Researchers studied why a proposed universal limit on chaotic growth can fail near some black holes. The bound, known as the Maldacena–Shenker–Stanford (MSS) chaos bound, links a rate of instability called the Lyapunov exponent to the black hole temperature. Using Hawking’s relation between temperature and surface gravity, the bound can be restated as a geometric inequality λ ≤ κ, where λ is the instability rate and κ is the surface gravity of the horizon.

The authors looked at the motion of particles on circular paths around many kinds of black holes. These included ordinary solutions of General Relativity and extensions such as shift-symmetric Horndeski (a scalar–tensor theory) and Einstein–Gauss–Bonnet gravity (a theory with higher-curvature terms). They computed Lyapunov exponents for timelike and lightlike circular orbits in static, spherically symmetric metrics and compared those instability scales with the surface gravity as black holes approach extremality (the limit in which surface gravity and temperature go to zero).

They find a simple geometric rule that controls when violations occur. The key is the relative position of the unstable circular orbit (often the photon-sphere, the radius where light can orbit) and the horizon. If, as the black hole is driven toward extremality, the unstable orbit stays outside the horizon, the instability scale λ can remain finite even though κ goes to zero. In that case the MSS inequality can be violated. But if the unstable orbit approaches and merges with the degenerate horizon, the strong gravitational time dilation near the horizon suppresses the local instability and λ decreases in step with κ, restoring the bound.

This result matters because it ties apparent violations of a quantum chaos bound to concrete geometric features of a spacetime. It gives a way to read off whether the MSS-type bound should hold from the photon-sphere and horizon structure, rather than attributing every violation to exotic modifications of gravity. The work thus clarifies when one should expect the bound to apply in classical orbital probes of black holes and when one should not.