Quasinormal Modes via Lyapunov Exponents

• Quasinormal modes (QNMs) are complex frequencies characterizing the ringdown of black holes, with their real part linked to orbital motion and the imaginary part to instability.
• The approach uses the eikonal limit to relate QNM frequencies to the angular frequency and Lyapunov exponent of circular null geodesics, providing a geometric optics view of perturbations.
• This methodology extends to higher dimensions and rotating spacetimes, highlighting both the power and limitations of applying geodesic dynamics to non-flat asymptotic geometries.

Quasinormal modes (QNMs) are the complex frequencies characterizing the ringdown response of black holes and other compact objects to external perturbations. In the eikonal (geometric optics) limit, the connection between QNMs and the stability properties of unstable null geodesics becomes precise: the real part of the QNM frequency is set by the angular frequency of the corresponding circular null geodesic, and the imaginary part by its instability timescale, measured by the principal Lyapunov exponent. This correspondence, which holds in arbitrary spacetime dimension for stationary, spherically symmetric, and asymptotically flat black holes, has been established to hold universally for a wide class of backgrounds, but is not generically extendable to anti–de Sitter (AdS) geometries (0812.1806).

1. Geodesic Dynamics and Their Role in QNMs

Unstable circular null geodesics—photon spheres—provide the underlying semiclassical structure controlling the propagation of high-frequency perturbations in black hole spacetimes. In the geometric optics approximation, the wave equation separates and localizes near these null geodesics. As a result, black hole perturbations in the eikonal limit can be thought of as massless particles orbiting on, and leaking from, the photon sphere.

Given an angular quantum number $l \gg 1$, the real part of the QNM frequency ($\omega_{\rm QNM}$) is dictated by the angular frequency ($\Omega_c$) of the unstable null geodesic at the photon sphere, and the imaginary part quantifies decay due to the instability of the orbit, encapsulated by the principal Lyapunov exponent ($\lambda$) through the relation:

$$\omega_{\rm QNM} = \Omega_c\,l - i(n + 1/2)|\lambda|,$$

where $n$ labels the overtone number, and $|\lambda|$ is the local divergence rate (inverse instability timescale).

2. Lyapunov Exponents: Instability of Circular Orbits

The Lyapunov exponent $\lambda$ measures the exponential growth rate of small radial perturbations to the circular orbit. For a geodesic with effective radial potential $V_r$ (in coordinates $(r, p_r)$), and under the assumption of a stationary, spherically symmetric, asymptotically flat metric:

$$ds^2 = f(r)\,dt^2 - \frac{dr^2}{g(r)} - r^2\,d\Omega_{d-2}^2,$$

the effective potential is

$$V_r = g(r)\left[\frac{E^2}{f(r)} - \frac{L^2}{r^2} - \delta_1\right],$$

where $\delta_1 = 1$ (timelike) or $0$ (null), $E$ and $L$ are the conserved energy and angular momentum.

For circular geodesics, $V_r = 0$ and $V_r' = 0$ at $r = r_c$, while the Lyapunov exponent is given by:

$\lambda = \sqrt{ \frac{V_r''}{2 \dot{t}^2} },$

where primes denote $r$-derivatives and $\dot{t}$ is the radial derivative of coordinate time along the null geodesic, $\dot{t} = E/f(r)$ (0812.1806).

A real, positive $\lambda$ signifies orbital instability; all QNM damping in the eikonal regime is governed by this instability timescale.

3. Eikonal QNM Formula and Universal Predictions

The direct correspondence between black hole “ringdown” and geodesic parameters in the eikonal limit is general and field-theory independent as long as the metric is stationary, spherically symmetric, and asymptotically flat. The principal QNM frequency formula reads:

$$\omega_{\rm QNM} = \Omega_c\,l - i(n + 1/2)|\lambda|,$$

with:

• $\Omega_c$ — the angular frequency of the unstable null orbit,
• $n$ — overtone number,
• $|\lambda|$ — Lyapunov exponent of null geodesic.

For Schwarzschild–Tangherlini (higher-dimensional) black holes, the ratio of $\Omega_c$ to $\lambda$ is:

$$\gamma = \frac{\Omega_c}{2\pi\lambda} = \frac{1}{2\pi\sqrt{d-3}},$$

quantifying the competing timescales of orbit and instability.

For generic stationary, spherically symmetric metrics, the explicit steps are:

1. Solve $V_r(r_c)=V_r'(r_c)=0$ for the circular null geodesic radius $r_c$.
2. Calculate $\Omega_c$ and the second derivative $V_r''$.
3. Evaluate $\lambda$ via the above formula.
4. Insert into the eikonal QNM frequency relation.

4. Implications for Higher Dimensions and Rotating Spacetimes

The paper extends the analysis to higher-dimensional black holes. For the Myers–Perry family of rotating black holes, analysis shows the following:

• For $d > 4$, all equatorial circular timelike geodesics in the Myers–Perry geometry become unstable—no stable circular orbits exist.
• For $d > 5$, the normalized Lyapunov exponent (${ \lambda }/{ \Omega_c }$) of equatorial null geodesics as a function of rotation parameter exhibits a local minimum. This suggests a qualitative transition analogous to a black hole–to–black brane changeover.

These features indicate that, with increasing dimension and/or rotation, the degree of instability in photon spheres is non-monotonic and highly sensitive to the spacetime parameters.

5. Mathematical Structure and Computational Workflow

The mathematical structure centers around the geodesic equation and the evaluation of the second derivative of the effective potential at the photon sphere. The computation proceeds as:

• Specify the line element $f(r), g(r)$.
• Find $r_c$ simultaneously solving $V_r=0$ and $V_r'=0$.
• Compute $V_r''$ and $\dot{t}$ at $r_c$.
• Use

$\lambda = \sqrt{ \frac{V_r''}{2 \dot{t}^2} }.$

• Insert $\lambda$ and $\Omega_c$ into:

$$\omega_{\rm QNM} = \Omega_c\,l - i(n + 1/2)|\lambda|.$$

This approach is independent of the chosen field equations, applicable to arbitrary spacetime dimension, and robust for any spherically symmetric, stationary, asymptotically flat geometry.

6. Restrictions and Breakdown of the Correspondence

The universality of the QNM–Lyapunov exponent relation relies critically on the assumption of asymptotic flatness. In anti–de Sitter (AdS) backgrounds:

• The boundary at infinity introduces reflecting conditions, which critically change the QNM spectrum.
• The simple geodesic instability argument, and hence the eikonal correspondence, fails; numerical and analytic studies display a different scaling of QNM damping with the horizon radius.
• The Lyapunov/geodesic picture still provides local instability information but does not capture the global QNM spectrum.

Thus, extension to AdS, or spacetimes with nontrivial asymptotics, requires a more sophisticated approach incorporating global boundary conditions.

7. Summary and Practical Deployment

The use of Lyapunov exponents to determine QNM damping rates offers a direct, geometric interpretation of black hole ringdown. The real QNM frequency is set by the angular frequency of the photon sphere, and the imaginary component by the rate of instability (Lyapunov exponent) of that orbit. This approach is highly efficient computationally and agrees well with both WKB and numerical calculations across a broad array of models and dimensions. In higher-dimensional and rotating black holes, the method further reveals instabilities invisible in lower-dimensional or non-rotating settings.

The framework is not only of theoretical value but also practical for modeling QNM spectra in numerical and semi-analytic relativity, providing a bridge between geometric optics and wave perturbation calculations in black hole spacetimes (0812.1806).