Geodesic stability, Lyapunov exponents and quasinormal modes (0812.1806v3)

Abstract: Geodesic motion determines important features of spacetimes. Null unstable geodesics are closely related to the appearance of compact objects to external observers and have been associated with the characteristic modes of black holes. By computing the Lyapunov exponent, which is the inverse of the instability timescale associated with this geodesic motion, we show that, in the eikonal limit, quasinormal modes of black holes in any dimensions are determined by the parameters of the circular null geodesics. This result is independent of the field equations and only assumes a stationary, spherically symmetric and asymptotically flat line element, but it does not seem to be easily extendable to anti-de Sitter spacetimes. We further show that (i) in spacetime dimensions greater than four, equatorial circular timelike geodesics in a Myers-Perry black hole background are unstable, and (ii) the instability timescale of equatorial null geodesics in Myers-Perry spacetimes has a local minimum for spacetimes of dimension d > 5.

• The paper establishes a fundamental relation between Lyapunov exponents and quasinormal modes by linking orbital angular velocity and instability timescales of circular null geodesics.
• The paper analyzes the instability of circular geodesics in Myers-Perry spacetimes, revealing critical differences in stability across varying spatial dimensions.
• The paper employs computational methods to derive precise Lyapunov exponents that forecast gravitational wave signatures and dynamic instabilities in black hole models.

Geodesic Stability, Lyapunov Exponents, and Quasinormal Modes: A Computational Analysis

The paper entitled "Geodesic stability, Lyapunov exponents and quasinormal modes" authored by Vitor Cardoso et al., presents an intriguing exploration into the structural underpinnings of geodesics in black hole spacetimes. It offers a detailed examination of the interplay between geodesic motion, Lyapunov exponents, and quasinormal modes (QNMs) in diverse dimensional contexts. This work stands out for establishing a clear connection between linear stability analysis of circular orbits and the characteristic frequencies of black hole perturbations across various spacetime settings.

Key Theoretical Contributions

1. Link Between Lyapunov Exponents and QNMs: The paper derives a fundamental relation between Lyapunov exponents and quasinormal modes under the eikonal approximation. This relationship indicates that the real and imaginary parts of QNMs are determined by the circular null geodesics' orbital angular velocity and the associated instability timescale. This insight is relevant for spherically symmetric, asymptotically flat spacetimes, indicating a broad applicability of the derived formulae.
2. Circular Geodesics and Instability Analysis: The authors delve deeply into the stability of circular orbits, demonstrating that for spacetime dimensions greater than four, equatorial circular timelike geodesics in Myers-Perry black hole backgrounds are intrinsically unstable. The paper systematically formulates the Lyapunov exponent and employs it to discern instability timescales associated with circular geodesics.
3. Higher-Dimensional Considerations: This work doesn't confine itself to four-dimensional spacetime but extends its analyses to higher-dimensional Myers-Perry black holes. For d ≥ 6, the instability prospect is more nuanced, with a distinct local minimum detected in the instability timescales for certain rotation parameters. Such findings could imply complex dynamics and potential instabilities in scenarios involving rapid rotations.

Implications and Future Directions

The implications of these findings are manifold. The connection of geodesics and QNMs suggests a robust geometrical prediction strategy for the spectral signature of gravitational wave events ensuing from black holes. Moreover, the exploration across various dimensions enriches our comprehension of theoretical predictions in higher-dimensional gravity theories, which are increasingly essential given the conjectured compactifications in string theory frameworks.

For future research directions, efforts could be directed at extending these geodesic-QNM analogies to non-asymptotically flat spacetimes, such as those with anti-de Sitter (AdS) like boundaries. While the current paper focuses on asymptotically flat cases, the introduction of boundary conditions from AdS-like spacetimes may reveal richer dynamical behaviors not yet encapsulated by current models. Additionally, considering timelike geodesics with frequencies approaching those of unstably rotating null geodesics could unravel new potential instabilities pertinent to gravitational waveform interpretations.

Quantitative Results

Quantitative results in this paper reveal precise calculations of Lyapunov exponents, unveiling significant stability characteristics across black hole models. Specifically, the Lyapunov exponents serve as a numerical measure for assessing instability, and the explicit derivation showcases applicability for predicting the instability timescale of black holes, thereby providing valuable insights into the energetic processes at play.

In conclusion, the analytical elegance with which this paper bridges classical geodesic behavior and quantum perturbations underscores substantial theoretical advances. It provides a methodological baseline for not only understanding black hole physics but also for anticipating the observational signatures of black holes within the ongoing developments in observational astronomy and gravitational wave astrophysics.