Bounds for Lyapunov exponent of circular light orbits in black holes

TL;DR

This paper investigates universal bounds on the Lyapunov exponent of unstable circular light orbits near black holes, proposing new bounds that relate to generalized surface gravity and have implications for quasinormal modes, shadow size, and entropy.

Contribution

It introduces new universal bounds for Lyapunov exponents of photon spheres in static, spherically symmetric black holes satisfying Einstein's equations and energy conditions.

Findings

Bounds relate Lyapunov coefficient to generalized surface gravity.

Constraints on quasinormal mode imaginary parts in the eikonal regime.

Connections between Lyapunov exponent, shadow size, and horizon entropy.

Abstract

Chaotic systems near black holes satisfy a universal bound, $\lambda \leq \kappa_H$ linking the Lyapunov coefficient $\lambda$ associated with unstable orbits to surface gravity $\kappa_H$ of the event horizon. A natural question is whether this bound is satisfied by unstable circular null geodesics in the vicinity of black holes. However, there are known cases where this bound is violated. It is intriguing to ask whether there exists an alternative universal bound that is valid in such situations. We show that for any spherically symmetric, static black hole that satisfies Einstein's equations and the dominant energy condition, there exist other universal bounds relating the Lyapunov coefficient to a generalized notion of surface gravity at the photon sphere. As applications, we show how these bounds also constrain the imaginary part of quasinormal modes in the eikonal regime and how the Lyapunov coefficient relates to the shadow size and the entropy of the horizon.