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

Abstract

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.

Figures (8)

• Figure 1: Plots of the largest Lyapunov exponent for Reissner-Nordström black holes with charges $Q = 0.4, 0.5, 0.6, 0.8, 0.9$, and the extremal case $Q = 1.0$. The exponent initially fluctuates but stabilizes to distinct saturation values, indicating reduced chaotic behavior as the extremal limit is approached.
• Figure 2: Largest Lyapunov exponent specifically for the extremal Reissner-Nordström black hole ($Q = M = 1.0$). Significant initial fluctuations settle into a moderate saturation value (0.017518), demonstrating subtle yet persistent chaotic dynamics in the extremal configuration.
• Figure 3: Poincaré sections $(r, p_{r})$ for the Reissner--Nordström spacetime. Orbits are recorded at $\theta = 0$ with $p_{\theta} > 0$. Parameters: $M = 1$, $K_{r} = 100$, $K_{\theta} = 25$, $E = 50$, $r_{c} = 4.3$, and $\theta_{c} = 0$. For a moderate charge ($Q = 0.6$), the section shows visible distortions and scattered points near the edges, indicating mild chaotic behavior. As $Q$ increases to $0.8$ and $0.9$, the KAM tori become progressively smoother and more regular. At the extremal value ($Q = 1.0$), the phase space is dominated by well-formed nested tori, signifying the restoration of integrable motion. This trend is consistent with the Lyapunov exponent analysis, where the degree of chaos decreases as the charge increases and the horizon moves farther from the confined region.
• Figure 4: Evolution of the largest Lyapunov exponent for Kerr black holes with varying rotation parameters $a = 0.1, 0.5, 0.6, 0.8, 0.9$, and the extremal limit $a = 1.0$. The saturation values increase with spin, showing stronger chaotic behavior as the black hole approaches extremality.
• Figure 5: Lyapunov exponent for the extremal Kerr black hole ($a = M = 1.0$). Noticeable fluctuations initially appear before stabilizing at a significantly higher saturation value (0.0185), highlighting enhanced chaotic and unstable dynamics in the fastest spinning Kerr black hole scenario.