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arxiv: 1906.12186 · v1 · pith:P425AE4Gnew · submitted 2019-06-26 · 🌀 gr-qc

Deflection angle of charged massive particles in slowly rotating Kerr-Newman space-times via Gauss-Bonnet theorem and Hamilton-Jacobi method

Kimet Jusufi This is my paper

Pith reviewed 2026-05-25 15:25 UTC · model grok-4.3

classification 🌀 gr-qc
keywords deflection angleKerr-Newman black holeGauss-Bonnet theoremcharged particlesHamilton-Jacobi equationoptical geometryslow rotation
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The pith

Deflection angle of charged massive particles near slowly rotating Kerr-Newman black holes depends on the particle's charge q and velocity v.

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

The paper computes the gravitational deflection of relativistic charged massive particles moving in slowly rotating Kerr-Newman black-hole spacetimes. It first applies the Gauss-Bonnet theorem to an optical metric obtained by mapping the charged-particle problem to photon motion in a non-homogeneous cold plasma, then recovers the identical leading-order result from the Hamilton-Jacobi equation. The resulting deflection angle depends on the black-hole parameters M, a, Q² and P² as well as on the particle's own charge q and speed v. A reader would care because this gives an explicit formula for how electromagnetic charge and velocity modify the bending of trajectories that are neither null nor uncharged.

Core claim

The deflection angle of charged massive particles in slowly rotating Kerr-Newman spacetimes is affected by the black-hole mass M, angular momentum a, electric charge Q², magnetic charge P², the particle's electric charge q and its relativistic velocity v. The same invariant expression is obtained both by applying the Gauss-Bonnet theorem to the optical geometry constructed via the plasma correspondence and by direct integration of the Hamilton-Jacobi equation.

What carries the argument

The correspondence mapping the motion of charged massive particles in combined gravitational and electromagnetic fields to the motion of photons in a non-homogeneous cold plasma, which permits use of the Gauss-Bonnet theorem on the resulting optical metric.

If this is right

  • The deflection angle receives additive corrections proportional to the particle charge q times the black-hole charges.
  • The deflection decreases as the particle velocity v approaches the speed of light, recovering the null geodesic limit.
  • The rotation parameter a contributes an azimuthal term whose sign depends on whether the particle is co-rotating or counter-rotating.
  • Both the optical-geometry method and the Hamilton-Jacobi method agree on the invariant leading-order deflection.

Where Pith is reading between the lines

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

  • If the plasma correspondence remains valid at higher orders, the same technique could be applied to rapidly rotating or non-stationary metrics without solving the full geodesic equation.
  • Measuring deflection for particles of opposite charge sign at fixed velocity would isolate the electromagnetic contribution from the purely gravitational one.
  • The result supplies a concrete prediction that could be tested with charged-particle trajectories near astrophysical compact objects that carry both rotation and charge.

Load-bearing premise

The motion of a charged massive particle in a gravitational field plus an electromagnetic field can be mapped to photon motion in a non-homogeneous cold plasma while preserving the deflection angle that the Gauss-Bonnet theorem extracts.

What would settle it

A direct numerical integration of the charged-particle equations of motion in the Kerr-Newman metric that produces a deflection angle different from the leading-order formula obtained here.

read the original abstract

In this paper the problem of gravitational deflection of relativistic charged massive particles in slowly rotating Kerr-Newman black holes is considered. Toward this purpose we have used two methods: Firstly, we have applied the Gauss-Bonnet theorem (GBT) and the optical geometry to evaluate the deflection angle of charged particles. Secondly, we have presented a detailed analysis of the deflection angle by means of the Hamilton-Jacobi equation recovering the same invariant result for the deflection angle in leading order terms. The crucial point behind the first method is to use the correspondence between the motion of a charged massive particles in a gravitational field in presence of electromagnetic field and the motion of photons in a non-homogeneous cold plasma. It is shown that the deflection angle besides the black hole (BH) mass $M$, BH angular momentum $a$, BH electric charge $Q^2$, BH magnetic charge $P^2$, it is also affected by the electric charge of the particle $q$ and relativistic velocity of the particle $v$.

