Geodesic stability, Lyapunov exponents and quasinormal modes

TL;DR

This work shows that in the eikonal limit the quasinormal mode spectrum of black holes, in any dimension and for static spherically symmetric spacetimes, is determined by the properties of unstable circular null geodesics, encapsulated by the angular velocity Ω_c and Lyapunov exponent λ. It provides a simple general formula for the principal Lyapunov exponent in terms of the second radial potential derivative and demonstrates the QNM relation ω_QNM = Ω_c l - i (n+1/2) |λ|, with consistent checks against WKB results and explicit results for Schwarzschild-Tangherlini black holes and near-extremal SdS. The paper then extends the analysis to rotating Myers-Perry black holes, showing all equatorial timelike circular orbits are unstable for d > 4 and revealing a dimension-dependent behavior of the Lyapunov exponent at large rotation that echoes QNM trends. These findings offer a geometric-optics perspective on ringdown and illuminate how geodesic stability properties govern observable black hole oscillations across a broad class of spacetimes, while highlighting limitations in AdS contexts and opportunities for extending the geodesic-QNM correspondence beyond spherical symmetry.

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.

Figures (2)

• Figure 1: Ratio of the timelike orbital frequency $\Omega$ to the orbital frequency $\Omega_c$ of corotating (left) and counterrotating (right) null geodesics as a function of $r/r_c$, with $r_c$ the radius of the null geodesic.
• Figure 2: Dimensionless instability exponents $\lambda/\Omega_c$ as a function of rotation for several spacetime dimensions $d$. We use units such that $\mu=2$. Solid lines refer to corotating orbits, dashed lines to counterrotating orbits.