Geodesic stability, Lyapunov exponents and quasinormal modes

TL;DR

This paper links the instability of null geodesics to black hole quasinormal modes, showing that in the eikonal limit, these modes are determined by geodesic parameters across dimensions, with implications for black hole stability.

Contribution

It demonstrates that quasinormal modes in any dimension are governed by circular null geodesics, independent of field equations, and analyzes geodesic stability in Myers-Perry black holes.

Findings

Quasinormal modes are determined by null geodesics in the eikonal limit.

Null geodesics in higher-dimensional black holes are unstable.

Instability timescales have a local minimum in dimensions greater than five.

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.