Bounds for Lyapunov exponent of circular light orbits in black holes

Emanuel Gallo; Thomas Mädler

doi:10.1140/epjc/s10052-025-14046-8

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2025) 85: 299
https://doi.org/10.1140/epjc/s10052-025-14046-8

Regular Article - Theoretical Physics

Bounds for Lyapunov exponent of circular light orbits in black holes

Emanuel Gallo¹^a and Thomas Mädler²

¹ FaMAF, UNC, Instituto de Física Enrique Gaviola (IFEG), CONICET, 5000, Córdoba, Argentina
² Escuela de Obras Civiles and Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Casilla 298-V, Avenida Ejército Libertador 441, Santiago, Chile

^a egallo@unc.edu.ar

Received: 19 December 2024
Accepted: 6 March 2025
Published online: 14 March 2025

Abstract

Chaotic systems near black holes satisfy a universal bound, $λ \leq κ_{H}$ $\lambda \le \kappa _H$ linking the Lyapunov coefficient $λ$ $\lambda$ associated with unstable orbits to surface gravity $κ_{H}$ $\kappa _H$ 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.

© The Author(s) 2025

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