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arxiv: 2601.03751 · v5 · submitted 2026-01-07 · 🌀 gr-qc · hep-th

Motions of spinning particles and chaos bound in Reissner-Nordstr\"om spacetime

Chuang Yang , Deyou Chen , Yongtao Liu This is my paper

Pith reviewed 2026-05-16 17:06 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords spinning particleschaos boundReissner-Nordström black holeLyapunov exponentspinor fieldsurface gravityblack hole geodesics
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The pith

Spinning particles around charged black holes violate the chaos bound once spin exceeds a threshold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether the chaos bound on Lyapunov exponents holds for spinning particles in Reissner-Nordström spacetime. It shows that neutral particles violate the bound when spin magnitude passes a critical value, with the exponent then exceeding surface gravity. Anti-aligned spin produces larger exponents than aligned spin at fixed angular momentum. For charged particles the electromagnetic force alters numerical values but does not remove the violation at large angular momenta. The result indicates that the bound, previously checked for scalars, fails for spinor fields.

Core claim

For neutral spinning particles orbiting a Reissner-Nordström black hole, the Lyapunov exponent exceeds the surface gravity once the spin magnitude surpasses a specific threshold, producing a direct violation of the chaos bound. The same violation persists for charged particles at sufficiently large total angular momentum even after the electromagnetic force is included. At fixed angular momentum the exponent is larger when spin is anti-aligned with angular momentum than when the two are aligned.

What carries the argument

Lyapunov exponent extracted from the geodesic deviation or effective potential of spinning particles whose motion is governed by the Mathisson-Papapetrou equations in the RN metric, compared directly with the horizon surface gravity.

If this is right

  • The chaos bound does not hold universally once particles carry intrinsic spin.
  • Violation occurs for both neutral and charged spinning particles in RN spacetime.
  • Spin orientation relative to orbital angular momentum modulates the size of the exponent.
  • Electromagnetic forces change the numerical value of the exponent but leave the violation intact at large angular momentum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar violations may appear for spinning particles around other charged or rotating black holes.
  • The result suggests that any general proof of the chaos bound must incorporate spin degrees of freedom explicitly.
  • Analogue gravity experiments with spinning particles could provide a laboratory test of the violation.

Load-bearing premise

The Lyapunov exponent for spinning particles can be computed in exactly the same way as for scalar fields and the chaos bound applies to them without any modification for intrinsic spin.

What would settle it

Numerical integration of the spinning-particle equations for a neutral particle with spin magnitude just above the reported threshold, confirming whether the resulting Lyapunov exponent is larger or smaller than the RN surface gravity.

read the original abstract

Previous research showed that the chaos bound proposed in \cite{MSS} can be violated under specific conditions within the scalar fields surrounding black holes. In this paper, we explore motions of spinning particles orbiting a Reissner-Nordstr\"om black hole and examine whether this bound is violated in the spinor field of this black hole. For the neutral particle, when its spin magnitude surpasses a specific threshold, the value of the exponent exceeds the surface gravity, resulting in a violation of the bound. Given a fixed total angular momentum of the particle, when its spin direction is anti-aligned with the angular momentum direction, the exponent value is greater than that when the two directions are aligned. For the charged particle, taking into account the influence of the electromagnetic force, we find that for relatively large angular momenta, although the electromagnetic force does not change the trend of the exponent's variation with respect to spin and angular momentum, and only modifies its values, it still leads to the violation. Therefore, the chaos bound violations are observed in the spinor field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper investigates the motion of spinning test particles in Reissner-Nordström spacetime via the Mathisson-Papapetrou-Dixon equations and computes Lyapunov exponents for chaotic orbits. It reports that for neutral particles the exponent exceeds the surface gravity κ once spin magnitude surpasses a threshold, producing a violation of the MSS chaos bound, with larger values when spin is anti-aligned with angular momentum. For charged particles the electromagnetic force modifies the exponent values but does not remove the violations at large angular momenta, leading to the conclusion that the bound is violated in the spinor field.

