Chaotic Dynamics in Extremal Black Holes: A Challenge to the Chaos Bound

TL;DR

This paper shows that extremal black holes exhibit persistent chaos with Lyapunov exponents that violate the MSS chaos bound, challenging previous assumptions about black hole dynamics at zero temperature.

Contribution

It provides a detailed analysis demonstrating that extremal black holes maintain residual chaos, contradicting the expectation that chaos vanishes at zero temperature.

Findings

Lyapunov exponent remains positive at extremality

Chaos persists and even intensifies in Kerr black holes with higher spin

Extremal black holes violate the MSS chaos bound

Abstract

We investigate chaotic dynamics in extremal black holes by analyzing the motion of massless particles in both Reissner-Nordstr\"{o}m and Kerr geometries. Two complementary approaches (i) taking the extremal limit of non-extremal solutions and (ii) working directly in the extremal background, yield consistent results. We find that, contrary to naive extrapolation of the Maldacena-Shenker-Stanford (MSS) chaos bound, the Lyapunov exponent remains positive even at zero temperature. For Reissner-Nordstr\"{o}m black holes, chaos diminishes but persists at extremality, while for Kerr black holes it strengthens with increasing spin. These results demonstrate that extremal black holes exhibit residual chaotic dynamics that violate the MSS bound, establishing them as qualitatively distinct dynamical phases of gravity.