decades, because, unlike more sophisticated numerical approaches, it is automatic for different effective potentials and mostly provides sufficient accuracy. At the same time, the seeming simplicity of the WKB approach resulted in appearance of a big number of partially misleading papers, where the WKB formula was used beyond its scope of applicability. Here we review various situations in which the WKB formula can or cannot bring us to reliable conclusions. As the WKB series converges only asymptotically, there is no mathematically strict criterium for evaluation of an error. Therefore, here we are trying to introduce a number of practical recipes instead and summarize cases in which higher WKB orders improve accuracy. We show that averaging of the Pade approximations, suggested first by J. Matyjasek and M. Opala [4], leads to much higher accuracy of the WKB approach, estimate the error and present the automatic code [5] which computes quasinormal modes and grey-body factors.

• The paper demonstrates that applying higher-order WKB approximations with Padé enhancements significantly refines quasinormal mode and grey-body factor calculations for black holes.
• The paper provides practical guidelines for error estimation, validated through empirical tests on Schwarzschild black holes.
• The paper also addresses method limitations while outlining its potential impact on gravitational wave data interpretation and alternative gravity tests.

Analyzing Quasinormal Modes and Grey-Body Factors with Higher Order WKB Approximations

The paper by R. A. Konoplya, A. Zhidenko, and A. F. Zinhailo explores the application of higher-order WKB approximations to compute quasinormal modes (QNMs) and grey-body factors for black holes. The WKB method, originally advanced by Schutz and Will, has gained traction due to its semi-analytic nature and its appeal in addressing the complex differential equations associated with QNMs. This analysis explores the practical implications of these theoretical advancements on both research and computation in gravitational physics.

The paper argues for the effectiveness of higher-order WKB methods, especially when expanded using Padé approximants to refine calculations of QNMs and grey-body factors. The problem is formulated in the context of linear perturbations of black holes, with these perturbations exhibiting characteristics—such as oscillation frequencies and damping rates—that are essential in testing the limits of gravitational theories and understanding the strong gravity regime.

Methodology Overview

The authors emphasize the significance of higher-order WKB calculations, which extend the seminal work by Schutz and Will, with further developments up to the 13th order. Although traditional approaches to QNMs often rely on numerical methods requiring specific solutions for different spacetime equations, the WKB method offers a more automated and flexible alternative. This, however, introduces challenges related to convergence, accuracy, and error estimation, which the paper addresses through practical guidelines and the incorporation of Padé approximants.

Key Numerical Results and Implications

1. Higher-Order Corrections: The paper highlights that while the WKB series remains asymptotic, its accuracy can be significantly improved by employing Padé approximants for capturing the asymptotic behavior of the series.
2. Error Estimation: The authors identify a lack of a mathematically strict criterion for error evaluation in WKB series, proposing practical methods instead. Their approach involves averaging Padé approximations to yield greater accuracy in determining QNMs.
3. Conducting Empirical Tests: Through computational tests, especially on the Schwarzschild black hole, the authors showcase that their method reduces errors substantially in dominant modes and higher overtones, illuminating paths for more precise analysis in theoretical predictions and experimental data interpretations.
4. Limitations and Future Directions: The paper does not shy away from addressing scenarios where the WKB remains ineffective, such as in the analysis of instability, superradiance, and quasi-resonances, which call for different methods or adaptations of the WKB approach.

Conclusions and Future Prospects

The paper's findings have significant implications for the theoretical and practical fields of gravitational wave research. The refined quasinormal mode predictions could enhance the accuracy of models used to interpret data from gravitational wave observatories like LIGO and VIRGO. This has potential applications in verifying alternative gravitational theories and improving our understanding of black hole dynamics.

Future work could focus on employing these methods on a broader range of black hole solutions and exploring more profound generalizations of Padé enhancements beyond spherically symmetric cases. By doing so, research could yield even more precise results, potentially contributing to new breakthroughs in gravitational physics, such as testing quantum gravity propositions and deepening insights into the nature of singularities and information loss.

In conclusion, this research represents a crucial step in enhancing semi-analytical methods for understanding complex gravitational phenomena, offering a framework that balances computational feasibility with theoretical rigor. The advancement in WKB techniques opens the door for further exploration and innovation in gravitational wave physics and quantum cosmology.