Geodesic stability, Lyapunov exponents, and quasinormal modes

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 time scale 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 time scale of equatorial null geodesics in Myers-Perry spacetimes has a local minimum for spacetimes of dimension $d\ge 6$.

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.