On the Maldacena-Shenker-Stanford chaos bound in black hole spacetimes: Saturation, apparent violations, and their geometric origin

Abstract

Reported violations of the Maldacena-Shenker-Stanford (MSS) chaos bound in black hole spacetimes raise the question of whether they reflect a genuine breakdown, a feature of particular geometries, or a limitation of the probe used to test the bound. We show that the violations are apparent and arise predominantly in the near-extremal regime, where the black hole develops an $AdS_2\times S^2$ throat, the surface gravity becomes parametrically small, while the Lyapunov exponent of an orbit separated from the throat remains nonzero. We establish a geometric criterion governing this behavior and demonstrate its resolution in two complementary ways: a charged-particle construction that localizes an otherwise excluded orbit within the throat and restores exact saturation, and an out-of-time-order correlator (OTOC) calculation in the Jackiw-Teitelboim (JT) throat, which identifies the near-horizon Lyapunov scale and shows why the photon-sphere exponent does not reproduce it when the orbit remains outside the throat. We illustrate the criterion for static and rotating black holes in Einstein, scalar-tensor, and higher-curvature gravity, representative of a broader class of geometries to which it applies. In the absence of external effects that displace the relevant trajectories from the throat, the MSS bound remains respected; the apparent violations instead reflect a geometric decoupling between the probe and the near-horizon region.