Geodesic stability, Lyapunov exponents, and quasinormal modes

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 ≥ 6.

Black holes die hard: Can one spin up a black hole past extremality?

“…For large E it tends to Lcrit E → 2 √ M − a, which also corresponds to the values of the null circular geodesic, as could be expected [13]. Eq.…”

“…In this case, it is sufficient to focus attention on the (co-rotating) circular null geodesic with r = r c as the geodesic with maximum possible impact parameter that can still be captured. This geodesic has [13] and for large a we get L E ∼ a [13]. Thus, from equation (30) we get…”

“…We start with the D = 5 case, which is simple enough that it allows an explicit solution [13]. Let's focus on co-rotating geodesics, since these are the only ones of significance here.…”

Quasinormal modes and Regge poles of the canonical acoustic hole

“…In the limit l n, it turns out that the spectrum of QN modes and RPs is determined by the properties of the null orbit [43][44][45][46]. The frequency Ω of the null orbit is…”

A Possible Failure of Determinism in General Relativity

“…Here, null particles can be trapped on circular unstable trajectories, defining the photon sphere. This region has a strong pull in the control of the decay of fluctuations and the spacetime's QNMs which have large frequency (i.e., large jReωj) [11,[31][32][33]. For instance, the decay time scale is related to the instability time scale of null geodesics near the photon sphere.…”