Bounds for Lyapunov exponent of circular light orbits in black holes

Abstract Chaotic systems near black holes satisfy a universal bound, $$\lambda \le \kappa _H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>κ</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:mrow> </mml:math> linking the Lyapunov coefficient $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> associated with unstable orbits to surface gravity $$\kappa _H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>κ</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:math> 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.