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

We investigate chaotic dynamics in extremal black holes by analyzing the motion of massless particles in both Reissner-Nordström 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-Nordström 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.

💡 Research Summary

In this work the authors investigate chaotic dynamics in extremal (zero‑temperature) black holes by studying the motion of mass‑less test particles in the near‑horizon regions of Reissner‑Nordström (RN) and Kerr spacetimes. Two complementary strategies are employed: (i) start from non‑extremal solutions and continuously tune the charge Q (for RN) or spin a (for Kerr) toward their extremal values, and (ii) work directly with the exact extremal metrics. To keep the particle trajectories from falling through the horizon, static harmonic potentials are added in the radial and angular directions; these serve only to stabilize the numerics and do not affect the intrinsic gravitational instability.

The equations of motion are derived from the Hamiltonian constraint H=0, and the total energy includes the gravitational part plus the harmonic confinement terms. Lyapunov exponents λ are extracted by evolving pairs of nearby trajectories with a fourth‑order Runge‑Kutta scheme and periodically re‑orthonormalising the separation vector. In the non‑extremal regime the authors recover the familiar proportionality λ∝T, consistent with the Maldacena‑Shenker‑Stanford (MSS) bound λ≤2πT/ħ.

When the extremal limit is taken, however, the surface gravity κ and thus the Hawking temperature vanish, yet the computed λ remains strictly positive. For RN black holes λ decreases as Q→M but never reaches zero; for Kerr black holes λ actually grows as the spin parameter a approaches the extremal value a→M. Consequently the MSS bound, which would predict λ≤0 at T=0, is violated. The residual Lyapunov exponent signals a “residual chaos” phase that persists even in the absence of thermal fluctuations.

The authors argue that this residual chaos is not an artifact of the artificial harmonic potentials—previous studies have shown that the radial instability of massless particles is insensitive to the detailed form of the confining potential—so the effect is rooted in the intrinsic near‑horizon geometry of extremal spacetimes. The results imply that extremal black holes constitute a distinct dynamical phase, separate from their non‑extremal counterparts, despite the continuity of thermodynamic quantities such as entropy.

Beyond the immediate finding, the paper highlights several broader implications. First, the MSS bound, originally derived for thermal quantum systems, may require refinement when applied to zero‑temperature gravitational backgrounds with non‑trivial microstate structure. Second, the enhanced chaos in extremal Kerr geometries suggests that rotation amplifies sensitivity to initial conditions, potentially affecting information scrambling, complexity growth, and holographic dual descriptions in the AdS/CFT context. Third, the methodology—combining analytic near‑horizon expansions with high‑precision numerical Lyapunov analysis—provides a template for exploring chaotic dynamics in other exotic spacetimes (e.g., higher‑dimensional or charged‑rotating solutions).

In conclusion, the study demonstrates that extremal black holes exhibit non‑vanishing Lyapunov exponents, thereby violating the naive MSS chaos bound. This residual chaotic behavior underscores the need to treat extremal horizons as qualitatively new dynamical objects and opens new avenues for probing the interplay between classical chaos, quantum information, and black‑hole thermodynamics. Future work is suggested to explore massive or charged probes, alternative confinement schemes, and the direct connection between residual chaos and microscopic BPS state counting.