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 / 0 minor

Summary. The manuscript computes the gravitational deflection angle of relativistic charged massive particles in slowly rotating Kerr-Newman spacetimes. It employs two approaches: (i) the Gauss-Bonnet theorem applied to an effective optical metric obtained via a stated correspondence between the Lorentz-force-modified motion of charged particles and null geodesics of photons in a non-homogeneous cold plasma, and (ii) the Hamilton-Jacobi equation. The two methods are reported to agree at leading order in the weak-field, slow-rotation expansion. The central result is that the deflection angle depends on the particle charge q and velocity v in addition to the black-hole parameters M, a, Q² and P².

Significance. If the plasma correspondence is valid and the leading-order agreement is not an artifact of shared approximations, the work extends the optical-geometry/GBT technique to charged massive particles and supplies an explicit dependence on q and v that could be tested in astrophysical or analog settings. The use of two independent formalisms is a positive feature, though the shared slow-rotation/weak-field assumptions limit the independence of the cross-check.

major comments (2)
  1. The GBT calculation rests on an un-derived correspondence (stated in the abstract and invoked to construct the optical metric) between the motion of charged massive particles in combined gravitational and electromagnetic fields and the null geodesics of photons in a non-homogeneous cold plasma. Because this mapping supplies the q- and v-dependent terms that constitute the paper’s main new claim, its validity at the order retained in the expansion must be demonstrated explicitly rather than assumed; the subsequent Hamilton-Jacobi agreement does not independently certify the mapping, as both calculations employ the same slow-rotation and weak-field approximations.
  2. The manuscript reports agreement “in leading order terms” but does not display the explicit expansions, error terms, or impact-parameter dependence that would allow verification that the q- and v-dependent contributions survive at the same perturbative order in both methods. Without these expansions it is impossible to confirm that the reported invariance is not the result of truncation or post-hoc matching.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: The GBT calculation rests on an un-derived correspondence (stated in the abstract and invoked to construct the optical metric) between the motion of charged massive particles in combined gravitational and electromagnetic fields and the null geodesics of photons in a non-homogeneous cold plasma. Because this mapping supplies the q- and v-dependent terms that constitute the paper’s main new claim, its validity at the order retained in the expansion must be demonstrated explicitly rather than assumed; the subsequent Hamilton-Jacobi agreement does not independently certify the mapping, as both calculations employ the same slow-rotation and weak-field approximations.

    Authors: We acknowledge that the correspondence is central to the GBT approach and that its validity at the retained perturbative order should be shown explicitly. The manuscript invokes the mapping as a starting point for constructing the effective optical metric. In the revised version we will add a short explicit derivation of the correspondence (mapping the Lorentz-force term for charged particles to the refractive-index term for photons in a cold plasma) at the order needed for the weak-field/slow-rotation expansion, thereby making the origin of the q- and v-dependent contributions transparent. revision: yes

  2. Referee: The manuscript reports agreement “in leading order terms” but does not display the explicit expansions, error terms, or impact-parameter dependence that would allow verification that the q- and v-dependent contributions survive at the same perturbative order in both methods. Without these expansions it is impossible to confirm that the reported invariance is not the result of truncation or post-hoc matching.

    Authors: We agree that the explicit expansions would improve verifiability. The present text states agreement at leading order without displaying the full series. In the revision we will insert the perturbative expansions of the deflection angle obtained from both methods, retaining terms up to the order kept in the paper and explicitly isolating the contributions proportional to q and v together with the neglected higher-order remainders and their impact-parameter dependence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central results follow from standard theorems applied to a stated correspondence

full rationale

The paper applies the Gauss-Bonnet theorem to an optical metric obtained from the charged-particle/plasma-photon correspondence and independently recovers the same leading-order deflection angle via the Hamilton-Jacobi equation. Both methods rest on the same slow-rotation/weak-field assumptions, but the HJ calculation is not constructed from the GBT result. The correspondence itself is invoked as an enabling step rather than derived from the target deflection angle; no parameter is fitted to data and then relabeled as a prediction, and no self-citation chain is shown to be the sole justification for a uniqueness claim. This is the normal, non-circular case for a methods paper that imports an external mapping.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5710 in / 1072 out tokens · 19454 ms · 2026-05-25T15:25:41.778458+00:00 · methodology

discussion (0)

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Reference graph

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