Bounds for Lyapunov exponent of circular light orbits in black holes

Abstract

Chaotic systems near black holes satisfy a universalbound, λ ≤ κH linking the Lyapunov coefficient λ associatedwith unstable orbits to surface gravity κH of the eventhorizon. A natural question is whether this bound is satisfiedby unstable circular null geodesics in the vicinity of blackholes. However, there are known cases where this bound isviolated. It is intriguing to ask whether there exists an alternativeuniversal bound that is valid in such situations. Weshowthat for any spherically symmetric, static black hole thatsatisfies Einstein’s equations and the dominant energy condition,there exist other universal bounds relating the Lyapunovcoefficient to a generalized notion of surface gravity at thephoton sphere. As applications, we show how these boundsalso constrain the imaginary part of quasinormal modes inthe eikonal regime and how the Lyapunov coefficient relatesto the shadow size and the entropy of the horizon.