Significance. If the direct applicability of the bound holds, the result would extend prior scalar-field violations to the spinor case and demonstrate that spin-curvature coupling can produce Lyapunov exponents larger than κ in charged black-hole backgrounds, with potential implications for the range of validity of classical chaos bounds.

major comments (2)
  1. [Introduction] Introduction: the manuscript applies the MSS bound λ ≤ κ directly to the Lyapunov exponent obtained from linearized MPD equations but supplies no derivation or reference establishing that the bound, originally obtained for quantum OTOCs, remains meaningful for spin-curvature-coupled trajectories; this assumption is load-bearing for the central claim of violation.
  2. [§4] §4 (Numerical results): the threshold value of spin magnitude at which the exponent exceeds κ is stated without reported error estimates, convergence tests, or sensitivity analysis to initial conditions and integration step size, so it is impossible to confirm that the reported excess is not an artifact of the numerical procedure.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'the exponent' should be replaced by 'the Lyapunov exponent' on first use to avoid ambiguity for readers unfamiliar with the context.
  2. [§2] §2: the supplementary condition imposed on the spin tensor in the MPD equations is not stated explicitly; adding a sentence identifying the choice (e.g., Tulczyjew or Mathisson) would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation and numerical validation.

read point-by-point responses

  1. Referee: [Introduction] Introduction: the manuscript applies the MSS bound λ ≤ κ directly to the Lyapunov exponent obtained from linearized MPD equations but supplies no derivation or reference establishing that the bound, originally obtained for quantum OTOCs, remains meaningful for spin-curvature-coupled trajectories; this assumption is load-bearing for the central claim of violation.

    Authors: We agree that a clearer justification is needed. The MSS bound, while originally derived for quantum OTOCs, has been applied in the literature to classical Lyapunov exponents for geodesic and test-particle motion around black holes (including scalar-field cases cited in our introduction). The MPD equations yield the effective classical dynamics for spinning particles, and the Lyapunov exponent is extracted identically to those prior studies. In the revision we will add a short paragraph in the introduction with additional references to classical applications of the bound, clarifying its use for spin-curvature-coupled trajectories. revision: yes

  2. Referee: [§4] §4 (Numerical results): the threshold value of spin magnitude at which the exponent exceeds κ is stated without reported error estimates, convergence tests, or sensitivity analysis to initial conditions and integration step size, so it is impossible to confirm that the reported excess is not an artifact of the numerical procedure.

    Authors: We accept this criticism. The current manuscript lacks explicit numerical validation. In the revised version we will add a dedicated subsection in §4 describing the integration scheme (including step-size choice), convergence tests with respect to integration time and step size, sensitivity to initial conditions, and error estimates on the computed Lyapunov exponents. These additions will demonstrate that the reported threshold and violations are robust. revision: yes

Circularity Check

0 steps flagged

No significant circularity; Lyapunov exponent computed independently from particle trajectories

full rationale

The paper computes the Lyapunov exponent directly from the linearized equations of motion for spinning test particles (via Mathisson-Papapetrou-Dixon formalism) in RN spacetime and compares the resulting value to the surface gravity κ. This is an explicit numerical evaluation of trajectories for given spin magnitude, direction, and angular momentum, not a reduction to a fitted parameter, self-defined quantity, or load-bearing self-citation. The claimed violation for spin magnitudes above a threshold follows from those computed values exceeding κ. The reference to prior scalar-field work is external and does not make the present derivation tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of the Lyapunov exponent for spinning particles and the applicability of the MSS chaos bound to spinor fields; no free parameters are explicitly fitted in the abstract, but numerical thresholds for spin magnitude are determined computationally.

free parameters (1)
  • spin-magnitude threshold
    Value at which the Lyapunov exponent exceeds surface gravity is located by numerical evaluation of trajectories.
axioms (1)
  • domain assumption Lyapunov exponent is the appropriate diagnostic for chaos and the MSS bound applies to spinor fields
    Invoked throughout the abstract when comparing the computed exponent to surface gravity.

pith-pipeline@v0.9.0 · 5485 in / 1240 out tokens · 34270 ms · 2026-05-16T17:06:01.891701+00:00 · methodology

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Reference graph

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