[Paper Review] Bound on the Lyapunov exponent in Kerr-Newman black holes via a charged particle

Naoto Kan, Bogeun Gwak|arXiv (Cornell University)|Sep 15, 2021

Black Holes and Theoretical PhysicsPhysics and Astronomy60 references46 citations

TL;DR

This paper investigates the Lyapunov exponent for a charged particle in Kerr-Newman black holes, showing that angular momentum of both the black hole and particle can violate the Maldacena-Shenker-Stanford bound on chaos. By analyzing the effective potential and its curvature at local maxima, the study finds that centrifugal forces from angular momentum increase chaotic behavior beyond the surface gravity limit, particularly in extremal and near-horizon regimes.

ABSTRACT

We investigate the conjecture on the upper bound of the Lyapunov exponent for the chaotic motion of a charged particle around a Kerr-Newman black hole. The Lyapunov exponent is closely associated with the maximum of the effective potential with respect to the particle. We show that when the angular momenta of the black hole and particle are considered, the Lyapunov exponent can exceed the conjectured upper bound. This is because the angular momenta change the effective potential and increase the magnitude of the chaotic behavior of the particle. Furthermore, the location of the maximum is also related to the value of the Lyapunov exponent and the extremal and non-extremal states of the black hole.

Motivation & Objective

• . The paper investigates whether the Maldacena-Shenker-Stanford bound on the Lyapunov exponent is violated in Kerr-Newman black holes.
• It examines the role of angular momentum in the black hole and the particle in modifying the effective potential and chaotic dynamics.
• The study focuses on how the effective potential's curvature at local maxima determines the Lyapunov exponent and whether it exceeds the surface gravity bound.
• It compares results across KN, Kerr, and Reissner-Nordström black holes, especially in near-horizon and extremal limits.
• The objective is to determine under what conditions the Lyapunov exponent exceeds the conjectured upper bound proportional to surface gravity.

Proposed method

• . The effective potential for a charged particle in a Kerr-Newman black hole is derived using the Lagrangian formalism and the static gauge.
• The Lyapunov exponent is calculated from the second derivative of the effective potential at its local maximum, approximating the system as an inverse harmonic oscillator.
• The surface gravity κ is computed from the black hole's mass, charge, and angular momentum to compare against the Lyapunov exponent.
• The analysis is performed in the near-horizon limit and for extremal cases, using asymptotic expansions in ϵ = r+ − r−.
• The bound λ ≤ κ is tested by comparing λ² and κ², with violations indicated when λ² > κ².
• Numerical and algebraic evaluations are conducted across different parameter regimes: KN, Kerr (Q=0), and RN (a=0) limits, with special attention to extremal and near-extremal cases.

Experimental results

Research questions

• RQ1 . Does the Lyapunov exponent for a charged particle in a Kerr-Newman black hole exceed the Maldacena-Shenker-Stanford bound, λ ≤ κ?
• RQ2 . How does the angular momentum of the black hole and the particle affect the effective potential and its curvature at the local maximum?
• RQ3 . Is the bound violated when the local maximum of the effective potential is located away from the horizon, particularly in extremal or near-extremal configurations?
• RQ4 . What is the behavior of the Lyapunov exponent in the near-horizon limit, and does the bound still hold when the maximum is close to the event horizon?
• RQ5 . Can a modified bound λ ≤ C₀κ or λ ≤ κ + C₁ with constant C₀ or C₁ avoid violations due to angular momentum?

Key findings

• . The Lyapunov exponent can exceed the surface gravity bound κ in Kerr-Newman black holes when the local maximum of the effective potential is not at the horizon, due to angular momentum contributions.
• . For the extremal Reissner-Nordström black hole, the bound is violated if the particle has finite angular momentum, as λ ∼ L² for large L.
• . In the near-horizon limit, the bound is satisfied for KN, Kerr, and RN black holes when the local maximum is close to the horizon, as λ² < κ² in the leading-order expansion.
• . The effective potential's curvature, which determines the Lyapunov exponent, becomes more negative (increasing chaos) due to centrifugal repulsion from angular momentum, especially in extremal cases.
• . The extremal Kerr black hole does not allow the local maximum to be at the horizon, so the bound cannot be tested at that point, but violations are found away from the horizon.
• . The study concludes that angular momenta of both the black hole and particle significantly alter the effective potential, enabling chaos beyond the κ-bound, particularly in extremal and non-extremal KN black holes with non-zero L.